Band-dominated operators with operator-valued coefficients, their Fredholm properties and finite

sections

Vladimir S. Rabinovich^{∗}, Steffen Roch, Bernd Silbermann

Abstract

The central theme of the present paper are band and band-dominated operators, i.e. norm limits of band operators. In the first part, we generalize the results from [24] and [25] concerning the Fredholm properties of band-dominated operators and the applicability of the finite section method to the case of operators with operator-valued coefficients. We characterize these properties in terms of the limit operators of the given band-dominated operator. The main objective of the second part is to apply these results to pseudodifferential operators on cones in ℝ^{n} which is possible after a suitable discretization.

1 Introduction

The investigation of several concrete classes of band and band-dominated operators has a long and interesting history the explanation of which would require its own paper. Let us mention only some of its highlights. There are good reasons to consider the papers by Krein [12] (1958) and Gohberg/Krein [10] (1958) as the foundation stones on which the further development is based. These papers initiated a series of investigations culminating in the seventies and eighties in the monographs by Gohberg/Feldman [8] (1971), Gohberg/Krupnik [11] (1973) and Böttcher/Silbermann [2] (1989).

Besides the study of concrete classes of band-dominated operators (such as Toeplitz and Wiener-Hopf operators and singular integral operators), there were attempts to consider band-dominated operators from a more general point of view. Basic steps towards a the general theory of band-dominated operators were gone by Simonenko [27] (1968), [28] (1967), who examined multidimensional convolution operators by means of his local theory. However, as one soon realized, an application of local principles in the forms proposed by Simonenko [26] (1965), Gohberg/Krupnik [11] (1973) or Allan/Douglas [2] requires, in some sense, a stabilization of the coefficients of the operator at infinity. Only Ben-Artzi/Gohberg [1] (1988) and Gohberg/Kaashoek [9] (1994) allow arbitrary bounded coefficients in their considerations.

∗Partially supported by the German Research Foundation (DFG) under Grant Nr. 436 RUS 17/67/98.

In the present paper, we also deal with the general case of bounded and not necessarily stabilized at infinity coefficients. Our presentation, which is quite different from the approaches of the quoted papers, is based on the method of limit operators.

Perhaps, limit operators were first introduced by Favard [7] (1928) for verifying the existence of almost-periodic solutions of ordinary differential equations. Muhamadiev [19] (1972), [20] (1981) applied these techniques to the study of the solvability of elliptic partial differential equations on ℝ^{n}. Later, Lange/Rabinovich (1985-1987) and Rabinovich (1988-1999) developed a general framework for applying the limit operators approach to pseudodifferential operators on non-compact manifolds as well as to the related boundary value problems [16, 17, 18, 23]. Let us also mention the recent papers by the authors [24, 25] (1998, 2000) where the Fredholmness and the stability of the finite section method for wide classes of band-dominated operators are studied by means of limit operators, and the paper by Böttcher, Karlovich and Rabinovich [3] (1997) where the limit operators prove to be an effective tool to describe the Fredholm spectrum of singular integral operators on Carleson curves acting on weighted L^{2}-spaces.

The first part of the present paper is devoted to a generalization of the results from [24, 25] to the case of band-dominated operators with operator-valued coefficients. We will introduce a class of band-dominated operator (discrete convolutions) with operator-valued coefficients and define limit operators for operators in this class. In terms of these limit operators, we give conditions for the Fredholmness and the local invertibility at infinity of band-dominated operators, and we derive necessary and sufficient conditions for the applicability of the finite section method.

In the second part we study several concrete classes of operators on ℝ^{n} such as pseudodifferential operators, convolution operators and operators in finite differences. We consider these operators as elements of a certain C^{∗}-subalgebra B of L(L^{2}(ℝ^{n})) and prove that, after a suitable discretization, the operators in B become unitarily equivalent to discrete band-dominated operators with operatorvalued coefficients. In this way, we obtain results on the Fredholmness of operators in B as well as on the applicability of the finite section method to these operators as immediate corollaries of the corresponding results for operator-valued band-dominated operators. In particular, we apply this approach to restrictions of pseudodifferential operators onto cones in ℝ^{n}.

2 Band-dominated operators with operator-val-ued coefficients

2.1 The C^{∗}-algebra of the band-dominated operators

Throughout this paper, let H be a Hilbert space with inner product ⟨., .⟩_{H}, L(H) the C^{∗}-algebra of the bounded linear operators on H and K(H) the ideal of the compact operators on H. Further, let H stand for the Hilbert space l^{2}(ℤ^{n}, H) of all vector-valued functions on ℤ^{n} with values in H provided with the inner product

⟨f, g⟩_{H} := ∑_{x∈ℤ^{n}}⟨f(x), g(x)⟩_{H}  (1)

and with the associated norm ∥f∥_{H} := ⟨f, f⟩_{H}, and write l^{∞}(ℤ^{n}, L(H)) for the space of all bounded L(H)-valued functions a on ℤ^{n} with norm

∥a∥_{l^{∞}(ℤ^{n},L(H))} = sup_{x∈ℤ^{n}} ∥a(x)∥_{L(H)}.

We agree upon identifying a function a ∈ l^{∞}(ℤ^{n}, L(H)) with the multiplication operator f ↦ af acting on l^{2}(ℤ^{n}, H). If a ∈ l^{∞}(ℤ^{n}) := l^{∞}(ℤ^{n}, L(ℂ)), then we abbreviate the operator of multiplication f ↦ af on l^{2}(ℤ^{n}, H) to a⊗I_{H}. Finally, for h ∈ ℤ^{n}, let τ_{h} stand for the shift operator (τ_{h}f)(x) := f(x − h) which is a unitary operator on l^{2}(ℤ^{n}, H).

Definition 2.1 A band operator on H = l^{2}(ℤ^{n}, H) is a finite sum of the form ∑_{α∈ℤ^{n}} a_{α}τ_{α} where a_{α} ∈ l^{∞}(ℤ^{n}, L(H)). A band-dominated operator is a uniform limit of a sequence of band operators.

The band-dominated operators on H form a C^{∗}-subalgebra of L(H) which we denote by A(H).

For an equivalent description of band-dominated operators, we introduce the space BUC(ℝ^{n}) of the bounded and uniformly continuous functions on ℝ^{n}. Given a function φ on ℝ^{n}, we denote its restriction onto ℤ^{n} by φ̂. The following proposition characterizes the algebra A(H) completely.

Proposition 2.2 The following assertions are equivalent for an operator A ∈ L(H) :

(a) A is band-dominated.

(b) If φ ∈ BUC(ℝ^{n}) and φ_{t}(x) := φ(t_{1}x_{1}, ..., t_{n}x_{n}), then

lim_{t→0} ∥(φ̂_{t} ⊗ I_{H})A − A(φ̂_{t} ⊗ I_{H})∥_{L(H)} = 0.

If these conditions are satisfied, then

lim_{t→0} ∥(φ̂_{t,h} ⊗ I_{H})A − A(φ̂_{t,h} ⊗ I_{H})∥_{L(H)} = 0

uniformly with respect h ∈ ℝ^{n}, where φ̂_{t,h}(x) = φ̂_{t}(x − h).

The proof is similar to the proof of Proposition 8 in [24] and can be found in [21].

Our next goal is to introduce an ideal J (H) of A(H) which will serve as a substitute for the ideal of the compact operators. Given a point η in the unit sphere S^{n−1} of ℝ^{n}, a neighborhood U ⊆ S^{n−1} of η, and a positive number R, we define

W_{R,U} := {x ∈ ℝ^{n} : |x| > R and x/|x| ∈ U}

and

W_{R} := {x ∈ ℝ^{n} : |x| > R}.

In what follows we think of S^{n−1} as the sphere S̃^{n−1} of the infinitely distant points of ℝ^{n}. Thus, the sets W_{R,U} and W_{R} form a system of neighborhoods of the point η ∈ S̃^{n−1} and of ∞, respectively, and a sequence h = (h_{j}) ⊂ ℝ^{n} tends to η ∈ S̃^{n−1} (resp. to ∞) if, for every R > 0 and every neighborhood U of η ∈ S^{n−1}, there is a j_{0} such that h_{j} ∈ W_{R,U} (resp. h_{j} ∈ W_{R}) for all j ≥ j_{0}.

Definition 2.3 Let J (H) refer to the set of all operators T ∈ L(H) such that, for every ε > 0, there is a neighborhood W_{r} of infinity such that

∥T (χ̂_{W_{R}} ⊗ I_{H})∥_{L(H)} < ε and ∥(χ̂_{W_{R}} ⊗ I_{H})T ∥_{L(H)} < ε.

Here and in what follows, χ_{M} refers to the characteristic function of the set M.

Proposition 2.4 (a) Let φ ∈ BUC(ℝ^{n}) with φ(0) = 0 and let T ∈ J (H). Then

lim_{t→0} ∥(φ̂_{t} ⊗ I_{H})T ∥ = lim_{t→0} ∥T (φ̂_{t} ⊗ I_{H})∥ = 0.

(b) J (H) is a closed two-sided ideal of A(H).

Proof. (a) Let φ ∈ BUC(ℝ^{n}) with φ(0) = 0 and T ∈ J (H). Then, for every R,

∥(φ̂_{t} ⊗ I_{H})T ∥ ≤ ∥(φ̂_{t} ⊗ I_{H})(I − χ̂_{W_{R}} ⊗ I_{H})T ∥ + ∥(φ̂_{t} ⊗ I_{H})(χ̂_{W_{R}} ⊗ I_{H})T ∥.

Given ε > 0, choose R_{0} > 0 such that ∥(χ̂_{W_{R_{0}}} ⊗ I_{H})T ∥ ≤ ε. Since, evidently,

lim_{t→0} ∥φ̂_{t}(I − χ̂_{W_{R_{0}}}) ⊗ I_{H}∥ = 0,

we can further find a t_{0} > 0 such that, for t ≥ t_{0},

∥(φ̂_{t} ⊗ I_{H})T ∥ ≤ ε(∥T ∥ + sup |φ(x)|).

Thus, lim_{t→0} ∥(φ̂_{t}⊗I_{H})T ∥ = 0, and in the same way one gets that ∥T (φ̂_{t}⊗I_{H})∥ → 0 as t → 0.

(b) It follows immediately from part (a) and Proposition 2.2 that J (H) is a closed subspace of A(H). For a proof of the ideal property, let A ∈ A(H) and T ∈ J (H), and choose φ ∈ BUC(ℝ^{n}) such that φ(x) = 0 if |x| ≤ 1 and φ(x) = 1 if |x| ≥ 2. Then

(φ̂_{t} ⊗ I_{H})AT = [(φ̂_{t} ⊗ I_{H}), A]T + A(φ̂_{t} ⊗ I_{H})T

where [A, B] refers to the commutator AB − BA of the operators A and B. As follows from Proposition 2.2,

lim_{t→0} ∥[(φ̂_{t} ⊗ I_{H}), A]T ∥ = 0

and, because of T ∈ J (H), we have lim_{t→0} ∥A(φ̂_{t} ⊗ I_{H})T ∥ = 0. Hence,

lim_{t→0} ∥(φ̂_{t} ⊗ I_{H})AT ∥ = 0,

which shows that AT ∈ J (H). In the same way one obtains T A ∈ J (H).

It is easy to see that J (H) is the smallest closed ideal of A(H) which contains all operators of multiplication by functions â⊗I_{H} where a ∈ BUC(ℝ^{n}) is a function such that lim_{x→∞} a(x) = 0. It is also evident that the inclusion K(H) ⊂ J (H) holds for every Hilbert space H.

Definition 2.5 An operator A ∈ L(H) is invertible at infinity if there exists an operator L ∈ L(H) and operators T_{1}, T_{2} ∈ J (H) such that

LA = I + T_{1} and AL = I + T_{2}.  (2)

In case H is a finite-dimensional Hilbert space, the inclusion K(H) ⊂ J (H) becomes an equality. Hence, in this setting, invertibility at infinity of an operator A ∈ L(H) is nothing than the common Fredholmness of A.

Proposition 2.6 Let the operator A ∈ A(H) be invertible at infinity, and let L be an operator as in (2). Then this operator also belongs to A(H).

Proof. Let φ ∈ BUC(ℝ^{n}). Without loss of generality we can suppose that φ(0) = 0. Then, as one easily checks,

[(φ̂_{t} ⊗ I_{H}), L] = L[A, (φ̂_{t} ⊗ I_{H})]L + Q_{t}

with lim_{t→0} ∥Q_{t}∥ = 0. By Proposition 2.2, L is band-dominated.

Let

π : A(H) → A(H)/J (H)

refer to the canonical ^{∗}-homomorphism. One can reformulate Proposition 2.6 by saying that a band-dominated operator A is invertible at infinity if and only if the coset π(A) is invertible in the quotient algebra A(H)/J (H).

We are still going to mention the analogous definitions and results for the local invertibility at an point η ∈ S̃^{n−1}. The proofs are similar to their counterparts concerning invertibility at infinity.

Definition 2.7 Given η ∈ S^{n−1}, let J_{η} denote the set of all operators T ∈ A(H) which are subject to the following condition: for each ε > 0, there is a neighborhood W_{R,U} of η ∈ S̃^{n−1} such that

∥T (χ̂_{W_{R,U}} ⊗ I_{H})∥_{L(H)} < ε and ∥(χ̂_{W_{R,U}} ⊗ I_{H})T ∥_{L(H)} < ε.

Proposition 2.8 J_{η} is a closed two-sided ideal of the C^{∗}-algebra A(H).

The ideal J_{η} can also be characterized as the smallest closed ideal of A(H) which contains all operators of multiplication by operator-valued functions â⊗I_{H} where now a ∈ BUC(ℝ^{n}) is such that lim_{x→η} a(x) = 0.

Definition 2.9 Let η ∈ S̃^{n−1}. An operator A ∈ L(H) is locally invertible at η if there exists an operator L_{η} ∈ L(H) as well as operators T_{1} and T_{2} in J_{η} such that

L_{η}A = I + T_{1} and AL_{η} = I + T_{2}.  (3)

Proposition 2.10 If the band-dominated operator A is locally invertible at the point η and if L_{η} is as in (3), then L_{η} also belongs to A(H).

If further π_{η} stands for the canonical ^{∗}-homomorphism from A(H) onto A(H)/J_{η} then, evidently, a band-dominated operator A ∈ A(H) is locally invertible at η ∈ S̃^{n−1} if and only if its coset π_{η}(A) is invertible in the quotient algebra A(H)/J_{η}.

2.2 Limit operators of band-dominated operators

The notion of limit operators which we are going to introduce in this subsection will provide us with an adequate tool for examining band-dominated operators.

Definition 2.11 Let h = (h_{j}) ⊂ ℤ^{n} be a sequence tending to infinity. An operator A_{h} is said to be a limit operator of A ∈ L(H) with respect to h if

lim_{j→∞} ∥(A_{h} − τ_{−h_{j}}Aτ_{h_{j}})(φ ⊗ I_{H})∥ = lim_{j→∞} ∥(φ ⊗ I_{H})(A_{h} − τ_{−h_{j}}Aτ_{h_{j}})∥ = 0

for every function φ ∈ l^{∞}(ℤ^{n}) with finite support.

An operator can possess only one limit operator with respect to a given sequence which justifies the notation A_{h} for the limit operator. Observe also that, if g is an (infinite) subsequence of h and if A_{h} exists, then A_{g} exists, too, and A_{g} = A_{h}.

