Spectral approximation of Wiener-Hopf operators with almost periodic generating function

Steffen Roch

Abstract

The paper deals with spectral approximation of Wiener-Hopf operators acting on L^{p}-spaces by their finite sections. The generating function of the Wiener-Hopf operators are supposed to be continuous plus almost periodic. Whereas the usual spectra of the finite sections drastically fail to converge to the spectrum of the Wiener-Hopf operator, it turns out that other spectral approximants, viz. the pseudospectra and the numerical ranges, do converge perfectly. The proof requires a modified approach to the finite section method for Wiener-Hopf operators which will be developed first. This note generalizes results obtained by Böttcher, Grudsky and Silbermann for the case of continuous generating functions.

1 Introduction

Let F stand for the operator of Fourier transform which acts on the rapidly decreasing function f on R by

(F f)(x) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−ixt}f(t) dt.

It is well known (see [14], Ch. IX.1) that the operator F extends by continuity to a unitary operator on the Hilbert space L^{2}(R) and that

(F^{∗}f)(x) = (F^{−1}f)(x) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{ixt}f(t) dt.

Thus, if a ∈ L^{∞}(R), then F aF^{−1} is a bounded operator on L^{2}(R). This operator is the so-called operator of Fourier convolution by a, and its restriction onto the subspace L^{2}(R^{+}) of L^{2}(R) is the Wiener-Hopf operator with generating function a. We denote these operators by C(a) and W (a), respectively. Hence, if χ_{+} is the function on R which is 1 for x ≥ 0 and 0 for x < 0, and if P stands for the operator of multiplication by this function, then

W (a) = P C(a)|_{L^{2}(R^{+})} = P F aF^{−1}|_{L^{2}(R^{+})}.

Given p ≥ 1, let L^{p}(R) refer to the Lebesgue space with norm

∥f∥_{p} := (∫_{−∞}^{∞} |f(t)|^{p} dt)^{1/p} ,

and let M_{p} denote the collection of all functions a ∈ L^{∞}(R) which have the following property: if f ∈ L^{2}(R) ∩ L^{p}(R), then C(a)f ∈ L^{p}(R) and ∥C(a)f∥_{p} ≤ c_{p}∥f∥_{p} with some constant c_{p} independent of f. Particularly, M_{2} = L^{∞}(R). If a ∈ M_{p}, then the operators C(a) : L^{2}(R) ∩ L^{p}(R) → L^{p}(R) as well as W (a) : L^{2}(R^{+})∩L^{p}(R^{+}) → L^{p}(R^{+}) extend by continuity to bounded operators on L^{p}(R) and L^{p}(R^{+}) which we again denote by C(a) and W (a), respectively.

Here are two basic examples of convolution and Wiener-Hopf operators on L^{p}-spaces:

Example 1. Let k ∈ L^{1}(R), and set

a(x) = \sqrt{2π}(F^{−1}k)(x) = ∫_{−∞}^{∞} e^{ixt}f(t) dt.

The function a is continuous on R and vanishes at infinity (i. e. lim a(x) = 0 as x → +∞ and x → −∞), and it belongs to M_{p} for every p ≥ 1. The convolution theorem (Theorem IX.3 in [14]) says that

(C(a)f)(x) = ∫_{−∞}^{∞} k(x − t)f(t) dt, x ∈ R,

for every function f ∈ L^{p}(R). Thus, C(a) is a classical convolution operator, and

(W (a)f)(x) = ∫_{0}^{∞} k(x − t)f(t) dt, x ∈ R^{+},

is a classical Wiener-Hopf operator in this case.

Example 2. For α ∈ R, let U_{α} denote the shift operator

U_{α} : L^{p}(R) → L^{p}(R), (U_{α}f)(x) = f(x − α).