If H is of finite dimension, the definition of a limit operator is equivalent to the following one: A_{h} is the limit operator of A with respect to h if

s-lim_{j→∞}τ_{−h_{j}}Aτ_{h_{j}} = A_{h}.

The following proposition summarizes basic properties of limit operators.

Proposition 2.12 Let A, B and A^{(m)} be operators in L(H), and let h ⊂ ℤ^{n} be a sequence tending to infinity.

(a) If A_{h} exists, then ∥A_{h}∥ ≤ ∥A∥.

(b) If the limit operators A_{h}, B_{h} exist, then (A + B)_{h} exists and (A + B)_{h} = A_{h} + B_{h}.

(c) If the limit operators A_{h}, B_{h} exist, then (AB)_{h} exists and (AB)_{h} = A_{h}B_{h}.

(d) If the limit operator A_{h} exists, then (A^{∗})_{h} exists and (A^{∗})_{h} = (A_{h})^{∗}.

(e) If A = lim_{m→∞} A^{(m)} and if the limit operators (A^{(m)})_{h} exist for all sufficiently large m, then A_{h} exists and A_{h} = lim_{m→∞}(A^{(m)})_{h}.

The elementary proof is omitted. Of particular interest is a special class of banddominated operators which we introduce in the following definition.

Definition 2.13 Let A_{0}(H) stand for the set of all operators A ∈ A(H) with the following property: every sequence h ∈ ℤ^{n} tending to infinity possesses a subsequence g such that the limit operator A_{g} exists.

Proposition 2.14 Let A ∈ A(H) be a uniform limit of band operators,

A = lim_{N→∞} ∑_{|α|≤N} a_{α}τ_{α},

and suppose that, for every coefficient a_{α} ∈ l^{∞}(ℤ^{n}, L(H)), for every x ∈ ℤ^{n} and for every sequence h ⊂ ℤ^{n} tending to infinity, there is a subsequence g = (g_{k}) of h such that the limit

lim_{k→∞} a_{α}(x + g_{k}) = (a_{α})_{g}(x)

exists. Then A belongs to A_{0}(H).

The proof of this proposition is the same as the proof of Proposition 12 in [24]. Observe also that A_{0}(H) is a C^{∗}-subalgebra of A(H) due to Proposition 2.12. In case H is a finite-dimensional Hilbert space, one can moreover show that the equality A(H) = A_{0}(H) holds (see Proposition 12 in [24] again).

Definition 2.15 Let η ∈ S̃^{n−1} and A ∈ L(H). The local operator spectrum of A at the point η is the set σ_{η}(A) of all limit operators A_{h} of A defined by the sequences h tending to η. The (global) operator spectrum of A is the collection σ(A) of all limit operators of A. Thus,

σ(A) = ∪_{η∈S̃^{n−1}}σ_{η}(A).

In particular, σ_{η}(A) = {0} if A ∈ J_{η}, and σ(A) = {0} if A ∈ J (H).

2.3 Local invertibility at infinity of band-dominated operators

The following theorems present the main results on local invertibility at infinity (and Fredholmness in the finite-dimensional case) of band-dominated operators in A_{0}(H).

Theorem 2.16 Let A ∈ A_{0}(H) and η ∈ S̃^{n−1}. The coset π_{η}(A) is invertible in A(H)/J_{η} if and only if all limit operators A_{g} ∈ σ_{η}(A) are invertible and if

sup_{A_{g}∈σ_{η}(A)} ∥A_{g}^{−1}∥ =: M_{η}(A) < ∞.

The proof of this theorem is based on the following two propositions which can be verified in a similar way as Propositions 14 and 15 in [24].

We will make use of the following notations. Let φ be a continuous real-valued function on ℝ^{n} which is equal to 1 on the cube [−1/3, 1/3]^{n} and equal to 0 outside the cube [−2/3, 2/3]^{n}. Given α ∈ ℤ^{n} and R > 0, define φ_{α}(x) := φ(x − α) and φ_{α,R}(x) := φ_{α}(x/R). A little thought shows that it is moreover possible to choose the function φ in such a way that the collection {φ_{α,R}}_{α∈ℤ^{n}} forms a partition of unity on ℝ^{n}, i.e. such that

∑_{α∈ℤ^{n}} φ_{α,R}(x) = 1 for every x.

Finally, let ψ ∈ C_{0}^{∞}(ℝ^{n}) be a function which is 1 on the support of φ.

Proposition 2.17 Let A ∈ A_{0}(H) and η ∈ S̃^{n−1}, and suppose there is a constant M > 0 such that, for every R > 0, there is a neighborhood W_{R} of η such that, for all α ∈ ℤ^{n} with α ∈ W_{R}, there are operators B_{α,R} and C_{α,R} with

∥B_{α,R}∥ ≤ M,  ∥C_{α,R}∥ ≤ M  (4)

and

B_{α,R}A(ψ̂_{α,R} ⊗ I_{H}) = (ψ̂_{α,R} ⊗ I_{H})AC_{α,R} = ψ̂_{α,R} ⊗ I_{H}.  (5)

Then there exist operators B, C ∈ A(H) and a neighborhood U of η such that

BA(χ_{U} ⊗ I_{H}) = (χ_{U} ⊗ I_{H}) and (χ_{U} ⊗ I_{H})AC = (χ_{U} ⊗ I_{H}).  (6)

Observe that, due to (6), the conditions of Proposition 2.17 imply the invertibility of the coset A + J_{η} in the C^{∗}-algebra A(H)/J_{η}.

Proposition 2.18 Let A ∈ A(H) and let h = (h_{m}) ⊂ ℤ^{n} be a sequence tending to infinity such that the limit operator A_{h} of A with respect to h exists and is invertible. Then, for every function φ̂ on ℤ^{n} with finite support, there is an m_{0} such that, for all m ≥ m_{0}, there are operators B_{m} and C_{m} in A(H) with

∥B_{m}∥ ≤ 2∥A_{h}^{−1}∥, ∥C_{m}∥ ≤ 2∥A_{h}^{−1}∥

and

B_{m}A(φ̂_{h_{m}} ⊗ I_{H}) = (φ̂_{h_{m}} ⊗ I_{H})AC_{m} = φ̂_{h_{m}} ⊗ I_{H}.

Proof of Theorem 2.16. Sufficiency. Suppose that A + J_{η} is not invertible in A(H)/J_{η}. Then Proposition 2.17 implies that, for every M > 0, there is an R > 0 such that, for every neighborhood U of η, there is a point α ∈ U such that there do not exist operators B_{α,R} and C_{α,R} with ∥B_{α,R}∥ ≤ M and ∥C_{α,R}∥ ≤ M for which (5) is satisfied. Hence, for the special choice M := 2M_{η}(A), one can find an R > 0 and a sequence h = (h_{m}) → η such that

B_{h_{m},R}A(ψ̂_{h_{m},R} ⊗ I_{H}) ≠ (ψ̂_{h_{m},R} ⊗ I_{H})AC_{h_{m},R} ≠ ψ̂_{h_{m},R} ⊗ I_{H}

for all operators B_{h_{m},R} and C_{h_{m},R} with ∥B_{h_{m},R}∥, ∥C_{h_{m},R}∥ ≤ 2M_{η}(A).

Without loss we can suppose that the sequence h defines a limit operator A_{h} ∈ σ_{η}(A) (in the opposite case we pass to a subsequence of h). If we replace in Proposition 2.18 the function ϕ(x) by ϕ(x/R) we arrive at a contradiction which proves the sufficiency part of Theorem 2.16.

Necessity. Let the coset A + J_{η} be invertible in A(H)/J_{η}. Then the a priori estimates

∥u∥ ≤ C(∥Au∥ + ∥T′u∥),  ∥u∥ ≤ C(∥A^{∗}u∥ + ∥T′′u∥),  (7)

hold with certain operators T′ and T′′ in J_{η}. Let ϕ ∈ C_{0}^{∞}(ℝ^{n}) be a function which is equal to 1 on a neighborhood of the origin, and set ϕ_{R}(x) := ϕ(x/R). As follows from (7), we have

∥(ϕ̂_{R} ⊗ I_{H})u∥ ≤ C(∥A(ϕ̂_{R} ⊗ I_{H})u∥ + ∥T′(ϕ̂_{R} ⊗ I_{H})u∥).  (8)

If now h = (h_{m}) is a sequence which tends to η ∈ S̃^{n−1} and for which the limit operator A_{h} exists, then (8) implies

∥(ϕ̂_{R} ⊗ I_{H})u∥ ≤ C(∥τ_{−h_{m}}Aτ_{h_{m}}(ϕ̂_{R} ⊗ I_{H})u∥ + ∥τ_{−h_{m}}T′τ_{h_{m}}(ϕ̂_{R} ⊗ I_{H})u∥). (9)

Passing to the limit m → ∞ in (9) and taking into account that T′ is in J_{η} we obtain the a priori estimate

∥A_{h}(ϕ̂_{R} ⊗ I_{H})u∥ ≥ C′∥(ϕ̂_{R} ⊗ I_{H})u∥ with C′ := 1/C > 0.  (10)

It is evident that

s-lim_{R→∞}(ϕ̂_{R} ⊗ I_{H}) = I_{H}.

Hence, letting R go to infinity in (10) we arrive at ∥A_{h}u∥ ≥ C′∥u∥. In the same way, we get the estimate ∥(A_{h})^{∗}u∥ ≥ C′∥u∥. The latter two estimates show that the limit operator A_{h} is invertible and that the norm of its inverse is not greater than C′ which is independent of h.

Theorem 2.19 For A ∈ A_{0}(H), the coset π(A) is invertible in A(H)/J (H) if and only if all limit operators A_{h} of A are invertible and if

sup_{A_{h}∈σ(A)} ∥A_{h}^{−1}∥ < ∞.

Proof. Let C(ℝ̃^{n}) denote the subspace of BUC(ℝ^{n}) which consists of all functions a such that the limits

lim_{r→∞} a(rη) =: ã(η)

exist for every η ∈ S^{n−1} and that this convergence is uniform with respect to η. Then ã : η ↦ ã(η) is a continuous function on S^{n−1}. Let further C(ℤ̃^{n}) stand for the set of all restrictions onto ℤ^{n} of functions in C(ℝ̃^{n}). It is easy to check that [aI, A] ∈ J (H) if A ∈ A(H) and a ∈ C(ℤ̃^{n}).

The maximal ideal space of the commutative C^{∗}-algebra C(ℤ̃^{n}) is homeomorphic to ℤ^{n} ∪ S̃^{n−1}, with the points x ∈ ℤ^{n} and η ∈ S̃^{n−1} corresponding to the ideals I_{x} := {â ∈ C(ℤ̃^{n}) : â(x) = 0} and I_{η} := {â ∈ C(ℤ̃^{n}) : lim_{r→∞} a(rη) = 0}, respectively. A little thought shows that J_{η} coincides with the smallest closed ideal of A(H) which contains all operators a ⊗ I_{H} where a ∈ I_{η} and that, furthermore,

J (H) = ∩_{η∈S̃^{n−1}}J_{η}.

Thus we are in a position where Allan’s local principle [2] applies. As follows from this principle, the coset π(A) is invertible in A(H)/J (H) if and only if the coset π_{η}(A) is invertible in A(H)/J_{η} for every η. It remains to recall the conditions for the invertibility of π_{η}(A) derived in Theorem 2.16 to get the assertion.

The conditions of Theorem 2.19 are necessary and sufficient for an operator A ∈ A(H) to be Fredholm if H is a finite-dimensional Hilbert space. In the general case, they are only necessary, and additional properties of A are needed to yield their sufficiency. We say that an operator A ∈ L(H) is locally Fredholm if, for every r > 0, there are operators R′_{r}, R′′_{r} ∈ L(H) and K′_{r}, K′′_{r} ∈ K(H) such that

R′_{r}A(φ̂_{r} ⊗ I_{H}) = (φ̂_{r} ⊗ I_{H}) + K′_{r}, (φ̂_{r} ⊗ I_{H})AR′′_{r} = (φ̂_{r} ⊗ I_{H}) + K′′_{r}

where φ_{r} is the characteristic function of the ball {x ∈ ℝ^{n} : |x| ≤ r}.

Theorem 2.20 Let A ∈ A_{0}(H) be a locally Fredholm operator. Then A is a Fredholm operator if and only if the conditions of Theorem 2.19 are fulfilled.

Proof. Let A ∈ A_{0}(H) be a locally Fredholm operator which is invertible at infinity, i.e. there are operators L ∈ A(H) and T ∈ J (H) such that LA = I_{H} +T (Definition 2.5). Set ψ̂_{r} := I − φ̂_{r} and B′_{r} := (ψ̂_{r} ⊗ I_{H})L + (φ̂_{r} ⊗ I_{H})R′_{r}. Then

B′_{r}A = (ψ̂_{r} ⊗ I_{H}) + (ψ̂_{r} ⊗ I_{H})T + φ̂_{r} ⊗ I_{H} + C′_{r} = I_{H} + (ψ̂_{r} ⊗ I_{H})T + C′_{r}

with a certain compact operator C′_{r}. If we choose r such that ∥(ψ̂_{r} ⊗ I_{H})T ∥ < 1, then I_{H} + (ψ̂_{r} ⊗ I_{H})T becomes invertible, and

(I_{H} + (ψ̂_{r} ⊗ I_{H})T)^{−1}B′A = I_{H} + (I_{H} + (ψ̂_{r} ⊗ I_{H})T)^{−1}C′_{r}.

Hence, (I_{H} +(ψ̂_{r} ⊗I_{H})T)^{−1}B′_{r} is a regularizer of A from the left hand side. In the same way, one verifies the existence of a right-sided regularizer. The necessity of the conditions of Theorem 2.20 is evident.

Example. Consider the smallest closed subalgebra A_{0}^{K}(H) of A_{0}(H) which contains all operators of multiplication by functions a in l_{∞}(ℤ^{n}, K(H)). It is evident that every operator A = I + B with B ∈ A_{0}^{K}(H) is locally Fredholm. Hence, A is a Fredholm operator if and only if the conditions of Theorem 2.19 are fulfilled.

2.4 Symbol calculus

Our next goal is a symbol calculus for the invertibility at infinity of operators in A_{0}(H). We recall the construction of the symbol from [24] with the necessary modifications caused by the circumstance that H can have infinite dimension.

We denote by M_{η} the set of all sequences h ⊂ ℤ^{n} which tend to η ∈ S̃^{n−1} and, given an operator A ∈ A_{0}(H), we let M_{η}(A) stand for the set of all sequences h ∈ M_{η} such that the limit operator A_{h} exists. A subset D_{η} of M_{η} will be called admissible if,

(a) every sequence h ∈ M_{η} possesses a subsequence g which belongs to D_{η},

(b) whenever h ∈ D_{η} and g is a subsequence of h, then g ∈ D_{η},

(c) whenever h ∈ D_{η} and g ∈ M_{η} is a sequence with h_{k} = g_{k} for all sufficiently large k, then g ∈ D_{η}.

Evidently, the set M_{η} is admissible, and so are all the sets M_{η}(A) with A ∈ A_{0}(H). In [24], we proved that the intersection of an at most countable set of admissible subsets of M_{η} is admissible again.

We let F_{H} stand for the set of all bounded functions X which are defined on a set D(X) ⊆ M_{η} and take values in L(H), and which own the following properties:

(d) the region D(X) of definition of X is admissible,

(e) if h ∈ D(X) and g is a subsequence of h, then X(h) = X(g),

(f) if h ∈ D(X) and g is a sequence with h_{k} = g_{k} for all sufficiently large k, then X(h) = X(g).

It is clear from these definitions that, for A ∈ A_{0}(H), the function

smb_{η}^{0}A : M_{η}(A) → L(H), h ↦ A_{h}  (11)

belongs to F_{H}.

We introduce operations on F_{H} as follows : Given X, Y ∈ F_{H}, the sum X +Y (resp.the product XY) is the function which is defined on D(X)∩D(Y) and takes at h ∈ D(X)∩D(Y) the value X(h)+Y (h) (resp. X(h)Y (h)). Further, if α ∈ ℂ, then αX is the function which is defined on D(X) and takes at h ∈ D(X) the value αX(h), and X^{∗} is the function which is defined on D(X) by (X^{∗})(h) := (X(h))^{∗}. Finally, we set

∥X∥ := sup{∥X(h)∥ : h ∈ D(X)},

and we introduce a relation on F_{H} by

X ∼ Y if and only if X|_{D(X)∩D(Y)} = Y |_{D(X)∩D(Y)}.

The relation ∼ is an equivalence relation on F_{H} which is compatible with the operations (see [24]). Let us also observe that, if W (X) ⊆ L(H) denotes the set of the values of the function X, then X ∼ Y implies that W (X) = W (Y). Due to this observation, it makes sense to define operations and a norm on the set F_{H}^{∼} of all equivalence classes X^{∼} of elements of F_{H} by

X^{∼} + Y^{∼} := (X + Y)^{∼}, X^{∼}Y^{∼} := (XY)^{∼}, αX^{∼} := (αX)^{∼}, (X^{∼})^{∗} := (X^{∗})^{∼}

and

∥X^{∼}∥ := ∥X∥.

It is straightforward to check that F_{H}^{∼} becomes a C^{∗}-algebra under these operations.

Definition 2.21 Let A ∈ A_{0}(H). The symbol smb_{η}A of A is the coset (smb_{η}^{0}A)^{∼} with smb_{η}^{0}A being defined by (11).

This definition is correct because M_{η}(A) is admissible and smb_{η}^{0}A belongs to F_{H} for all A ∈ A_{0}(H).

Corollary 2.22 The mapping smb_{η} : A_{0}(H) → F_{H}^{∼} is a ^{∗}-homomorphism.

With these notations, Theorem 2.16 can be restated as follows.

Theorem 2.23 An operator A ∈ A_{0}(H) is locally invertible at η ∈ S̃^{n−1} if and only if its symbol smb_{η}(A) is invertible in F_{H}.

If we define the (global) symbol smbA in a similar manner (simply by considering the set M of all sequences h tending to ∞ and by omitting the subscript η in the above definitions leading to the (local) symbol smb_{η}A), then Theorem 2.19 can be reformulated in the following manner.

Theorem 2.24 An operator A ∈ A_{0}(H) is invertible at infinity if and only if its symbol smb(A) is invertible in F_{H}.

We complete this theorem by the following norm equality which can be proved as the corresponding result (Corollary 3 in [24]) in the finite-dimensional setting.

Theorem 2.25 Let A ∈ A_{0}(H). Then ∥smb(A)∥_{F_{H}} = ∥π(A)∥_{A(H)/J (H)}.

2.5 Operators with slowly oscillating coefficients

In this section, we consider band-dominated operators with slowly oscillating coefficients. One peculiarity of the operators in this class is that their limit operators are shift invariant.

Definition 2.26 A function a ∈ l^{∞}(ℤ^{n}, L(H)) is slowly oscillating if

lim_{x→∞} ∥a(x + α) − a(x)∥_{L(H)} = 0 for all α ∈ ℤ^{n}.  (12)

We denote the class of the slowly oscillating operator-valued functions by SO_{H}, and we write A^{SO}(H) for the smallest C^{∗}-subalgebra of A(H) which contains all operators

A = ∑_{|α|≤m} a_{α}τ_{α} with a_{α} ∈ SO_{H}.  (13)

Proposition 2.27 Limit operators of operators in A^{SO}(H) are translation invariant.

Proof. First consider a single operator of multiplication by a function a ∈ SO_{H}. Let h = (h_{m}) be a sequence such that the limit operator (aI)_{h} exists. It is not hard to see that (aI)_{h} is an operator of multiplication again, say by the function a_{h}. Thus, for every x ∈ ℤ^{n},

lim_{m→∞} a(x + h_{m}) = a_{h}(x),

whereas on the other hand, due to (12),

lim_{m→∞}(a(x′ + h_{m}) − a(x′′ + h_{m})) = 0

for each fixed pair of points x′, x′′ ∈ ℤ^{n}. Thus, the function a_{h} does not depend on x.

Now let A be the norm limit lim_{m→∞} A_{m} where

A_{m} = ∑_{|α|≤m} a_{α}τ_{α} with a_{α} ∈ SO_{H}.

All limit operators of A_{m} are of the form

(A_{m})_{h} = ∑_{|α|≤m}(a_{α})_{h}τ_{α},

whence their translation invariance. Being the norm limit of the translation invariant operators (A_{m})_{h}, the limit operator A_{h} is translation invariant, too.

Thus, all limit operators of an operator A ∈ A^{SO}(H) belong to the smallest closed C^{∗}-subalgebra V_{H} of L(H) which contains all finite sums

B = ∑ a_{α}τ_{α} with a_{α} ∈ L(H).  (14)

We associate with every operator B of this special form its symbol

σ_{B} : t ↦ ∑_{|α|≤m} a_{α}t^{α} where t^{α} := t_{1}^{α_{1}} … t_{n}^{α_{n}}

which is a function from 𝕋^{n}, the n-dimensional torus, into L(H). It is easy to see that

∥B∥_{L(H)} = sup_{t∈𝕋^{n}} ∥σ_{B}(t)∥_{L(H)}.  (15)

Since the operators B of the form (14) lie densely in V_{H}, we can use (15) together with a continuity argument in order to associate with every operator A ∈ V_{H} a function σ_{A} ∈ C(𝕋^{n}, L(H)), called the symbol of A. It is well-known that an operator A ∈ V_{H} is invertible if and only if its symbol is invertible in C(𝕋^{n}, L(H)).

2.6 Invertibility at infinity of band-dominated operators in cones

In this section, we deal with the problem of local invertibility at infinity for banddominated operators on discrete cones in dimensions n = 1, 2 and 3. In case the Hilbert space H has a finite dimension, we derive necessary and sufficient conditions for a band-dominated operator (or an operator of discrete convolution) on a cone to be Fredholm. The results obtained are generalizations of the Fredholm criteria due to Gohberg-Feldman [8] and Simonenko [28].

The case n = 1. Consider the operator

A_{+} := (χ̂_{+} ⊗ I_{H})A(χ̂_{+} ⊗ I_{H}) : l^{2}(ℤ_{+}, H) → l^{2}(ℤ_{+}, H)  (16)

where χ̂_{+} is the characteristic function of the set ℤ_{+} of the positive integers and where the operator A belongs to A_{0}(H) with H := l^{2}(ℤ, H). We say that the operator (16) is locally invertible at infinity if the operator

B_{+} := (χ̂_{+} ⊗ I_{H})A(χ̂_{+} ⊗ I_{H}) + (1 − χ̂_{+}) ⊗ I_{H} : l^{2}(ℤ, H) → l^{2}(ℤ, H)

is locally invertible at infinity. Note that, in the case at hand, the unit sphere S̃^{0} consists of two points only, which we identify with ±∞, and that clearly σ(B_{+}) = σ_{+∞}(B_{+}) ∪ σ_{−∞}(B_{+}) = σ_{+∞}(A) ∪ {I}. Thus, applying Theorem 2.19 we immediately obtain

Theorem 2.28 Let H = l^{2}(ℤ, H) and A ∈ A_{0}(H). Then the operator A_{+} given by (16) is locally invertible at infinity if and only if all limit operators A_{g} ∈ σ_{+∞}(A) are uniformly invertible (i.e. if they are invertible and if the norms of their inverses are uniformly bounded).

The case n = 2. Let a, b ∈ ℤ, and let Ω be the closed segment [a, b] ⊂ ℝ. Further let V_{Ω} ⊂ ℝ^{2} refer to the angular domain determined by Ω, i.e.

V_{Ω} := {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{2} > 0 and x_{1}/x_{2} ∈ Ω},

and set V̂_{Ω} := V_{Ω} ∩ ℤ^{2}. As in case n = 1, we consider the operator

A_{V_{Ω}} := (χ̂_{V_{Ω}} ⊗ I_{H})A(χ̂_{V_{Ω}} ⊗ I_{H}) : l^{2}(V̂_{Ω}, H) → l^{2}(V̂_{Ω}, H),  (17)

and we say that this operator is locally invertible at infinity if the operator

B_{V_{Ω}} := A_{V_{Ω}} + (1 − χ_{V_{Ω}}) ⊗ I_{H} : l^{2}(ℤ^{2}, H) → l^{2}(ℤ^{2}, H)

is locally invertible at infinity.

Given x ∈ ℝ, we denote by η_{x} the point on the infinitely distant sphere S̃^{1} which is determined by the ray in ℝ^{2} starting at the origin and passing through the point (x, 1). It is easy to check that, if x is in the interior of Ω and if g ⊂ ℤ^{2} is a sequence tending to η_{x}, then the limit operator of χ̂_{V_{Ω}} ⊗ I_{H} with respect to g exists and is equal to I_{H}, whereas in case x ∈ ℝ \ Ω one has σ_{η_{x}}(χ̂_{V_{Ω}} ⊗ I_{H}) = {0}. If x = a, then

σ_{η_{a}}(χ̂_{V_{Ω}} ⊗ I_{H}) = {0, I_{H}, χ̂_{W (a)+v} ⊗ I_{H} : v ∈ ℤ^{2}}

where W (a) is the half-plane which is bounded by the straight line passing through the origin and the point (a, 1) and which contains V_{Ω} (cp. Proposition 6 in [24]). Analogously,

σ_{η_{b}}(χ̂_{V_{Ω}} ⊗ I_{H}) = {0, I_{H}, χ̂_{W (b)+v} ⊗ I_{H} : v ∈ ℤ^{2}}.

All limit operators of χ̂_{V_{Ω}} ⊗ I_{H} which correspond to other points in S̃^{1} are zero. Thus, in case x = a we get a decomposition of the local spectrum σ_{η_{a}}(A),

σ_{η_{a}}(A) = σ_{η_{a},−}(A) ∪ σ_{η_{a},+}(A) ∪ {σ_{η_{a},v}(A) : v ∈ ℤ^{2}},

where σ_{η_{a},−}(A), σ_{η_{a},+}(A) and σ_{η_{a},v}(A) contain all limit operators A_{g} ∈ σ_{η_{a}}(A) for which there is a subsequence h of g such that the limit operator of χ̂_{V_{Ω}} ⊗ I_{H} with respect to h exists and is equal to 0, I_{H} and to χ̂_{W (a)+v} ⊗ I_{H} with v ∈ ℤ^{2}, respectively. An analogous decomposition is given for σ_{η_{b}}(A). With these notations, we have the following theorem.

Theorem 2.29 Let H = l^{2}(ℤ^{2}, H) and A ∈ A_{0}(H). Then the operator A_{V_{Ω}} defined by (17) is locally invertible at infinity if and only if

(a) all limit operators A_{g} ∈ ∪_{x∈int Ω} σ_{η_{x}}(A) are uniformly invertible,

(b) all operators

(χ̂_{W (a)} ⊗ I_{H})A_{g}(χ̂_{W (a)} ⊗ I_{H}) : l^{2}(W (a), H) → l^{2}(W (a), H)

with A_{g} ∈ σ_{η_{a},0}(A) and all operators

A_{g} : l^{2}(ℤ^{2}, H) → l^{2}(ℤ^{2}, H)

belonging to σ_{η_{a},+}(A) are uniformly invertible, and if

(c) all operators

(χ̂_{W (b)} ⊗ I_{H})A_{g}(χ̂_{W (b)} ⊗ I_{H}) : l^{2}(W (b), H) → l^{2}(W (b), H)

with A_{g} ∈ σ_{η_{b},0}(A) and all operators

A_{g} : l^{2}(ℤ^{2}, H) → l^{2}(ℤ^{2}, H)

belonging to σ_{η_{b},+}(A) are uniformly invertible.

Proof. It is immediate from Theorem 2.16 and from the special form of the limit operators of the operator χ̂_{V_{Ω}} ⊗ I_{H} that the conditions of the theorem are necessary. For their suffiency we have to show that, under the conditions of the theorem, all operators

(χ̂_{W (x)+v} ⊗ I_{H})A_{g}(χ̂_{W (x)+v} ⊗ I_{H}) : l^{2}(W (x) + v, H) → l^{2}(W (x) + v, H)

with A_{g} ∈ σ_{η_{x},v}(A) and x ∈ {a, b} are uniformly invertible. We will do this for x = a, i.e. we let A_{g} ∈ σ_{η_{a},v}(A) with some v ∈ ℤ^{2}. Since

(χ̂_{W (a)+v} ⊗ I_{H})A_{g}(χ̂_{W (a)+v} ⊗ I_{H}) = τ_{v}(χ̂_{W (a)} ⊗ I_{H})τ_{−v}A_{g}τ_{v}(χ̂_{W (a)} ⊗ I_{H})τ_{−v},

it is clear that the operator (χ̂_{W (a)+v} ⊗ I_{H})A_{g}(χ̂_{W (a)+v} ⊗ I_{H}) is invertible on l^{2}(W (a) + v, H) if and only if the operator (χ̂_{W (a)} ⊗ I_{H})τ_{−v}A_{g}τ_{v}(χ̂_{W (a)} ⊗ I_{H}) is invertible on l^{2}(W (a), H). It is further evident that τ_{−v}A_{g}τ_{v} is the limit operator of A with respect to the sequence g + v := (g_{k} + v)_{k≥1} and that this operator belongs to σ_{η_{a},0}(A). Thus, condition (b) of the theorem implies the invertibility of the operator (χ̂_{W (a)+v} ⊗ I_{H})A_{g}(χ̂_{W (a)+v} ⊗ I_{H}).

The case n = 3. Let now Ω be a closed, bounded and convex polygon in ℝ^{2} with vertices u_{1}, u_{2}, …, u_{k} ∈ ℤ^{2}, and denote by V_{Ω} ⊂ ℝ^{3} the cone generated by Ω, that is

V_{Ω} = {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : x_{3} > 0 and (x_{1}/x_{3}, x_{2}/x_{3}) ∈ Ω}.

Further set u_{k+1} := u_{k} and u_{0} := u_{1} and write (u_{j}, u_{j+1}) for the open segment joining u_{j} to u_{j+1}. Moreover, let K_{u_{j}} refer to the angular domain in ℝ^{2} with vertex at u_{j} which is bounded by the rays starting at u_{j} and passing through u_{j−1} resp. u_{j+1} and which contains Ω, and let H_{j} ⊂ ℝ^{2} refer to the half plane, which is bounded by the straight line passing through u_{j} and u_{j+1} and which also contains Ω. Finally, set K_{u_{j}}^{0} := K_{u_{j}} − u_{j} and H_{j}^{0} := H_{j} − u_{j} (= the algebraic differences), and introduce the wedge

K_{u_{j}} := {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : (x_{1}, x_{2}) ∈ K_{u_{j}}^{0} + x_{3}u_{j}},

the half space

H_{j} := {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : (x_{1}, x_{2}) ∈ H_{j}^{0} + x_{3}u_{j}},

and the sets

V̂_{Ω} := V_{Ω} ∩ ℤ^{3}, K̂_{u_{j}} = K_{u_{j}} ∩ ℤ^{3} and Ĥ_{j} = H_{j} ∩ ℤ^{3}.