An elementary calculation yields

U_{α} = C(u_{α}) where u_{α}(x) = e^{iαx},

and since U_{α} is a bounded operator on L^{p}(R) with norm 1 we conclude that u_{α} ∈ M_{p} for every α and p. The associated Wiener-Hopf operator V_{α} := W (u_{α}) acts on L^{p}(R^{+}) via

(V_{α}f)(x) = \{ \begin{matrix}f(x − α) if & x ≥ max\{0, α\}, \\ 0 if & 0 ≤ x < max\{0, α\}. \end{matrix}

Defining a norm on M_{p} by ∥a∥_{M_{p}} := ∥C(a)∥_{L(L^{p}(R))} makes M_{p} to a Banach algebra. The closure in this algebra of the set of all functions a as in Example 1 will be denoted by C_{p}, and the closure in M_{p} of the set of all functions c_{1}u_{α_{1}}+ … + c_{k}u_{α_{k}} with c_{i} ∈ C and α_{i} ∈ R by AP_{p}. Both C_{p} and AP_{p} are closed subalgebras of M_{p}; the functions in C_{p} are continuous again and vanish at infinity, and the functions in AP_{p} are called almost periodic. What we are going to consider here is approximations of Wiener-Hopf operators W (a) by compressing W (a) onto the subspace L^{p}[0, t] of L^{p}(R^{+}), i. e. if χ_{t} denotes the function

χ_{t}(x) = \{ \begin{matrix}1 if & 0 ≤ x ≤ t, \\ 1 if & x > t \end{matrix}

(= the characteristic function of the interval [0, t]), and if P_{t} stands for the operator of multiplication by this function, then W (a) will be approximated by the operators

W_{t}(a) := P_{t}W (a)|_{L^{p}[0, t]}.

Clearly, the operators P_{t} converge strongly to the identity operator on L^{p}(R^{+}) as t → ∞, and so the finite sections W_{t}(a) of W (a) converge strongly to W (a).

The question we would like to discuss here is whether the spectra of the finite sections W_{t}(a) do approximate the spectrum of the Wiener-Hopf operator W (a) in some sense. If p = 2, and if the function a is real-valued, then the answer is affirmative: The limiting set of the spectra of the W_{t}(a) (i.e. the set of all partial limits of sequences of the form (λ_{t})_{t>0} with λ_{t} in the spectrum of W_{t}(a)) is exactly the spectrum of W (a). For non-selfadjoint operators, this convergence can fail drastically even in the simplest cases: If V_{−1} is the backward shift operator W (u_{−1}) on L^{2}(R^{+}), then the spectrum of V_{−1} is the closed unit disk, whereas the spectrum of every of its finite sections P_{t}V_{−1}P_{t} reduces to the singleton {0} (because (P_{t}V_{−1}P_{t})^{k} is zero for k > t).

Thus, we will have to modify our question appropriately: We will look for certain sets which can take the place of the usual spectrum of an operator and are closely related with the spectrum, and which exhibit much better convergence properties than common spectra. It turns out that spectral approximants with the desired properties indeed exist, and we will examine two of them: the εpseudospectra, and the numerical ranges. The precise definitions of these sets will be given in Section 3 below.

The approach to the convergence both of the pseudospectra and of the numerical ranges is based on a very precise knowledge on the finite section method for the operators under consideration. We say that the finite section method applies to the operator A ∈ L(L^{p}(R^{+})), or that this method is stable, if there is a t_{0} such that the finite sections P_{t}AP_{t} are invertible for all t ≥ t_{0} and that the norms of their inverses are uniformly bounded.

It is well known (see [4]) that the finite section method applies to the Wiener-Hopf operator W (a) with a being a generating function in AP_{p} + C_{p} if and only if the operator W (a) is invertible. But this result is not sufficient for our purposes for at least two reasons: First, as we have learned in the Hilbert space case (compare [19]), we need a stability criterion which holds not only for single sequences, but for sequences running through a whole Banach algebra of approximation sequences. So what we will do in Section 2 is to construct a certain Banach algebra A of approximation sequences which especially includes the sequences (W_{t}(a))_{t>0} of the finite section method we are interested in, and we will derive a stability criterion in the following form: A sequence (A_{t}) ∈ A is stable if and only if two associated operators, say S_{0}(A_{t}) and S_{1}(A_{t}), are invertible. In the Hilbert space or C^{∗}-case, this information is already all we need in order to verify the announced convergence results both for the pseudospectra and the numerical ranges. The point is that, in this case, the operators S_{0} and S_{1} can be defined as strong limits:

S_{0}(A_{t}) := s-lim_{t→∞}A_{t} and S_{1}(A_{t}) := s-lim_{t→∞}U_{−t}A_{t}U_{t},  (1)

thus they are ∗-homomorphisms, and this fact almost automatically involves that the algebra A under consideration is ∗-isomorphic to the C^{∗}-algebra of all ordered pairs of operators (S_{0}(A_{t}), S_{1}(A_{t})). However, this conclusion is no longer possible in the L^{p}-setting if p ≠ 2. Thus, our second goal in the following section will be to identify the algebra A up to an isometry with a Banach algebra of ordered pairs of operators.

2 The finite section method revisited

Throughout what follows let 1 < p < ∞ and 1/p + 1/q = 1. Let F stand for the Banach algebra of all bounded functions (A_{t})_{t>0} of operators A_{t} ∈ L(L^{p}(R)) (it will be more convenient in what follows to work with operators acting on the whole real line rather than on the interval [0, t]) provided with pointwise operations and the supremum norm. The set G consisting of all sequences (G_{t}) in F with lim ∥G_{t}∥ = 0 forms a closed two-sided ideal of the algebra F. Hence, the quotient algebra F/G is a Banach algebra again (with respect to the inherited operations and norm), and Kozak’s theorem tells us that a sequence (A_{t}) ∈ F is stable if and only if its coset (A_{t}) + G is invertible in this quotient algebra. So what we are left with is invertibility problems for certain cosets in F/G.

It turns out that these invertibility problems can be studied by localization via Allan’s local principle, and we will use this principle in order to localize the projections P_{t} (i.e. the sequence (P_{t}), considered as being quite complicated, will be locally replaced by sequences which are simpler in some sense, namely constant). This particular localization is possible for cosets in F/G which commute with all cosets of the form

(f_{t}I) + G where f ∈ C(Ṙ) and f_{t}(x) := f(x/t).  (2)

Herein, C(Ṙ) stands for the set of all continuous functions f on the real line with the property that the two limits lim_{x→+∞} f(x) and lim_{x→−∞} exist and coincide, and fI refers to the operator of multiplication by the function f, acting on L^{p}(R). The collection of all cosets of the form (2) will be denoted by C. It is not hard to check that C is a closed subalgebra of F/G and that ∥(f_{t}I)+G∥_{F/G} = ∥f∥_{∞}. Since we will moreover need the existence of the strong limits (1), we are naturally led to the consideration of the subset B of F/G consisting of all cosets (A_{t}) + G such that

((A_{t}) + G)((f_{t}I) + G) = ((f_{t}I) + G)((A_{t}) + G)  (3)

for all functions f ∈ C(Ṙ) and that the strong limits

S_{0}(A_{t}) := s-lim_{t→∞}A_{t} and S_{1}(A_{t}) := s-lim_{t→∞}U_{−t}A_{t}U_{t}

exist. Observe that the operators S_{0}(A_{t}), S_{1}(A_{t}) ∈ L(L^{p}(R)) depend on the coset (A_{t}) + G only.

The proof of the following proposition is standard and will be omitted.

Proposition 1 The set B is a closed subalgebra of F/G which is inverse closed.

The inverse closedness means that, whenever a coset b ∈ B is invertible in F/G, then its inverse b^{−1} already belongs to B. Thus, for cosets in B, the stability problem reduces to an invertibility problem in B.

Let furthermore A refer to the smallest closed subalgebra of F/G which contains all cosets of the form

(P_{t}W (a)P_{t} + α(I − P_{t})) + G with a ∈ AP_{p} + C_{p} and α ∈ C  (4)

where now P_{t}W (a)P_{t} + α(I − P_{t}) is viewed of as an operator which acts on the L^{p}-space over the real line. Thus, instead of the finite section sequence (W_{t}(a)), we will be concerned with the sequence (P_{t}W (a)P_{t} + (I − P_{t})) which evidently is stable if and only the sequence (W_{t}(a)) is so.