Our next goal is to determine the limit operators of the operator of multiplication by the function χ̂_{V_{Ω}} ⊗ I_{H}.

Given x ∈ ℝ^{2}, let η_{x} stand for the point on the sphere S̃^{2} of the infinitely distant points which lies on the ray starting at the origin and passing through (x, 1) ∈ ℝ^{3}. A little thought shows that

σ_{η_{x}}(χ̂_{V_{Ω}} ⊗ I_{H}) = {I_{H}} if x is in the interior of Ω  (18)

and

σ_{η_{x}}(χ̂_{V_{Ω}} ⊗ I_{H}) = {0} if x ∈ ℝ^{2} \ Ω.  (19)

Further, if x belongs to the segment (u_{j}, u_{j+1}), then

σ_{η_{x}}(χ̂_{V_{Ω}} ⊗ I_{H}) = {0, I_{H}, χ̂_{H_{j}+v} ⊗ I_{H} : v ∈ ℤ^{3}}  (20)

and, if x is the vertex u_{j}, then σ_{η_{u_{j}}} (χ̂_{V_{Ω}} ⊗ I_{H}) is the set

{0, I_{H}, χ̂_{H_{j−1}+v} ⊗ I_{H}, χ̂_{H_{j}+v} ⊗ I_{H}, χ̂_{K_{u_{j}} +v} ⊗ I_{H} : v ∈ ℤ^{3}}.  (21)

Now let H = l^{2}(ℤ^{3}, H) and A ∈ A_{0}(H). We say that the operator

A_{V_{Ω}} := (χ̂_{V_{Ω}} ⊗ I_{H})A(χ̂_{V_{Ω}} ⊗ I_{H}) : l^{2}(V̂_{Ω}, H) → l^{2}(V̂_{Ω}, H)  (22)

is locally invertible at infinity if the operator

B_{V_{Ω}} := A_{V_{Ω}} + (1 − χ̂_{V_{Ω}}) ⊗ I_{H} : l^{2}(ℤ^{3}, H) → l^{2}(ℤ^{3}, H)

is locally invertible at infinity.

If x belongs to the boundary of Ω then we get, in analogy to the case n = 2, decompositions of the local spectra σ_{η_{x}}(A) as follows. For x belonging to the segment (u_{j}, u_{j+1}) we have

σ_{η_{x}}(A) = σ_{η_{x},−}(A) ∪ σ_{η_{x},+}(A) ∪ {σ_{η_{x},v}(A) : v ∈ ℤ^{3}}

where σ_{η_{x},−}(A), σ_{η_{x},+}(A) and σ_{η_{x},v}(A) contain all limit operators A_{g} ∈ σ_{η_{x}}(A) for which there is a subsequence h of g such that the limit operator of χ̂_{V_{Ω}} ⊗ I_{H} with respect to h exists and is equal to 0, I_{H} and χ̂_{H_{j}+v} ⊗ I_{H}, respectively. Similarly, if x is the vertex u_{j}, then

σ_{η_{u_{j}}} (A) = σ_{η_{u_{j}} ,−}(A) ∪ σ_{η_{u_{j}} ,+}(A) ∪ {σ_{η_{u_{j}} ,−,v}(A) : v ∈ ℤ^{3}}
∪{σ_{η_{u_{j}} ,+,v}(A) : v ∈ ℤ^{3}} ∪ {σ_{η_{u_{j}} ,v}(A) : v ∈ ℤ^{3}}

where σ_{η_{u_{j}} ,−}(A), σ_{η_{u_{j}} ,+}(A), σ_{η_{u_{j}} ,−,v}(A), σ_{η_{u_{j}} ,+,v}(A) and σ_{η_{u_{j}} ,v}(A) consist of all limit operators A_{g} ∈ σ_{η_{u_{j}}} (A) for which there is a subsequence h of g such that the limit operator of χ̂_{V_{Ω}} ⊗ I_{H} with respect to h exists and is equal to 0, I_{H}, χ̂_{H_{j−1}+v} ⊗ I_{H}, χ̂_{H_{j}+v} ⊗ I_{H} and χ̂_{K_{u_{j}} +v} ⊗ I_{H}, respectively. With these notations we have the following result.

Theorem 2.30 Let H = l^{2}(ℤ^{3}, H) and A ∈ A_{0}(H). Then the operator A_{V_{Ω}} defined by (22) is locally invertible at infinity if and only if

(a) all limit operators A_{g} ∈ ∪_{x∈intΩ}σ_{η_{x}}(A) are uniformly invertible on l_{2}(ℤ^{3}, H),

(b) for every j and every x ∈ (u_{j}, u_{j+1}), all operators

(χ̂_{H_{j}} ⊗ I_{H})A_{g}(χ̂_{H_{j}} ⊗ I_{H}) : l^{2}(H_{j}, H) → l^{2}(H_{j}, H)

with A_{g} ∈ σ_{η_{x},0}(A) and all operators

A_{g} : l^{2}(ℤ^{3}, H) → l^{2}(ℤ^{3}, H)

belonging to σ_{η_{x},+}(A) are uniformly invertible.

(c) for every x = u_{j}, all operators

(χ̂_{K_{u_{j}}} ⊗ I_{H})A_{g}(χ̂_{K_{u_{j}}} ⊗ I_{H}) : l^{2}(K_{u_{j}}, H) → l^{2}(K_{u_{j}}, H)

with A_{g} ∈ σ_{η_{u_{j}} ,0}(A), all operators

(χ̂_{H_{j−1}} ⊗ I_{H})A_{g}(χ̂_{H_{j−1}} ⊗ I_{H}) : l^{2}(H_{j−1}, H) → l^{2}(H_{j−1}, H)

with A_{g} ∈ σ_{η_{u_{j}} ,−,0}(A), all operators

(χ̂_{H_{j}} ⊗ I_{H})A_{g}(χ̂_{H_{j}} ⊗ I_{H}) : l^{2}(H_{j}, H) → l^{2}(H_{j}, H)

with A_{g} ∈ σ_{η_{u_{j}} ,+,0}(A), and all operators

A_{g} : l^{2}(ℤ^{3}, H) → l^{2}(ℤ^{3}, H)

belonging to σ_{η_{u_{j}} ,+}(A) are uniformly invertible.

This result follows in the same way from Theorem 2.19 and from the identities (18)-(20) as its predecessor.

2.7 The finite section method for operator-valued banddominated operators

In this section we are going to derive necessary and sufficient conditions for the stability of the finite section method for band-dominated operators. Somewhat surprisingly, it will turn out that this problem is very closely related to the topic of the preceding section.

Let H = l^{2}(ℤ^{n}, H) and n = 1, 2. Given an operator A ∈ A_{0}(H) we consider its finite sections

𝔄_{m} := (χ̂_{Ω_{m}} ⊗ I_{H})A(χ̂_{Ω_{m}} ⊗ I_{H}) : l^{2}(Ω_{m}, H) → l^{2}(Ω_{m}, H),

where Ω is an interval with endpoints in ℤ in case n = 1 and a convex polygon with vertices in ℤ^{2} if n = 2, and where Ω_{m} := mΩ for m ∈ ℤ_{+}. Further we suppose that 0 is an interior point of Ω, which guarantees that

s-lim_{m→∞}χ̂_{Ω_{m}} ⊗ I_{H} = I_{H}.

What we are interested in is whether the finite section method is stable, i.e. whether the finite sections 𝔄_{m} of A are invertible for all sufficiently large m and whether the norms of their inverses are uniformly bounded:

sup_{m≥m_{0}} ∥𝔄_{m}^{−1}∥_{L(L^{2}(Ω_{m},H))} < ∞.

It is well-known and easy to see that, if the sequence (𝔄_{m}) is stable and if the operator A is invertible, then the equations

𝔄_{m}φ_{m} = (χ̂_{Ω_{m}} ⊗ I_{H})f

possess unique solutions φ_{m} ∈ l^{2}(Ω_{m}, H) for all sufficiently large m and all right hand sides f ∈ l^{2}(ℤ^{n}, H), and these solutions converge in the norm of l^{2}(ℤ^{n}, H) to the solution φ of the equation Aφ = f.

The following proposition relates the stability problem for the sequence (𝔄_{m}) of the finite sections of an operator on l^{2}(ℤ^{n}, H) to a local invertibility problem for an operator acting on l^{2}(ℤ^{n+1}, H). We will use the notations introduced in the preceding section.

Proposition 2.31 (a) Let A ∈ L(l^{2}(ℤ^{n}, H)). The sequence (𝔄_{m}) is stable if and only if the operator

B_{V_{Ω}} := (χ̂_{V_{Ω}} ⊗ I_{H})(A ⊗ I_{l^{2}})(χ̂_{V_{Ω}} ⊗ I_{H})

acting on l^{2}(V̂_{Ω}, H) is invertible at infinity.

(b) If A ∈ A_{0}(l^{2}(ℤ^{n}, H)), then B_{V_{Ω}} + (1 − χ̂_{V_{Ω}}) ⊗ I_{H} ∈ A_{0}(l^{2}(ℤ^{n+1}, H)).

Observe that A ⊗ I_{l^{2}} acts as a diagonal operator on l^{2}(V̂_{Ω}, H).

Proof. (a) We identify l^{2}(V̂_{Ω}, H) with the orthogonal sum ⊕_{m=1}^{∞}l^{2}(Ω_{m}, H) and, hence, B_{V_{Ω}} with the operator ⊕_{m=1}^{∞}𝔄_{m}, i.e. for x ∈ Ω and m ∈ ℤ_{+},

(B_{V_{Ω}}f)(x, m) = (𝔄_{m}f)(x).  (23)

Let the sequence (𝔄_{m}) be stable, and let m_{0} be an integer such that the operators 𝔄_{m} are invertible for m ≥ m_{0}. Set C_{m} := 𝔄_{m}^{−1} if m ≥ m_{0} and C_{m} := χ̂_{Ω_{m}} ⊗ I_{H} in case m < m_{0}. Our assumption guarantees that (C_{m}) is a bounded sequence; hence, D := ⊕C_{m} is a bounded operator on l^{2}(V̂_{Ω}, H). It is easy to check that D is an inverse of the operator B_{V_{Ω}} modulo the ideal J ∩L(l^{2}(V̂_{Ω}, H)) of L(l^{2}(V̂_{Ω}, H)). Thus, B_{V_{Ω}} is invertible at infinity.

For the reverse implication, suppose B_{V_{Ω}} = ⊕𝔄_{m} is invertible at infinity, i.e. there exist operators D in L(l^{2}(V̂_{Ω}, H)) and J_{1}, J_{2} in J ∩ L(l^{2}(V̂_{Ω}, H)) such that DB_{V_{Ω}} = I + J_{1} and B_{V_{Ω}}D = I + J_{2}. Multiplying these equalities from both sides by S_{m} := ⊕_{k=1}^{∞}P_{k} where P_{k} = 0 if k ≠ m and P_{m} = χ̂_{Ω_{m}} ⊗ I_{H}, one easily obtains the invertibility of 𝔄_{m} for all sufficiently large m as well as the inequality ∥𝔄_{m}^{−1}∥ ≤ 2∥D∥. Hence, the sequence (𝔄_{m}) is stable.

The assertion (b) follows simply from Proposition 2.14 which characterizes the band-dominated operators in A_{0}.

Thus, to get a stability criterion for the sequence (𝔄_{m}), one has to apply the Theorem 2.19 to the operator B_{V_{Ω}}, and this application essentially requires to compute all limit operators of B_{V_{Ω}}.

To formulate the desired stability result in case n = 1 let us assume for simplicity that Ω = [−1, 1] (which is no essential restriction), abbreviate the projection operator χ̂_{Ω_{m}} ⊗ I_{H} to P_{m}, define Q_{m} := I − P_{m}, let Π_{−1} : l^{2}(ℤ; H) → l^{2}(ℤ; H) stand for the projection operator which sends the sequence (…, x_{−1}, x_{0}, x_{1}, …) into (…, x_{−1}, 0, 0, …), and set Π_{1} := I − Π_{−1}. Further, if χ_{H_{1}} and χ_{H_{−1}} refer to the characteristic functions of the half planes

H_{1} := {(x, y) : y ≥ x} and H_{−1} := {(x, y) : y ≥ −x},

then

χ̂_{H_{−1}} ⊗ I_{H} = ⊕_{m≥1}τ_{−m}Π_{−1}τ_{m} and χ̂_{H_{1}} ⊗ I_{H} = ⊕_{m≥1}τ_{−m}Π_{1}τ_{m}.

Finally, for every bounded sequence (A_{n}) of operators we abbreviate the operator ⊕_{n≥1}A_{n} to Op(A_{n}).

Theorem 2.32 Let A ∈ A_{0}(l^{2}(ℤ, H)) and let Ω = [−1, 1]. Then the finite section method (P_{n}AP_{n} + Q_{n}) is stable if and only if the operator A is invertible, if the operators

Π_{−1}A_{s}Π_{−1} + Π_{1}  (24)

are invertible for all limit operators A_{s} ∈ σ_{−1}(A), if the operators

Π_{1}A_{s}Π_{1} + Π_{−1}  (25)

are invertible for all limit operators A_{s} ∈ σ_{1}(A), and if the norms of the inverses of all operators in (24) and (25) are uniformly bounded.

Proof. The necessity of the invertiblity of A is a simple consequence of the stability of the sequence (P_{m}AP_{m} + Q_{m}) which holds for every operator A. Let us check that all operators in (24) are necessarily invertible. The proof for (25) is analogous. Let s ⊂ ℤ be a sequence tending to −1 ∈ S^{0} such that the limit operator A_{s} exists. Then h_{m} := (s_{m}, −s_{m}) defines a sequence in ℤ^{2} which tends to exp(iφ) ∈ S^{1} with φ = 3π/4, and the limit operators Op(A)_{h} and Op(P_{m})_{h} exist and are equal to

Op(A_{s}) and χ̂_{H_{−1}}I = Op(τ_{−m}Π_{−1}τ_{m})_{m≥0},

respectively. Thus, σ(Op(P_{m}AP_{m} + Q_{m})) contains all operators

τ_{−m}Π_{−1}τ_{m}A_{s}τ_{−m}Π_{−1}τ_{m} + τ_{−m}Π_{1}τ_{m}

with m ≥ 0. In particular, it contains the operator Π_{−1}A_{s}Π_{−1} + Π_{1} and, thus, all operators as in (24). The proof for the operators in (25) is analogous, and the uniform boundedness of the inverses of all of these operators is a consequence of Theorem 2.16.

Now suppose that the operator A as well as all operators in (24) and (25) are invertible and that the norms of their inverses are uniformly bounded. We have to show that all limit operators Op(P_{n}AP_{n} +Q_{n}) are invertible and that the norms of their inverses are uniformly bounded.

If h ∈ ℤ^{2} is a sequence tending to infinity such that the limit operator Op(P_{m})_{h} exists and is equal to 0, and if g is a subsequence of h such that the limit operators Op(A)_{g} exist, too, then

Op(P_{m}AP_{m} + Q_{m})_{g} = I

and nothing is to prove. If Op(P_{m})_{h} exists and is equal to I, and if g is as above, then either

Op(P_{m}AP_{m} + Q_{m})_{g} = Op(A_{s}) or Op(P_{n}AP_{m} + Q_{m})_{g} = Op(V_{−c}AV_{c}I)

with a sequence s ∈ ℤ tending to infinity or with a constant c ∈ ℤ. In both cases, the invertibility of Op(P_{m}AP_{m} + Q_{m})_{g} follows from the invertibility of A, and it is also clear that

∥(Op(P_{m}AP_{m} + Q_{m})_{g})^{−1}∥ ≤ ∥A^{−1}∥.

It remains to consider sequences h which tend to exp(iφ) ∈ S̃^{1} with φ = π/4 and φ = 3π/4 and for which Op(P_{m})_{h} exists and is not equal to 0 or I. Let, for example, φ = 3π/4 and let g be a subsequence of h such that the limit operator Op(A)_{g} exists. Then the first component of g = (s, t) is a sequence in ℤ which tends to −∞ (or, equivalently, to −1 ∈ S̃^{0}) and for which the limit operator A_{s} exists. Thus, there is an k ∈ ℤ such that, with U_{k} := Op(τ_{k}),

Op(P_{m}AP_{m} + Q_{m})_{g} =

= U_{−k}(χ̂_{H_{−1}} ⊗ I_{H})U_{k}Op(A_{s})U_{−k}(χ̂_{H_{−1}} ⊗ I_{H})U_{k} + U_{−k}((1 − χ̂_{H_{−1}}) ⊗ I_{H})U_{k}
= U_{−k}Op(τ_{−m}Π_{−1}τ_{m})U_{k}Op(A_{s})U_{−k}Op(τ_{−m}Π_{−1}τ_{m})U_{k} +
+ U_{−k}Op(τ_{−m}Π_{1}τ_{m})U_{k}
= Op(τ_{−m−k}Π_{−1}τ_{m+k})Op(A_{s})Op(τ_{−m−k}Π_{−1}τ_{m+k}) + Op(τ_{−m−k}Π_{1}τ_{m+k})
= Op(τ_{−m−k}(Π_{−1}τ_{m+n}A_{s}τ_{−m−k}Π_{−1} + Π_{1})τ_{m+n}).

Clearly, this operator on l^{2}(ℤ^{2}, H) is invertible if and only if all operators

Π_{−1}τ_{m+k}A_{s}τ_{−m−k}Π_{−1} + Π_{1} with k, m ∈ ℤ

are invertible and if the norms of their inverses are uniformly bounded. But this invertibility is an obvious consequence of the invertibility of all operators in (24) since operator spectra are shift invariant as we already observed.

Our next goal is the stability of the finite section method (P_{n}AP_{n} + Q_{n}) in case A is band-limited on l^{2}(ℤ^{2}, H) and Ω is a convex polygon with vertices in ℤ^{2} and with 0 as its interior point.

Let, as above, u_{1}, u_{2}, …, u_{k} ∈ ℤ^{2} denote the vertices of Ω, set u_{k+1} := u_{1} and u_{0} := u_{1}, and abbreviate {x ∈ ℝ^{2} : x = (1 − t)u_{j} + tu_{j+1} with t ∈ (0, 1)} to (u_{j}, u_{j+1}). For j = 1, 2, …, k, let H_{j} refer to the half-plane which is bounded by the straight line through u_{j} and u_{j+1} and which contains the point 0, and let K_{j} refer to the angle with vertex at u_{j} which is bounded by the straight lines through u_{j} and u_{j−1} resp. u_{j} and u_{j+1} and which contains the point 0. Further, set H_{j}^{0} := H_{j} − u_{j} and K_{j}^{0} := K_{j} − u_{j} (= the algebraic difference).

Besides these notations for subsets of the plane ℝ^{2}, we need some notations for their spatial counterparts. If Ω is a polygon, then the operator Op(P_{m}) turns out to be the characteristic function of a pyramid in ℝ^{3} with its vertex in the origin. The faces of this pyramid are in correspondence with the edges of Ω. We let F_{j} stand for the half-space of ℝ^{3} which is bounded by the plane through the points 0, u_{j} and u_{j+1} and which contains the pyramid. Similarly, the edges of the pyramid are in correspondence with the vertices of Ω; the edge corresponding to u_{j} will be denoted by G_{j}.

Thus, the operator Op(P_{m}) is the product of characteristic functions of halfspaces which arise from a rotation of the characteristic function of the upper half-space, χI := χ̂_{R}_{+}^{3}I, where ℝ_{+}^{3} := {(x, y, z) : z > 0}. If S_{+}^{2}, S_{−}^{2} and S_{0}^{2} stand for the northern and southern hemisphere and for the equator of the sphere S̃^{2} of the infinite points of ℝ^{3}, respectively, and if U_{m} denotes the shift operator τ_{(0, 0, m)} for m ∈ ℤ, then one has (see also the previous section)

σ_{ξ}(χI) = \{ \begin{matrix}\{I\} & if & ξ ∈ S_{+}^{2} \\ \{0\} & if & ξ ∈ S_{−}^{2} \\ \{0, I, U_{−m}χU_{m} : m ∈ ℤ\} & if & ξ ∈ S_{0}^{2}. \end{matrix}  (26)

Next we have to determine the local operator spectra of Op(P_{m}). Given a point x ∈ Ω, the ray {(kx, k) : k ≥ 0} ⊆ ℝ^{3} points to an element of the sphere S̃^{2} which we denote by x^{(2)}. As a consequence of (26) one gets:

• If ξ ∈ S^{2} is not of the form x^{(2)} for some x ∈ Ω, then σ_{ξ}(Op(P_{m})) = {0}.

• If x is in the interior of Ω, then σ_{x^{(2)}}(Op(P_{m})) = {I}.

• If x ∈ (u_{j}, u_{j+1}), then σ_{x^{(2)}}(Op(P_{m})) consists of the operators 0, I and of the characteristic functions of the shifted half-planes U_{−k}χ̂_{F_{j}}U_{k} with k ∈ ℤ.

• If x = u_{j}, then σ_{x^{(2)}}(Op(P_{n})) consists of the operators 0 and I, the characteristic functions of the shifted half-planes

U_{−k}χ̂_{F_{j−1}}U_{k} and U_{−l}χ̂_{F_{j}}U_{l} with k, l ∈ ℤ,

and of the characteristic functions of the shifted dihedrals

U_{−k}χ̂_{F_{j−1}}U_{k}U_{−l}χ̂_{F_{j}}U_{l} with k, l ∈ ℤ.

The projections χ̂_{H_{j}^{0}}I will be abbreviated to Π_{j} in what follows. Finally, given a point x in the boundary ∂Ω of Ω, the ray {kx : k ≥ 0} ⊆ ℝ^{2} points to an element of the sphere S̃^{1} which we denote by x^{(1)}.

Theorem 2.33 Let A ∈ A_{0}(l^{2}(ℤ^{2}, H)) and let Ω be a convex polygon with 0 as its interior point and with vertices in ℤ^{2}. Then the finite section method (P_{n}AP_{n} + Q_{n}) is stable if and only if the operator A is invertible, if the operators

Π_{j}A_{s}Π_{j} + (I − Π_{j})  (27)

are invertible for all limit operators A_{s} ∈ σ_{x^{(1)}}(A) and for all x ∈ (u_{j}, u_{j+1}), if the operators

Π_{j}Π_{j−1}A_{s}Π_{j−1}Π_{j} + (I − Π_{j}Π_{j−1})  (28)

are invertible for all limit operators A_{s} ∈ σ_{x^{(1)}}(A) and for all x = u_{j}, and if the norms of the inverses of all operators in (27) and (28) are uniformly bounded.

Proof. Most parts of the proof run completely parallel to the proof of Theorem 2.32. The only essential difference is that now, at points x = u_{j}, we have (besides the trivial operators 0 and I) two kinds of limit operators of Op(P_{m}), namely shifted half-plane projections and shifted dihedral projections. So what we have to verify for every fixed vertex x = u_{j} of Ω is, for instance, that the invertibility of all operators in (28) implies the invertibility of the operators

U_{−k}(χ̂_{F_{j}} ⊗ I_{H})U_{k}Op(A_{s})U_{−k}(χ̂_{F_{j}} ⊗ I_{H})U_{k} + U_{−k}((1 − χ̂_{F_{j}}) ⊗ I_{H})U_{k}

for every k and every limit operator A_{s} ∈ σ_{x^{(1)}}(A) together with the uniform boundedness of their inverses.

Since χ̂_{F_{j}} ⊗ I_{H} = Op(τ_{−m}Π_{j}τ_{m}), this is equivalent to the invertibility of all operators

τ_{−m−n}Π_{j}τ_{m+n}A_{s}τ_{−m−n}Π_{j}τ_{m+n} + τ_{−m−n}(I − Π_{j})τ_{m+n}.

Employing, as in the proof of preceding theorem, the shift invariance of the local spectra, we see that this is further equivalent to the invertibility of all operators

Π_{j}A_{s}Π_{j} + (I − Π_{j}) with A_{s} ∈ σ_{x^{(1)}}(A).  (29)

The invertibility of this operator for a given operator A_{s} ∈ σ_{x^{(1)}}(A) can be shown as follows: Choose a sequence h ∈ ℤ^{2} such that

τ_{−h(m)}Π_{j−1}τ_{h(m)} → I_{H} strongly as m → ∞

but

τ_{−h(m)}Π_{j}τ_{h(m)} = Π_{j} for all m.

Since every shifted limit operator of A is a limit operator of A again, we conclude from the invertibility of all operators in (28) that the operators

Π_{j}Π_{j−1}τ_{h(m)}A_{s}τ_{−h(m)}Π_{j−1}Π_{j} + (I − Π_{j}Π_{j−1})

are invertible for all n and that the norms of their inverses are uniformly bounded. Then also the operators

Π_{j}τ_{−h(m)}Π_{j−1}τ_{h(m)}A_{s}τ_{−h(m)}Π_{j−1}τ_{h(m)}Π_{j} + (I − Π_{j}τ_{−h(m)}Π_{j−1}τ_{h(m)})  (30)

are invertible for all n, and the norms of their inverses are uniformly bounded, too. Letting n go to infinity, these operators converge strongly to the operator (29). Since also the adjoints of the operators (30) converge strongly, the invertibility of the operator (29) follows from standard arguments. Finally, the Banach-Steinhaus theorem entails that ∥(Π_{j}A_{s}Π_{j} +(I −Π_{j}))^{−1}∥ is bounded from above by the same constant as the norms of the inverses of the operators (30).

It is evident that an analogous result holds for the finite section method for banddominated operators on l^{2}(ℤ^{n}, H) with n > 2 if the corresponding projections P_{m} are defined with respect to convex polytopes having 0 as interior point and vertices in ℤ^{n}.

The results of this section were derived in [25] for finite-dimensional Hilbert spaces. The proofs given here are essentially those from [25] with the only difference that here we have the infinite-dimensional version of Theorem 2.19 at our disposal.

3 Local invertibility at infinity and stability of the finite section method for classes of operators on ℝ^{n}

In this section we are going to introduce and investigate a C^{∗}-algebra, B, which contains important classes of operators on L^{2}(ℝ^{n}) such as pseudodifferential operators in the Hörmander class OP S_{0,0}^{0}, Fourier convolution operators as well as operators in finite differences. Again we will study the problems of local invertibility at infinity, Fredholmness and applicability of the finite section method for these operators.

3.1 The C^{∗}-algebra B of operators on L^{2}(ℝ^{n})

The following definition is the analogue to the characterization of band-dominated operators on l^{2}(ℤ^{n}, H) given in Proposition 2.2.

Definition 3.1 We say that an operator A ∈ L(L^{2}(ℝ^{n})) belongs to the class B if, for every function φ ∈ BUC(ℝ^{n}),

lim_{t→0} ∥[A, φ_{t,h}]∥_{L(L^{2}(ℝ^{n}))} = 0  (31)

uniformly with respect to h ∈ ℝ^{n}, where

φ_{t,h}(x) := φ_{t}(x − h) and φ_{t}(x) := φ(t_{1}x_{1}, t_{2}x_{2}, …, t_{n}x_{n}).

Proposition 3.2 B is a C^{∗}-subalgebra of L(L^{2}(ℝ^{n})).

Proof. Let A, B ∈ B. Then, evidently, A + B ∈ B. Further, the identity

[AB, φ_{t,h}] = A[B, φ_{t,h}] + [A, φ_{t,h}]B

shows that lim_{t→0} ∥[AB, φ_{t,h}]∥ = 0 uniformly with respect to h, whence AB ∈ B. Analogously, the identity [A^{∗}, φ_{t,h}] = [φ_{t,h}, A]^{∗} implies A^{∗} ∈ B. It is also clear that B is a closed subalgebra of L(L^{2}(ℝ^{n})).

By I we denote the set of all operators A in L(L^{2}(ℝ^{n})) such that

lim_{R→∞} ∥Aχ_{R}∥ = lim_{R→∞} ∥χ_{R}A∥ = 0

where χ_{R}(x) is 1 if |x| ≥ R and 0 if |x| < R.

Proposition 3.3 I is a closed two-sided ideal of the C^{∗}-algebra B.

We omit the proof which is similar to the proof of Proposition 2.4. It is further evident that K(L^{2}(ℝ^{n})) ⊂ I.

An operator A ∈ B is said to be invertible at infinity if there is an operator L ∈ L(L^{2}(ℝ^{n})) and operators T_{1}, T_{2} ∈ I such that

LA = I + T_{1} and AL = I + T_{2}.  (32)

Proposition 3.4 If A ∈ B is invertible at infinity and L is an operator as in (32), then L belongs to B, too.

The proof is similar to the proof of Proposition 2.6.

With every point η ∈ S̃^{n−1}, we associate the set I_{η} of all operators T ∈ B which own the following property: for every ε > 0, there is a neighborhood W_{R,U} of η such that ∥T χ_{W_{R,U}} ∥ < ε and ∥χ_{W_{R,U}} T ∥ < ε. It is easy to see that I_{η} is a closed ideal in B, and that

I = ∩_{η∈S̃^{n−1}}I_{η}.

Again we call an operator A ∈ B invertible at the infinitely distant point η if there is an operator L_{η} ∈ L(L^{2}(ℝ^{n})) as well as operators T_{1}, T_{2} ∈ I_{η} such that

L_{η}A = I + T_{1} and AL_{η} = I + T_{2}.  (33)

Proposition 3.5 If A ∈ B is invertible at the infinitely distant point η and L_{η} is an operator as in (33), then L_{η} also belongs to B.

The simple proof follows from the identity

[L_{η}, φ_{t,h}] = L_{η}φ_{t,h}(AL_{η} − T_{2}) − (L_{η}A − T_{1})φ_{t,h}L_{η} = L_{η}[φ_{t,h}, A]L_{η} + Q_{t}

where Q_{t} := T_{1}φ_{h,t}L_{η} − L_{η}φ_{h,t}T_{2} tends to 0 in the norm of L(L^{2}(ℝ^{n})) as t → 0.