Observe that Wiener-Hopf operators with generating function in AP_{p} + C_{p} are integro-difference operators. For, if

a(x) = ∑_{r=0}^{∞} c_{r}e^{iα_{r}x} + \sqrt{2π}(F^{−1}k)(x)

with ∑ |c_{k}| < ∞ and k ∈ L^{1}(R), then

(W (a)f)(x) = ∑_{r=0}^{∞} c_{r}f(x − α_{r}) + ∫_{0}^{∞} k(x − t)f(t)dt.

The following result offers the way of localizing the invertibility problem for cosets in the algebra A and, thus, the stability problem for the finite section method for Wiener-Hopf integro-difference operators.

Theorem 1 The algebra A is a closed subalgebra of B.

Particularly, every coset (A_{t}) + G in A commutes with every coset (2), and the strong limits S_{0}(A_{t}) as well as S_{1}(A_{t}) exist. Before proving Theorem 1, we will formulate a theorem which together with Theorem 4 below describes the main results of this section.

Theorem 2 A coset (A_{t}) + G ∈ A is invertible in B (and thus in F/G) if and only if the operators S_{0}(A_{t}) and S_{1}(A_{t}) are invertible.

Proof of Theorem 1. Let us start with the commutator property (3). Clearly, the multiplication operators f_{t}I commute with the projections P_{t}, and they also commute with the operator P of multiplication by the characteristic function of the non-negative semi-axis; so it is sufficient to verify (3) in case A_{t} is the convolution operator C(a) where a ∈ AP_{p} or a ∈ C_{p}.

In case a ∈ AP_{p} it is enough to show that

∥f_{t}C(u_{α}) − C(u_{α})f_{t}I∥ = ∥f_{t}U_{α} − U_{α}f_{t}I∥ = ∥U_{−α}f_{t}U_{α} − f_{t}I∥ → 0

as t → ∞, which on its hand is a simple consequence of the uniform continuity of the function f.

For a ∈ C_{p}, the proof is a little bit more involved. Let f ∈ C(Ṙ). Since

C(a)f_{t}I − f_{t}C(a) = C(a)(f_{t} − f(∞))I − (f_{t} − f(∞))C(a)

we can suppose that f(∞) = 0, and since every continuous function which vanishes at infinity can by approximated by finitely supported functions, we can further assume that f has finite support. Moreover, splitting f into its real and imaginary part, and employing the fact that the continuously differentiable functions are dense in the continuous functions, we can suppose without loss that f is a real-valued function with supp f ⊆ [−r, r] and ∥f′∥ ≤ C with some positive constants r and C.

Further, it is evident from the definition of the class C_{p} that the operator C(a) with a ∈ C_{p} can be approximated as closely as desired by operators V (k) of the form (V (k)y)(t) := ∫_{−∞}^{∞} k(t − s)y(s)ds with k ∈ L^{1}(R). Every function in L^{1} can be approximated in the L^{1}-norm by a compactly supported and bounded function, and so we can assume without loss that C(a) = V (k) with supp k ⊆ [−r, r] and ∥k∥_{∞} ≤ C.

Let χ stand for the characteristic function of the interval [−r, r] and define χ_{t}(x) := χ(x/t). For proving that ∥V (k)f_{t}I − f_{t}V (k)∥ → 0 as t → ∞ is is sufficient to check the convergence

∥χ_{t}V (k)f_{t}I − f_{t}V (k)∥ → 0 as t → ∞.  (5)

Indeed, if this is shown for every f and k as above, then taking adjoints yields that ∥V (k)f_{t}I − f_{t}V (k)χ_{t}I∥ → 0, and multiplying from the left hand side by χ_{t} shows that ∥f_{t}V (k)χ_{t}I − χ_{t}V (k)f_{t}I∥ → 0 as t → ∞. Hence,

∥V (k)f_{t}I − f_{t}V (k)∥ ≤ ∥V (k)f_{t}I − f_{t}V (k)χ_{t}I∥
+∥f_{t}V (k)χ_{t}I − χ_{t}V (k)f_{t}I∥ + ∥χ_{t}V (k)f_{t}I − f_{t}V (k)∥ → 0.

So we are left with verifying (5). For all functions x ∈ L^{p}(R) we have

∥χ_{t}V (k)f_{t}x − f_{t}V (k)x∥_{L^{p}}^{p}

= ∫_{−∞}^{∞} |χ(u/t) ∫_{−∞}^{∞} k(u − s)f(s/t)x(s)ds − f(u/t) ∫_{−∞}^{∞} k(u − s)x(s)ds|^{p} du

= ∫_{−∞}^{∞} χ(u/t) |∫_{−∞}^{∞} k(u − s)(f(s/t) − f(u/t))x(s)ds|^{p} du

≤ ∫_{−∞}^{∞} χ(u/t) (∫_{−∞}^{∞} |k(u − s)(f(s/t) − f(u/t))|^{q}ds)^{p/q} (∫_{−∞}^{∞} |x(s)|^{p}ds) du

= ∫_{−∞}^{∞} χ(u/t) (∫_{−∞}^{∞} |k(u − s)(f(s/t) − f(u/t))|^{q}ds)^{p/q} du ∥x∥_{L^{p}}^{p}.

Furthermore,

∫_{−∞}^{∞} χ(u/t) (∫_{−∞}^{∞} |k(u − s)(f(s/t) − f(u/t))|^{q}ds)^{p/q} du

= ∫_{−rt}^{rt} (∫_{u−r}^{u+r} |k(u − s)|^{q}|f(s/t) − f(u/t)|^{q}ds)^{p/q} du

≤ C^{p} ∫_{−rt}^{rt} (∫_{u−r}^{u+r} |\frac{f(s/t) − f(u/t)}{s/t − u/t}|^{q} |\frac{s − u}{t}|^{q} ds)^{p/q} du

≤ C^{2p}t^{−p} ∫_{−rt}^{rt} (∫_{u−r}^{u+r} |s − u|^{q}ds)^{p/q} du = C^{2p}t^{−p} ∫_{−rt}^{rt} (∫_{−r}^{r} |s|^{q}ds)^{p/q} du

= C^{2p}t^{−p}2rt (∫_{−r}^{r} |s|^{q}ds)^{p/q} ≤ C̃t^{1−p} → 0 as t → ∞

thus verifying our claim (5) and the commutator property.

It remains to check the existence of the strong limits S_{0}(A_{t}) and S_{1}(A_{t}) for the sequences (A_{t}) in A. Because the operators S_{0} and S_{1} are continuous algebra homomorphisms, it is sufficient to check this for the sequences (P), (P_{t}) and (C(a)) in place of (A_{t}) what is readily done. What results is that

S_{0}(P) = S_{0}(P_{t}) = P, S_{0}(C(a)) = S_{1}(C(a)) = C(a), S_{1}(P) = I, S_{1}(P_{t}) = I − P

which finishes the proof.

Now we turn over to the proof of Theorem 2. By the definition of B, the algebra C of all cosets (f_{t}I)_{t>0} + G with f ∈ C(Ṙ) is a closed subalgebra in the center of B. It is elementary to check that this algebra is isometrically isomorphic to the algebra C(Ṙ), the maximal ideal space of which can be identified with the onepoint compactification Ṙ := R ∪{∞} of the real line. Under these identifications, the points s ∈ R correspond to the ideals I_{s} := {(f_{t}I)+G with f(s) = 0}, and the point ∞ corresponds to the ideal I_{∞} := {(f_{t}I)+G with f(∞) := lim_{t→∞} f(t) = 0} of C.

Given s ∈ Ṙ and (A_{t}) + G ∈ B, let J_{s} denote the smallest closed ideal of the algebra B which contains the ideal I_{s} of C, and we write Φ_{s}(A_{t}) for the local coset ((A_{t}) + G) + J_{s} of (A_{t}) + G in the quotient algebra B/J_{s}. Allan’s local principle (see [11], Theorem 1.5) then states that the coset (A_{t}) + G is invertible in B if and only if the local cosets Φ_{s}(A_{t}) are invertible in B/J_{s} for every s ∈ Ṙ.