3.2 Discretization of operators in L(L^{2}(ℝ^{n}))

Let I_{0} denote the cube [0, 1)^{n} and χ_{0} the characteristic function of I_{0}, and set H_{0} := L^{2}(I_{0}). There is an obvious isometry, G, from L^{2}(ℝ^{n}) onto the Hilbert space H_{0} := l^{2}(ℤ^{n}, H_{0}) which sends a function u ∈ L^{2}(ℝ^{n}) to a sequence Gu = ((Gu)_{α})_{α∈ℤ^{n}} by

(Gu)_{α}(x) := χ_{0}(x)u(x + α) where α ∈ ℤ^{n} and x ∈ I_{0}.  (34)

The inverse operator G^{−1} is given by

(G^{−1}ψ)(x) = ∑_{α∈ℤ^{n}} χ_{0}(x − α)ψ_{α}(x − α) where x ∈ ℝ

with the series converging in the norm of L^{2}(ℝ^{n}). Proposition 3.6 The algebra B is ^{∗}-isomorphic to B_{G} = GBG^{−1} which is a C^{∗}-subalgebra of the algebra A(H_{0}). Proof. Let A ∈ B. We have to show that A_{G} := GAG^{−1} belongs to A(H_{0}). The condition A ∈ B is equivalent to

lim_{t→0} ∥[A, φ_{t,h}I]∥_{L(L^{2}(ℝ^{n}))} = 0  (35)

for arbitrary φ ∈ BUC(ℝ^{n}) and uniformly with respect to h ∈ ℝ^{n}. Consequently,

lim_{t→0} ∥[A_{G}, (φ_{t}I)_{G}]∥_{L(H_{0})} = 0,  (36)

where (φ_{t}I)_{G} is the operator of multiplication by the function

(φ_{t})_{G}(α, x) = φ(t · (x + α)) with α ∈ ℤ^{n}, x ∈ I_{0}

and where t · (x + α) := (t_{1}(x_{1} + α_{1}), …, t_{n}(x_{n} + α_{n})). Further, the condition φ ∈ BUC(ℝ^{n}) guarantees that

lim_{t→0} sup_{x∈I_{0}} |φ(t · (x + α)) − φ(t · α)| = 0,

whence

lim_{t→0} ∥(φ_{t}I)_{G} − φ̂_{t}I∥_{L(H_{0})} = 0.  (37)

The equalities (36) and (37) imply

lim_{t→0} ∥[A_{G}, φ̂_{t}I]∥_{L(H_{0})} = 0  (38)

for arbitrary functions φ ∈ BUC(ℝ^{n}). Thus, by Proposition 2.2, the operator A_{G} = GAG^{−1} is band-dominated on A(H_{0}). Proposition 3.7 One has I_{G} := GIG^{−1} = J (H_{0}) and (I_{η})_{G} := GI_{η}G^{−1} = J_{η}. The proof is similar to the proof of Proposition 3.2.

3.3 A C^{∗}-algebra of pseudodifferential operators

The algebra B contains some important classes of concrete operators as we will point out now. Let a = a(x, ξ) be a function in the class S_{0,0}^{0}, i.e. suppose there are constants C_{αβ} such that

|∂_{x}^{β}∂_{ξ}^{α}a(x, ξ)| ≤ C_{αβ}  (39)

for all multi-indices α = (α_{1}, …, α_{n}) and β = (β_{1}, …, β_{n}). We say that A = a(x, D) is a pseudodifferential operator in the Hörmander class OP S_{0,0}^{0} if there is a function a ∈ S_{0,0}^{0}, the so-called symbol of the operator A, such that

(Au)(x) = a(x, D)u(x) = ∫_{ℝ^{n}} ∫_{ℝ^{n}} a(x, ξ)e^{i(x−y,ξ)}u(y) dy dξ, u ∈ C_{0}^{∞}(ℝ^{n}).

Sometimes, we will denote the symbol of A by σ_{A}.

The well-known Calderon-Vaillancourt Theorem ([29]) says that the operators in OP S_{0,0}^{0} can be extended to bounded operators on L^{2}(ℝ^{n}) and that

∥Au∥_{L(L^{2}(ℝ^{n}))} ≤ C ∑_{|α|+|β|≤m} sup_{(x,ξ)∈ℝ^{2n}} |∂_{x}^{β}∂_{ξ}^{α}a(x, ξ)|  (40)

for every A ∈ OP S_{0,0}^{0}.

Proposition 3.8 The class OP S_{0,0}^{0} is contained in the algebra B.

Proof. Let φ be a function in BUC(ℝ^{n}). This function can be uniformly approximated by functions in C_{b}^{∞}(ℝ^{n}), the space of all infinitely differentiable functions which are bounded together with all of their derivatives. Indeed, choose a function ϕ ∈ C_{0}^{∞}(ℝ^{n}) such that ∫_{ℝ^{n}} ϕ(x)dx = 1 and define, for every δ > 0, the function

φ_{⟨δ⟩}(x) := δ^{−n} ∫_{ℝ^{n}} ϕ((x − y)/δ)φ(y) dy.

It is an elementary exercise to check that the functions φ_{⟨δ⟩} belong to C_{b}^{∞}(ℝ^{n}) and that they converge uniformly on ℝ^{n} to φ as δ tends to 0.

Consequently, what we have to verify for the proof of the inclusion OP S_{0,0}^{0} ⊂ B is that

lim_{t→0} ∥[A, φ_{t,h}I]∥_{L(L^{2}(ℝ^{n}))} = 0 for every function φ ∈ C_{b}^{∞}(ℝ^{n})

uniformly with respect to h. According to the formula for the symbol of the product of pseudodifferential operators we have

σ_{Aφ_{t,h}}(x, ξ) = (2π)^{−n} ∫∫ _{ℝ^{2n}} a(x, ξ + η) φ_{t,h}(x + y) e^{−i(y,η)} dy dη  (41)

where the double integral exists in the sense of oscillatory integrals. Further, by the Lagrange formula,

φ_{t,h}(x + y) = φ_{t,h}(x) + t ∑_{j=1}^{n} (∫_{0}^{1} ∂_{x_{j}}φ(t(x + h + θy)) dθ) y_{j}.  (42)

Substituting (42) into (41) we obtain

σ_{Aφ_{t,h}}(x, ξ) = a(x, ξ)φ_{t,h}(x)+

∑_{j=1}^{n} \frac{t_{j}}{(2π)^{n}} ∫_{0}^{1} (∫∫ _{ℝ^{2n}}(−i∂_{ξ_{j}})a(x, ξ)∂_{x_{j}}φ(t · (x + h + θy))e^{−i(y,η)} dy dη) dθ.(43)

The elementary machinery of estimating oscillatory integrals (see, e.g., ([21])) shows that the symbol of the double oscillatory integral in (43) belongs to the class S_{0,0}^{0} uniformly with respect to t, h ∈ ℝ^{n} and θ ∈ [0, 1]. Thus, the norm of the operator defined by the second term in (43) tends to 0 uniformly with respect to h ∈ ℝ^{n} if t → 0. This observation implies that

lim_{t→0} ∥[A, φ_{t,h}I]∥ = 0  (44)

for every function φ ∈ C_{b}^{∞}(ℝ^{n}). In view of the proved density of C_{b}^{∞}(ℝ^{n}) in BUC(ℝ^{n}), the equality (44) holds for every function φ in BUC(ℝ^{n}).

We denote by P the closure in L(L^{2}(H)) of the class OP S_{0,0}^{0}. By the preceding proposition, P is C^{∗}-subalgebra of B.

3.4 Limit operators of operators in L(L^{2}(ℝ^{n}))

In this section, we will develop the concept of limit operators for operators acting on L^{2}(ℝ^{n}).

Definition 3.9 Let A ∈ L(L^{2}(ℝ^{n})), and let h = (h_{m}) ⊂ ℤ^{n} be a sequence tending to infinity. We say that A_{h} is the limit operator of A defined by the sequence h if

A_{h} = s-lim_{m→∞}V_{−h_{m}}AV_{h_{m}}

where V_{α} is the shift operator (V_{α}u)(x) = u(x − α) on L^{2}(ℝ^{n}).

In complete analogy to Proposition 2.12, one has the following basic properties of limit operators of operators on L^{2}(ℝ^{n}).

Proposition 3.10 Let h be a sequence tending to infinity and let A, B and A^{(m)} be operators on L(L^{2}(ℝ^{n})).

(a) If the limit operator A_{h} exists, then ∥A_{h}∥ ≤ ∥A∥.

(b) If the limit operators A_{h}, B_{h} exist, then (A + B)_{h} exists and (A + B)_{h} = A_{h} + B_{h}.

(c) If the limit operators A_{h}, B_{h} exist, then (AB)_{h} exists and (AB)_{h} = A_{h}B_{h}.

(d) If A_{h} exists, then also (A^{∗})_{h} exists and (A^{∗})_{h} = (A_{h})^{∗}.

(e) If A = lim_{m→∞} A^{(m)} and if the limit operators (A^{(m)})_{h} exist for all sufficiently large m, then A_{h} exists and A_{h} = lim_{m→∞}(A^{(m)})_{h}.

Proposition 3.11 Let A ∈ B and let A_{h} be a limit operator of A. Then A_{h} also belongs to B.

Proof. Let φ ∈ BUC(ℝ^{n}) and q ∈ ℝ^{n}. By the Banach-Steinhaus Theorem,

∥[A_{h}, φ_{t,q}I]∥ ≤ lim inf_{m→∞} ∥[V_{−h_{m}}AV_{h_{m}}, φ_{t,q}I]∥ = lim inf_{m→∞} ∥[A, φ_{t,q+h_{m}}I]∥

where, according to (31),

lim_{t→0} ∥[A, φ_{t,q+h_{m}}I]∥_{L(L^{2}(ℝ^{n}))} = 0

uniformly with respect to q + h_{m}. Hence,

lim_{t→0} ∥[A, φ_{t,q}I]∥_{L(L^{2}(ℝ^{n}))} = 0

uniformly with respect to q ∈ ℝ^{n} which implies the assertion.

Given η ∈ S̃^{n−1}, we denote by σ_{η}(A) the set of all limit operators of A ∈ L(L^{2}(ℝ^{n})) which are defined by sequences h ⊂ ℤ^{n} tending to η, and we set

σ(A) := ∪_{η∈S̃^{n−1}}σ_{η}(A).

Proposition 3.12 (a) Let T ∈ I_{η} and h ⊂ ℤ^{n} a sequence tending to η ∈ S̃^{n−1}. Then the limit operator T_{h} exists and is equal to 0.

(b) Let T ∈ I and h ⊂ ℤ^{n} sequence tending to infinity. Then the limit operator T_{h} exists and is equal to 0.

Proof. (a) Let h = (h_{m}) and let u ∈ L^{2}(ℝ^{n}) be a function with bounded support. Then, for every neighborhood W_{R,U} of η, there exists an m_{0} such that supp (V_{h_{m}}u) ⊂ W_{R,U} if only m ≥ m_{0}. Thus, s-lim_{m→∞}V_{−h_{m}}T V_{h_{m}} = 0. Assertion (b) in an immediate consequence of (a).

Definition 3.13 Let B_{0} denote the set of all operators A ∈ B which own the folowing property: every sequence h tending to infinity possesses a subsequence g for which the limit operator A_{g} exists.

A little thought reveals that B_{0} is actually a C^{∗}-subalgebra of B.

Proposition 3.14 The algebra P of pseudodifferential operators is contained in B_{0}.

Proof. Let h = (h_{m}) be a sequence tending to infinity, and let A = a(x, D) ∈ OP S_{0,0}^{0}. We consider the sequence of the functions (x, ξ) ↦ a(x + h_{m}, ξ). This sequence is bounded in the topology of C^{∞}(ℝ^{2n}). Hence, there is a subsequence g of h such that the sequence (a(x+g_{m}, ξ))_{m≥1} converges in the topology of C^{∞}(ℝ^{2n}) (i.e. uniformly on compact subsets of ℝ^{2n} together with all partial derivatives). It is easy to check that the limit function a_{g}(x, ξ) of that subsequence belongs to S_{0,0}^{0}. We claim that the operator a_{g}(x, D) is the limit operator of A defined by the sequence g.

Indeed, let φ(x, ξ) ∈ C_{0}^{∞}(ℝ^{2n}) be a function which is 1 in a neighborhood of the origin, define φ_{R}(x, ξ) := φ(x/R, ξ/R), and observe that

V_{−g_{m}}AV_{g_{m}} = a(x + g_{m}, D).  (45)

Then, as follows from the composition formula for pseudodifferential operators (see, e.g., [21]),

lim_{R→∞} ∥(a(x + g_{m}, D) − a_{g}(x, D))φ_{R}(x, D)∥ = 0.  (46)

Since s-lim_{R→∞}φ_{R}(x, D) = I due to our assumption, the formulas (45) and (46) combine to give

s-lim_{R→∞}V_{−g_{m}}AV_{g_{m}} = a_{g}(x, D).

Thus, the limit operator A_{g} exists and is equal to a_{g}(x, D).

Finally we consider the discretizations of operators in B_{0}.

Proposition 3.15 If A ∈ B_{0}, then A_{G} := GAG^{−1} belongs to A_{0}(H_{0}). In particular, if g is a sequence tending to η ∈ S̃^{n−1} for which both limit operators A_{g} and (A_{G})_{g} exist, then

(A_{G})_{g} = (A_{g})_{G}.  (47)

Proof. Recall from Proposition 3.6 that the operator A_{G} belongs to the algebra A(H_{0}) where H_{0} = l^{2}(ℤ^{n}, L^{2}(H_{0})). Thus, what we have to show for the first assertion is that every sequence h tending to η possesses a subsequence g such that the limit operator (A_{G})_{g} exists.

The operator A_{G} maps the sequence u = (u_{β})_{β∈ℤ^{n}} ∈ H_{0} to a sequence A_{G}u = ((A_{G}u)_{α})_{α∈ℤ^{n}} as follows:

(A_{G}u)_{α}(x) = χ_{0}(x) ∑_{β∈ℤ^{n}} V_{−α}AV_{β}u_{β}(x) = χ_{0}(x) ∑_{γ∈ℤ^{n}} V_{−α}AV_{α−γ}u(α − γ, x) (48)

with the occurring series converging in the norm of L^{2}(ℝ^{n}). Thus, if h = (h_{m}) tends to η, then

(τ_{−h_{m}}A_{G}τ_{h_{m}}u)α(x) = χ_{0}(x) ∑_{β∈ℤ^{n}} V_{−α+h_{m}}AV_{β+h_{m}}u_{β}(x)
= χ_{0}(x) ∑_{γ∈ℤ^{n}} V_{−α+h_{m}}AV_{α+h_{m}−γ}u_{α−γ}(x).  (49)

We fix α, γ ∈ ℤ^{n} and consider the operator χ_{0}V_{−(α+h_{m})}AV_{α+h_{m}−γ}χ_{0}I. Since A belongs to B_{0} by assumption, there is a subsequence g = (g_{m}) = (h_{m_{k}}) of h such that the limit operator

A_{g} = s-lim_{k→∞}V_{−h_{m_{k}}} AV_{h_{m_{k}}}

exists. Hence, also the following limit in L(L^{2}(I_{0})),

lim_{k→∞} χ_{0}V_{−(α+h_{m_{k}})}AV_{α+h_{m_{k}} −γ}χ_{0}I = χ_{0}V_{−α}A_{g}V_{α−γ}χ_{0}I,

exists. Thus, if we replace in (49) the numbers h_{m} by h_{m_{k}} and let k go to infinity, then we arrive at

((A_{G})_{g}u)α(x) = χ_{0} ∑_{γ∈ℤ^{n}} V_{−α}A_{g}V_{α−γ}χ_{0}u_{α−γ}(x).

The right hand side of this equality is just the discretization of the operator A_{g}. Consequently, A_{G} belongs to A_{0}(H_{0}), and the identity (47) holds.

3.5 Local invertibility at infinity

Here we consider the problem of local invertibility at infinity of operators in the algebra B_{0}. We will establish these conditions again in terms of limit operators. All proofs will follows by having recourse to the results for band-dominated operators on l^{2}(ℤ^{n}, L^{2}(H_{0})).

3.5.1 Local invertibility at infinity of operators in B_{0}

We start with the analogue of Theorem 2.16 for operators in B_{0}.

Theorem 3.16 An operator A ∈ B_{0} is locally invertible at the point η ∈ S̃^{n−1} if and only if all limit operators A_{h} ∈ σ_{η}(A) are invertible and if

sup_{A_{h}∈σ_{η}(A)} ∥A_{h}^{−1}∥ < ∞.

Proof. Let A_{G} = GAG^{−1} be the discretization of A . From Proposition 3.15 we know that A_{G} belongs to A_{0}(H_{0}). Thus, Theorem 2.16 provides us with a criterion for the local invertibility of A_{G} at η in terms of limit operators of A_{G}. Since A is invertible at η if and only if A_{G} is invertible at η, the proof of the theorem follows immediately from the identity (47).

Theorem 3.17 An operator A ∈ B_{0} is invertible at infinity if and only if all limit operators A_{h} ∈ σ(A) are invertible and if

sup_{A_{h}∈σ(A)} ∥A_{h}^{−1}∥ < ∞.

Taking into account Proposition 3.15, this result follows from Theorem 2.19 in the same way as the preceding theorem followed from Theorem 2.16.

We say that an operator A ∈ B_{0} is locally Fredholm if, for all r > 0, there are operators B′_{r}, B′′_{r} ∈ L(L^{2}(ℝ^{n})) and operators K′_{r}, K′′_{r} ∈ K(L^{2}(ℝ^{n})) such that

B′_{r}Aψ_{r}I = ψ_{r}I + K′_{r} and ψ_{r}AB′′_{r} = ψ_{r}I + K′′_{r}

where ψ_{r} is the characteristic function of the set {x ∈ ℝ^{n} : |x| ≤ r}.

Theorem 3.18 Let A ∈ B_{0} be a locally Fredholm operator. Then A is a Fredholm operator if and only A is invertible at infinity (i.e. if the conditions of Theorem 3.17 are fulfilled).

This can be proved in the same way as Theorem 2.20.

Example. Let OP S_{1,0}^{0} denote the set of all pseudodifferential operators with symbols in S_{1,0}^{0}, the latter class being the set of all infinitely differentiable functions a = a(x, ξ) which satisfy the estimates

|∂_{x}^{β}∂_{ξ}^{α}a(x, ξ)| ≤ C_{αβ}(1 + |ξ|)^{−|α|}

for all multi-indices α and β. An operator A = a(x, D) ∈ OP S_{1,0}^{0} is called uniformly elliptic if

lim inf_{ξ→∞} inf_{x∈ℝ^{n}} |a(x, ξ)| > 0.

It is well-known (see, e.g., [29]) that an operator A = a(x, D) ∈ OP S_{1,0}^{0} is locally Fredholm if it is uniformly elliptic. Hence, if A = a(x, D) ∈ OP S_{1,0}^{0} is uniformly elliptic, then the uniform invertibility of all limit operators of A is a necessary and sufficient condition for A to be a Fredholm operator.

3.5.2 Local invertibility at infinity of pseudo-differential operators with slowly varying symbols

A pseudodifferential operator A ∈ OP S_{0,0}^{0} with symbol a ∈ S_{0,0}^{0} is a pseudodifferential operator with slowly varying symbol if

lim_{x→∞} sup_{ξ} |∂_{x_{j}}a(x, ξ)| = 0 for all j = 1, …, n.  (50)

The class of all pseudodifferential operators with slowly varying symbol will be denoted by OP S̃_{0,0}^{0} in what follows. Note that (50) implies that

lim_{x→∞} sup_{ξ} |∂_{x}^{β}∂_{ξ}^{α}a(x, ξ)| = 0 for all multi-indices α, β.  (51)

Proposition 3.19 Let A = a(x, D) ∈ OP S̃_{0,0}^{0}. Then the limit operators of A are translation invariant.

Proof. Let h = (h_{m}) be a sequence which tends to infinity and suppose that lim_{m→∞} a(x + h_{m}, ξ) =: a_{h}(x, ξ) in the topology of C^{∞}. We will show that the function a_{h} = a_{h}(x, ξ) does not depend on x. Indeed, for arbitrary x′, x′′ ∈ ℝ^{n} we get

a_{h}(x′, ξ) − a_{h}(x′′, ξ) = ∑_{j=1}^{n} (∫_{0}^{1} ∂_{x_{j}}a_{h}((1 − θ)x′ + θx′′, ξ) dθ) (x′_{j} − x′′_{j}) (52)

where

∂_{x_{j}}a_{h}((1 − θ)x′ + θx′′, ξ) = lim_{m→∞} ∂_{x_{j}}a((1 − θ)x′ + θx′′ + h_{m}, ξ) = 0  (53)

due to (50). As follows from (52) and (53), the symbol of the limit operator A_{h} is independent of x, whence the translation invariance of A_{h}.

Let P^{SO} denote the closure in L(L^{2}(ℝ^{n})) of the set OP S̃_{0,0}^{0}. This set forms a C^{∗}-subalgebra of L(L^{2}(ℝ^{n})). It is also clear that the limit operators of operators in P^{SO} are translation invariant again. Further, the set

J_{0} = {a(x, ξ) ∈ C_{b}(ℝ^{2n}) : lim_{x→∞} sup_{ξ∈ℝ^{n}} |a(x, ξ)| = 0}

proves to be a closed ideal of the C^{∗}-algebra C_{b}(ℝ^{2n}). To every operator A = a(x, D) ∈ OP S̃_{0,0}^{0}, we associate its symbol at infinity as the coset s_{∞}(A) := a(x, ξ) + J_{0} ∈ C_{b}(ℝ^{2n})/J_{0}. The following estimate has been derived in [22]:

∥s_{∞}(A)∥_{C_{b}(ℝ^{2n})/J_{0}} ≤ ∥A∥.  (54)

This estimate, together with a simple approximation argument, enables us to associate with every operator A ∈ P^{SO} its symbol at infinity, s_{∞}(A) ∈ C_{b}(ℝ)/J_{0}. As follows from [22] and [23], the condition of uniform invertibility of all limit operators of A ∈ P^{SO} has the following form.

Proposition 3.20 Let A ∈ P^{SO}. Then all limit operators A_{h} ∈ σ(A) are uniformly invertible if and only if the coset s_{∞}(A) is invertible in C_{b}(ℝ^{2n})/J_{0}.

The invertibility of a coset in C_{b}(ℝ^{2n})/J_{0} can be effectively checked: The coset a(x, ξ) + J_{0} is invertible in C_{b}(ℝ^{2n})/J_{0} if and only if

lim inf_{x→∞} sup_{ξ∈ℝ^{n}} |a(x, ξ)| > 0.

3.6 Local invertibility at infinity of operators in cones

In this section we formulate criteria for the invertibility at infinity of operators on cones in ℝ^{n}.

3.6.1 Local invertibility at infinity of operators in cones with smooth cross-section

Let Ω be an open convex domain in ℝ^{n} with smooth (C^{2}-) boundary and let V_{Ω} refer to the cone

V_{Ω} := {(x, x_{n+1}) ∈ ℝ^{n} × ℝ : x_{n+1} > 0 and x/x_{n+1} ∈ Ω}.

We will study the local invertibility at infinity of the operator

χ_{V_{Ω}}Aχ_{V_{Ω}}I : L^{2}(V_{Ω}, H) → L^{2}(V_{Ω}, H)

for operators A in B_{0}. By definition, this means that the operator

B_{V_{Ω}} := χ_{V_{Ω}}Aχ_{V_{Ω}}I + (I − χ_{V_{Ω}}I) : L^{2}(ℝ^{n+1}, H) → L^{2}(ℝ^{n+1}, H)

is locally invertible at infinity.

It is not hard to determine the limit operators of χ_{V_{Ω}}I. For, let us recall some notations. Given x ∈ ℝ^{n}, let η_{x} ∈ S̃^{n} denote the infinitely distant point which lies on the ray in ℝ^{n+1} starting at the origin and passing through the point (x, 1). Simple calculations give

σ_{η_{x}}(χ_{V_{Ω}}I) = {I} if x ∈ Ω,  (55)

σ_{η_{x}}(χ_{V_{Ω}}I) = {0} if x ∈ ℝ^{n} \ Ω̄.  (56)

Now let x ∈ ∂Ω. We denote by H_{x} the half space in ℝ^{n} which is bounded by the tangent plane to ∂Ω at the point x and which contains Ω. Further, set H_{x}^{0} := H_{x} − x, and let H_{x} refer to the half space in ℝ^{n+1} generated by H_{x}^{0}, i.e.

H_{x} := {(x, x_{n+1}) ∈ ℝ^{n} × ℝ : x ∈ H_{x}^{0} + x_{n+1}x}.

Then

σ_{η_{x}}(χ_{V_{Ω}}I) = {0, I, χ_{H_{x}+v} : v ∈ ℝ^{n+1}} if x ∈ ∂Ω.  (57)

For x ∈ ∂Ω, we have the following decomposition of the local spectra σ_{η_{x}}(A),

σ_{η_{x}}(A) = σ_{η_{x},−}(A) ∪ σ_{η_{x},+}(A) ∪ {σ_{η_{x},v}(A) : v ∈ ℝ^{n+1}}

where σ_{η_{x},−}(A), σ_{η_{x},+}(A) and σ_{η_{x},v}(A) contain all limit operators A_{g} ∈ σ_{η_{x}}(A) for which there is a subsequence h of g such that the limit operator of χ_{V_{Ω}}I with respect to h exists and is equal to 0, I and χ_{H_{x}+v}I, respectively.

Theorem 3.21 Let A ∈ B_{0}. The operator

χ_{V_{Ω}}Aχ_{V_{Ω}}I : L^{2}(V_{Ω}, H) → L^{2}(V_{Ω}, H)

is locally invertible at infinity if and only if

(a) all limit operators A_{h} ∈ ∪_{x∈Ω}σ_{η_{x}}(A) are uniformly invertible on L^{2}(ℝ^{n+1}, H),

(b) for every x ∈ ∂Ω, all operators

χ_{H_{x}}A_{g}χ_{H_{x}}I : L^{2}(H_{x}) → L^{2}(H_{x})

with A_{g} ∈ σ_{η_{x},0}(A) and all operators

A_{g} : L^{2}(ℝ^{n+1}) → L^{2}(ℝ^{n+1})

in σ_{η_{x},+}(A) are uniformly invertible.

The proof follows straightforwardly from Theorem 3.17 and from the identities (55) - (57).

Again, we say that χ_{V_{Ω}}Aχ_{V_{Ω}}I is a locally Fredholm operator if, for every r > 0, there are operators R′_{r}, R′′_{r} ∈ L(L^{2}(V_{Ω}, H)) and K′_{r}, K′′_{r} ∈ K(L^{2}(V_{Ω}, H)) such that

R′_{r}(χ_{V_{Ω}}Aχ_{V_{Ω}})χ_{r}I = χ_{r}I + K′_{r} and χ_{r}(χ_{V_{Ω}}Aχ_{V_{Ω}})R′′_{r} = χ_{r}I + K′′_{r},

where χ_{r} is the characteristic function of the ball {x ∈ ℝ^{n+1} : |x| ≤ r}.

Theorem 3.22 Let A ∈ B_{0}, and let χ_{V_{Ω}}Aχ_{V_{Ω}}I be a locally Fredholm operator. Then A is a Fredholm operator if and only if the conditions of Theorem 3.21 are fulfilled.

3.6.2 Local invertibility at infinity of operators in cones with edges

Here we consider operators on cones which can possess a finite number of conical points. For simplicity, we restrict ourselves to the case n = 3 which allows us to have recourse to the results derived in Section 2.6.

Let Ω be an open convex domain in ℝ^{2} the boundary ∂Ω of which is smooth (say, C^{2} again) up to a finite set M of singular points which are supposed to be conical points. Recall that a point x ∈ ∂Ω is conical if there is a neighborhood U_{x} ⊂ ℝ^{2} of that point such that Ω ∩ U_{x} = K_{x} ∩ U_{x} where K_{x} is a cone in ℝ^{2} with vertex at x.

As in the previous subsection, we let V_{Ω} denote the cone generated by Ω,

V_{Ω} := {(x, x_{3}) ∈ ℝ^{2} × ℝ : x_{3} > 0 and x/x_{3} ∈ Ω}.

What we are interested in is the local invertibility at infinity of the operator

A_{V_{Ω}} := χ_{V_{Ω}}Aχ_{V_{Ω}} : L^{2}(V_{Ω}, H) → L^{2}(V_{Ω}, H) where A ∈ B_{0}  (58)

which means, by definition, that the operator

B_{V_{Ω}} := χ_{V_{Ω}}Aχ_{V_{Ω}} + (I − χ_{V_{Ω}}) : L^{2}(ℝ^{3}, H) → L^{2}(ℝ^{3}, H)

is locally invertible at infinity.

We start with determining the limit operators of χ_{V_{Ω}}. As earlier, if x is a point in ℝ^{2}, we denote by η_{x} the point on the sphere S̃^{2} which lies on the ray in ℝ^{3} starting at the origin and passing through the point (x, 1) ∈ ℝ^{2} × ℝ. Clearly,

σ_{η_{x}}(χ_{V_{Ω}}) = {I} if x ∈ Ω and σ_{η_{x}}(χ_{V_{Ω}}) = {0} if x ∈ ℝ^{2} \ Ω̄.  (59)

Now let x ∈ ∂Ω \ M. We denote by H_{x} the half space in ℝ^{2} which is bounded by the tangent to ∂Ω at the point x and which contains Ω. Further, set H_{x}^{0} := H_{x}−x, and let H_{x} refer to the half space in ℝ^{3} generated by H_{x}^{0},

H_{x} := {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : (x_{1}, x_{2}) ∈ H_{x}^{0} + x_{3}x}.

A simple calculation yields

σ_{η_{x}}(χ_{V_{Ω}}I) = {0, I, χ_{H_{x}+v}I : v ∈ ℝ^{3}}.  (60)

It remains to consider the case when x ∈ M is a conical point. In this case, we denote by K_{x} the cone in ℝ^{2} which satisfies Ω ∩ U_{x} = K_{x} ∩ U_{x} for some neighbourhood U_{x} ⊆ ℝ^{2} of x, and we set K_{x}^{0} := K_{x} −x. Further, we let K_{x} stand for the wedge

K_{x} := {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : (x_{1}, x_{2}) ∈ K_{x}^{0} + x_{3}x}.