It is immediate from the definitions that

Φ_{s}(P) = \{ \begin{matrix}Φ_{s}(0) if & s < 0 \\ Φ_{s}(P) if & s = 0 \\ Φ_{s}(I) if & s > 0 \\ Φ_{s}(P) if & s = ∞ \end{matrix}

and

Φ_{s}(P_{t}) = \{ \begin{matrix}Φ_{s}(0) if & s ∈ Ṙ \ [0, 1] \\ Φ_{s}(P) if & s = 0 \\ Φ_{s}(I) if & s ∈ (0, 1) \\ Φ_{s}(I − P + P_{t}) if & s = 1. \end{matrix}

Thus, if we let

A := ∑_{i} ∏_{j} W (a_{ij}) + α(I − P) and A_{t} := ∑_{i} ∏_{j} W_{t}(a_{ij}) + α(I − P_{t}) (6)

with a_{ij} ∈ AP_{p} + C_{p} and α ∈ C (and with both operators being considered as acting on the L^{p}-space over the real line again), then

Φ_{s}(A_{t}) = αΦ_{s}(I) for s ∈ Ṙ \ [0, 1]
Φ_{0}(A_{t}) = Φ_{0}(∑_{i} ∏_{j} P_{t}C(a_{ij})P_{t} + α(1 − P_{t}))
= Φ_{0}(∑_{i} ∏_{j} P C(a_{ij})P + α(I − P)) = Φ_{0}(A),
Φ_{s}(A_{t}) = Φ_{s}(∑_{i} ∏_{j} P_{t}C(a_{ij})P_{t} + α(I − P_{t}))
= Φ_{s}(∑_{i} ∏_{j} C(a_{ij})) for s ∈ (0, 1),
Φ_{1}(A_{t}) = Φ_{1}(∑_{i} ∏_{j} P_{t}C(a_{ij})P_{t} + α(I − P_{t}))
= Φ_{1}(∑_{i} ∏_{j}(I − P + P_{t})C(a_{ij})(I − P + P_{t}) + α(P − P_{t}))

= Φ_{1}(∑_{i} ∏_{j} U_{t}(I − P)U_{−t}C(a_{ij})U_{t}(I − P)U_{−t} + αU_{t}P U_{−t})
= Φ_{1}(U_{t}(∑_{i} ∏_{j}(I − P)C(a_{ij})(I − P) + αP)U_{−t}),

where we used the translation invariance U_{−t}C(a)U_{t} = C(a), holding for every L^{p}-multiplier a in the latter equality. Set

Ã := ∑_{i} ∏_{j}(I − P)C(a_{ij})(I − P) + αP.

Proposition 2 Let A, Ã, A_{t} be as above. If A and Ã are invertible, then the sequence (A_{t}) is stable.

Proof. The stability of (A_{t}) is equivalent to the invertibility of the coset (A_{t})+G in F/G and, thus, in B due to Proposition 1. By the local principle, it remains to check the invertibility of Φ_{s}(A_{t}) for all s ∈ Ṙ.

If A is invertible, then evidently α ≠ 0, and so there is nothing to prove in case s ∉ [0, 1]. For s ∈ {0, 1} one has

Φ_{0}(A_{t}) = Φ_{0}(A) resp. Φ_{1}(A_{t}) = Φ_{1}(U_{t}ÃU_{−t})  (7)

which involves invertibility in these cases, too. (Take into account that the coset (U_{t}ÃU_{−t}) + G = (∑_{i} ∏_{j}(I − P + P_{t})C(a_{ij})(I − P + P_{t}) + α(P − P_{t})) + G belongs to B as we have seen in the proof of Theorem 1.) If, finally, s ∈ (0, 1), then Φ_{s}(A_{t}) = Φ_{s}(Â) with Â := ∑_{i} ∏_{j} C(a_{ij}) , and it remains to prove that the invertibility of A implies the invertibility of Â.

One easily checks that the strong limit

S(A) := s-lim_{t→∞} U_{−t}AU_{t}  (8)

exists and that S(A) = Â. If now A is invertible (in L(L^{2}(R))), then A is also invertible in the closed subalgebra L(L^{p}(R)) consisting of all operators B for which the strong limit S(B) exists, and since S is an algebra homomorphism from this algebra into L(L^{p}(R)), we conclude that Â = S(A) is invertible. Since the coset (Â) + G belongs to B, we are done.