Finally, we choose points x_{+}, x_{−} ∈ (∂Ω ∩ U_{x}) \ {x} which lie on different vertices of K_{x}. For these points, we define H_{x_{±}} as before. Then, as in Section 2.6,

σ_{η_{x}}(χ_{V_{Ω}}I) = {0, I, χ_{H_{x_{−}}+v}I, χ_{H_{x_{+}}+v}I, χ_{K_{x}+v}I : v ∈ ℝ^{3}}.  (61)

If x belongs to ∂Ω, then again we have decompositions of the local spectra σ_{η_{x}}(A) as follows. For x ∈ ∂Ω \ M,

σ_{η_{x}}(A) = σ_{η_{x},−}(A) ∪ σ_{η_{x},+}(A) ∪ {σ_{η_{x},v}(A) : v ∈ ℝ^{3}}

where σ_{η_{x},−}(A), σ_{η_{x},+}(A) and σ_{η_{x},v}(A) contain all limit operators A_{g} ∈ σ_{η_{x}}(A) for which there is a subsequence h of g such that the limit operator of χ_{V_{Ω}}I with respect to h exists and is equal to 0, I and χ_{H_{x}+v}I, respectively. Similarly, if x ∈ M, then

σ_{η_{x}}(A) = σ_{η_{x},−}(A) ∪ σ_{η_{x},+}(A) ∪ {σ_{η_{x},−,v}(A) : v ∈ ℝ^{3}}
∪{σ_{η_{x},+,v}(A) : v ∈ ℝ^{3}} ∪ {σ_{η_{x},v}(A) : v ∈ ℝ^{3}}

where σ_{η_{x},−}(A), σ_{η_{x},+}(A), σ_{η_{x},−,v}(A), σ_{η_{x},+,v}(A) and σ_{η_{x},v}(A) consist of all limit operators A_{g} ∈ σ_{η_{x}}(A) for which there is a subsequence h of g such that the limit operator of χ_{V_{Ω}}I with respect to h exists and is equal to 0, I, χ_{H_{x_{−}}+v}I, χ̂_{H_{x_{+}}+v}I and χ̂_{K_{x}+v}I, respectively. With these notations, we have the following result.

Theorem 3.23 For A ∈ B_{0}, the operator A_{V_{Ω}} defined by (58) is locally invertible at infinity if and only if

(a) all limit operators A_{g} ∈ ∪_{x∈Ω}σ_{η_{x}}(A) are uniformly invertible on L^{2}(ℝ^{3}),

(b) for every x ∈ ∂Ω \ M, all operators

χ_{H_{x}}A_{g}χ_{H_{x}}I : L^{2}(H_{x}) → L^{2}(H_{x})

with A_{g} ∈ σ_{η_{x},0}(A) and all operators

A_{g} : L^{2}(ℝ^{3}) → L^{2}(ℝ^{3})

belonging to σ_{η_{x},+}(A) are uniformly invertible.

(c) for every x ∈ M, all operators

χ_{K_{x}}A_{g}χ_{K_{x}}I : L^{2}(K_{x}) → L^{2}(K_{x})

with A_{g} ∈ σ_{η_{x},0}(A), all operators

χ_{H_{x_{−}}}A_{g}χ_{H_{x_{−}}}I : L^{2}(H_{x_{−}}) → L^{2}(H_{x_{−}})

with A_{g} ∈ σ_{η_{x},−,0}(A), all operators

χ_{H_{x_{+}}}A_{g}χ_{H_{x_{+}}}I : L^{2}(H_{x_{+}}) → L^{2}(H_{x_{+}})

with A_{g} ∈ σ_{η_{x},+,0}(A), and all operators

A_{g} : L^{2}(ℝ^{3}) → L^{2}(ℝ^{3})

belonging to σ_{η_{x},+}(A) are uniformly invertible.

This theorem can be proved in the same manner as Theorem 2.30 by taking into account Theorem 3.17.

Defining the local Fredholmness of an operator as before, we also get the following generalization of Theorem 3.22.

Theorem 3.24 Let A ∈ B_{0} and let χ_{V_{Ω}}Aχ_{V_{Ω}}I be a locally Fredholm operator. Then χ_{V_{Ω}}Aχ_{V_{Ω}}I is a Fredholm operator if and only if the conditions of Theorem 3.23 are fulfilled.

The proof is similar to the proof of Theorem 2.20.

3.7 The finite section method for operators in cones

Here we present two approaches to the study of the finite section method for operators on cones. The first one works for finite sections of a special shape such as rectangles or cubes in which case the discretization of the approximation operators 𝔄 yields immediately the situation considered in Section 2.7. The second approach mimics exactly our methods from Section 2.7 which requires some continuity assumptions for the function t ↦ 𝔄_{t}.

3.7.1 First approach

Let us consider once more the finite section method, but now for operators A in B_{0} = B_{0}(L^{2}(ℝ^{2})). For simplicity, we let Ω denote the square [−1, 1] × [−1, 1], set Ω_{m} := mΩ = [−m, m] × [−m, m] for m a positive integer, and consider the finite sections of A,

𝔄_{m} := χ_{Ω_{m}}Aχ_{Ω_{m}}I : L^{2}(Ω_{m}) → L^{2}(Ω_{m}).

Since, evidently,

G(χ_{Ω_{m}}I)G^{−1} = χ̂_{Ω_{m}} ⊗ I,

it is clear that the sequence (χ_{Ω_{m}}Aχ_{Ω_{m}}I)_{m≥1} is stable if and only if the sequence

((χ̂_{Ω_{m}} ⊗ I)A_{G}(χ̂_{Ω_{m}} ⊗ I))_{m≥1}  (62)

with A_{G} := GAG^{−1} is stable. Further, since A_{G} belongs to A_{0}(l^{2}(ℤ^{2}, L^{2}(I_{0}))) due to Proposition 3.15, the stability criterion established in Theorem 2.33 applies immediately to the sequence (62) under consideration. What results is the following theorem where we use the notations introduced in Section 2.7.

Theorem 3.25 Let A ∈ B_{0} = B_{0}(L^{2}(ℝ^{2})) and let Ω := [−1, 1]^{2}. Then the finite section method (χ_{Ω_{m}}Aχ_{Ω_{m}}I) is stable if and only if the operator A is invertible, if the operators

χ_{H_{j}^{0}}A_{s}χ_{H_{j}^{0}}I : L^{2}(H_{j}^{0}) → L^{2}(H_{j}^{0})  (63)

are invertible for all limit operators A_{s} ∈ σ_{x^{(1)}}(A) and for all x ∈ (u_{j}, u_{j+1}), if the operators

χ_{H_{j}^{0}∩H_{j−1}^{0}}A_{s}χ_{H_{j}^{0}∩H_{j−1}^{0}}I : L^{2}(H_{j}^{0} ∩ H_{j−1}^{0}) → L^{2}(H_{j}^{0} ∩ H_{j−1}^{0})  (64)

are invertible for all limit operators A_{s} ∈ σ_{x^{(1)}}(A) and for all x = u_{j}, and if the norms of the inverses of all operators in (63) and (64) are uniformly bounded.

The proof follows simply from Theorem 2.33 by taking into account the identity (47) saying that (A_{G})_{g} = (A_{g})_{G} whenever one of these limit operators exist.

3.7.2 Second approach

Let Ω ⊂ ℝ^{n} be an open convex domain with the point 0 in its interior, and let A ∈ B_{0}. We consider the problem of stability of the operator family (𝔄_{t})_{t∈ℝ_{+}} where 𝔄_{t} := χ_{Ω_{t}}Aχ_{Ω_{t}}I : L^{2}(Ω_{t}) → L^{2}(Ω_{t}) and Ω_{t} := tΩ. As above, stability of the family (𝔄_{t}) means that there exist a t_{0} > 0 such that the operators 𝔄_{t} are invertible for all t ≥ t_{0} and and that

sup_{t≥t_{0}} ∥𝔄_{t}^{−1}∥_{L(L^{2}(Ω_{t}))} < ∞.

As in Section 2.7 we relate the stability problem for (𝔄_{t}) with the problem of local invertibility at infinity of the operator

χ_{V_{Ω}}(A ⊗ I_{ℝ})χ_{V_{Ω}}I : L^{2}(V_{Ω}) → L^{2}(V_{Ω}).

Here, V_{Ω} again refers to the cone in ℝ^{n+1} generated by Ω. aIn what follows we will need the unitary operators Φ_{t} : L^{2}(Ω) → L^{2}(Ω_{t}), t > 0, acting by

(Φ_{t}u)(x) := t^{n/2}u(x/t), x ∈ Ω_{t}

and having

(Φ_{t}^{−1}u)(x) = t^{−n/2}u(tx), x ∈ Ω

as their inverse operators. Let A be a bounded operator on L^{2}(ℝ^{n}) and A_{t} := χ_{Ω}Φ_{t}^{−1}AΦ_{t}χ_{Ω}I.

Proposition 3.26 Let the function ℝ_{+} ∋ t ↦ A_{t} ∈ L(L^{2}(Ω)) be continuous in the norm topology. Then the operator family (𝔄_{t}) with 𝔄_{t} := χ_{Ω_{t}}Aχ_{Ω_{t}}I : L^{2}(Ω_{t}) → L^{2}(Ω_{t}) is stable if and only if the operator χ_{V_{Ω}}(A⊗I_{ℝ})χ_{V_{Ω}}I : L^{2}(V_{Ω}) → L^{2}(V_{Ω}) is locally invertible at infinity.

Proof. We define a unitary operator Ψ : L^{2}(Ω × ℝ_{+}) → L^{2}(V_{Ω}) by

(Ψu)(x, t) := t^{n/2}u(x/t, t) for t > 0, x ∈ Ω_{t},

the inverse of which is given by

(Ψ^{−1}u)(x, t) = t^{−n/2}u(tx, t) for t > 0, x ∈ Ω.

It is easy to see that

Ψ^{−1}χ_{V_{Ω}}(A ⊗ I_{sR})χ_{V_{Ω}}Ψ = χ_{Ω×ℝ_{+}}Φ_{t}^{−1}AΦ_{t}χ_{Ω×ℝ_{+}}I.

Now let χ_{V_{Ω}}(A ⊗ I_{ℝ})χ_{V_{Ω}}I : L^{2}(V_{Ω}) → L^{2}(V_{Ω}) be locally invertible at infinity. Then there exist positive constants R and C such that

∫_{R}^{∞} ∥χ_{Ω}Φ_{t}^{−1}AΦ_{t}χ_{Ω}u(x, t)∥_{L^{2}(Ω)}^{2}dt ≥ C^{2} ∫_{R}^{∞} ∥u(x, t)∥_{L^{2}(Ω)}^{2}dt  (65)

and

∫_{R}^{∞} ∥χ_{Ω}Φ_{t}^{−1}A^{∗}Φ_{t}χ_{Ω}u(x, t)∥_{L^{2}(Ω)}^{2}dt ≥ C^{2} ∫_{R}^{∞} ∥u(x, t)∥_{L^{2}(Ω)}^{2}dt.  (66)

Given t_{0} > R, choose a function φ ∈ C_{0}^{∞}(ℝ) which has its support in a sufficiently small neighborhood of t_{0} and for which ∥φ∥_{L^{2}(ℝ)} = 1. The inequalities (65) and (66) imply

∥χ_{Ω}Φ_{t_{0}}^{−1}AΦ_{t_{0}}χ_{Ω}(v(x) ⊗ φ(t))∥_{L^{2}(Ω×(R,∞))}^{2} ≥ \frac{C^{2}}{4} ∥v(x) ⊗ φ(t)∥_{L^{2}(Ω×(R,∞))}^{2}.

Analogously, one gets

∥χ_{Ω}Φ_{t_{0}}^{−1}A^{∗}Φ_{t_{0}}χ_{Ω}(v(x) ⊗ φ(t))∥_{L^{2}(Ω×(R,∞))}^{2} ≥ \frac{C^{2}}{4} ∥v(x) ⊗ φ(t)∥_{L^{2}(Ω×(R,∞))}^{2}.

These estimates show that

∥χ_{Ω_{t}}Aχ_{Ω_{t}}v∥_{L^{2}(Ω_{t})}^{2} ≥ \frac{C^{2}}{4} ∥v∥_{L^{2}(Ω_{t})}^{2} for all t > R

and

∥χ_{Ω_{t}}A^{∗}χ_{Ω_{t}}v∥_{L^{2}(Ω_{t})}^{2} ≥ \frac{C^{2}}{4} ∥v∥_{L^{2}(Ω_{t})}^{2} for all t > R

with the constant C independent of t > R. Thus, the family (χ_{Ω_{t}}A^{∗}χ_{Ω_{t}}I) is stable.

Let, conversely, this family be stable. Thus, there exist constants R > 0 and C > 0 such that, for every function v(x, t) ∈ C_{0}^{∞}(Ω × ℝ_{+}),

∥χ_{Ω}Φ_{t}^{−1}AΦ_{t}χ_{Ω}v(x, t)∥_{L^{2}(Ω)}^{2} ≥ C∥v(x, t)∥_{L^{2}(Ω)}^{2} whenever t > R

and

∥χ_{Ω}Φ_{t}^{−1}A^{∗}Φ_{t}χ_{Ω}v(x, t)∥_{L^{2}(Ω)}^{2} ≥ C∥v(x, t)∥_{L^{2}(Ω)}^{2} whenever t > R.

Integrating these estimates over the interval (R, ∞) we arrive at the estimates (65) and (66). After a change of the variables, these estimates show the local invertibility at infinity of the operator χ_{V_{Ω}}(A ⊗ I_{ℝ})χ_{V_{Ω}}I : L^{2}(V_{Ω}) → L^{2}(V_{Ω}) which proves our claim.

Example. Let A = Op(a) be a pseudodifferential operator with symbol a ∈ S_{1,0}^{0}, that is

|∂_{x}^{β}∂_{ξ}^{α}a(x, ξ)| ≤ C_{αβ}(1 + |ξ|)^{−|α|}  (67)

for all multi-indices α and β. In this case, the function t ↦ A_{t} is norm continuous on ℝ_{+}. Indeed, we have

A_{t} = χ_{Ω}Op(ψ(x)a(tx, ξ/t))χ_{Ω}I

where ψ ∈ C_{0}^{∞}(ℝ^{n}) and ψ(x) = 1 in a neighborhood of Ω. Applications of the estimates (67) and of the Calderon-Vaillancourt Theorem yield the continuity of the function t ↦ A_{t}.

More general, if the operator A belongs to the C^{∗}-algebra generated by pseudo-differential operators in the class OP S_{1,0}^{0}, then the function t ↦ A_{t} is also norm continuous as one easily checks.

Given x ∈ ℝ^{n} \ {0}, write η′_{x} for the point in S̃^{n−1} which corresponds to the ray starting in 0 and passing through x. Further, given an open convex domain Ω with smooth boundary ∂Ω and with 0 in its interior, and given a point x ∈ ∂Ω, let ℍ_{x} refer to the half-space bounded by the tangent space to Ω at the point x which contains Ω, and set ℍ_{x}^{0} := ℍ_{x} − x.

Theorem 3.27 Let A ∈ B_{0} and suppose that the function ℝ_{+} ∋ t ↦ A_{t} ∈ L(L^{2}(Ω)) is continuous. Then the family (𝔄_{t})_{t>0} is stable if and only if

(a) the operator A : L^{2}(ℝ^{n}) → L^{2}(ℝ^{n}) is invertible,

(b) all operators

χ_{ℍ_{x}^{0}}A_{h}χ_{ℍ_{x}^{0}}I : L^{2}(ℍ_{x}^{0}) → L^{2}(ℍ_{x}^{0})

where x ∈ ∂Ω and where A_{h} ∈ σ_{η′_{x}}(A) are uniformly invertible.

The proof follows from Theorem 3.21 and Proposition 3.26.

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Authors’ addresses:

Vladimir S. Rabinovich, Department of Mechanics and Mathematics, Rostov State University, Ul. Sorge 5, 344104 Rostov-na-Donu, Russia E-mail: rabinov@ns.unird.ac.ru The same author also in: Instituto Politecnico Nacional ESIME – Zacatenco, SEPI, Ed.1, 2-do piso, Av. IPN, Mecico D.F., 07738 Mexico, Mexico E-mail: rabinov@maya.esimez.ipn.mx

Steffen Roch, Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, 64289 Darmstadt, FRG E-mail: roch@mathematik.tu-darmstadt.de

Bernd Silbermann, Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, FRG E-mail: bsilberm@mathematik.tu-chemnitz.de