We finally prove the converse of Proposition 2.

Proposition 3 Let A, Ã, A_{t} be as above. If (A_{t}) is stable then A and Ã are invertible.

Proof. If (A_{t}) is stable, then (A_{t}) + G ∈ A is invertible in F/G and, thus, in B. Let (B_{t}) + G be the inverse of (A_{t}) + G in B, i.e.

((A_{t}) + G)((B_{t}) + G) = ((B_{t}) + G)((A_{t}) + G) = (I) + G.

Applying the homomorphisms S_{0} and S_{1} to both sides of this equality and taking into account that

S_{0}((A_{t}) + G) = A and S_{1}((A_{t}) + G) = Ã.

(compare the end of the proof of Theorem 1) we get the desired invertibility.

To finish the proof of Theorem 2, observe that the cosets of the form (A_{t}) + G with A_{t} := ∑_{i} ∏_{j} W_{t}(a_{ij}) + α(I − P_{t}) lie densely in A and that all occurring mappings (as Φ_{s}, S, S_{0}, S_{1}) are continuous algebra homomorphisms, and so a simple continuity argument yields the assertion of Theorem 2.

Let us consider the simplest case where A = W (a) + (I − P) and a ∈ AP_{p} + C_{p}. Then Ã = (I −P)C(a)(I −P)+P, and with the (anti-linear) operator (Jf)(t) := \overline{f(−t)} one gets

JÃJ = J((I − P)C(a)(I − P) + P)J = P C(ā)P + (I − P),

and invoking the invertibility criterion for Wiener-Hopf operators with generating function in AP_{p} + C_{p} (see Theorem 9.22 in [4] and Sections VII. 2 - 3 in [10]) one sees that the invertibility of A involves that of Ã in this special case. We so arrive at the classical theorem by Gohberg, Feldman, Coburn and Douglas:

Theorem 3 Let a ∈ AP_{p} + C_{p}. Then the finite section method (W_{t}(a)) is stable if and only if the operator W (a) is invertible.

As already mentioned, Theorem 2 is not our final goal. Here is what we really need:

Theorem 4 The Banach algebra A is isometrically isomorphic to the smallest closed subalgebra of L(L^{p}(R)) × L(L^{p}(R)) which contains all ordered pairs (S_{0}(A_{t}), S_{1}(A_{t})) where (A_{t}) + G runs through A. The isomorphism is given by the map

(A_{t}) + G ↦ (S_{0}(A_{t}), S_{1}(A_{t})).

Herein, the Cartesian product L(L^{p}(R))×L(L^{p}(R)) is provided with componentwise operations and the supremum norm.

The proof of Theorem 4 is essentially based on the concept of KMS-algebras introduced in [3]. Let B be a Banach algebra and C be a central subalgebra of B. Recall that the algebra B is called KMS with respect to C if the following conditions are satisfied:

(a) There is an involution on C which makes C, with respect to the norm inherited from B, to a C^{∗}-algebra.

(b) For all b ∈ B and all c_{1}, c_{2} ∈ C with supp c_{1} ∩ supp c_{2} = ∅ one has

∥b(c_{1} + c_{2})∥ ≤ max{∥bc_{1}∥, ∥bc_{2}∥}.

By the support of an element c of the commutative C^{∗}-algebra C we mean the support of the Gelfand transform of c which is a complex-valued function.

Theorem 5 (a) The algebra B is KMS with respect to its central subalgebra C.

(b) For all cosets (A_{t}) + G ∈ B,

∥(A_{t}) + G∥_{F/G} = max_{s∈Ṙ} ∥Φ_{s}(A_{t})∥_{B/J_{s}}.

The proof of assertion (a) proceeds completely analogous to the proof of Theorem 3.8. in [11], but the technical details are much simpler than there: In the present setting, the algebra C is isometrically isomorphic to C(Ṙ) (as already mentioned) and, thus, a C^{∗}-algebra, and also the verification of the second KMS property is less involved since the only ideal which is factored out is the ideal G of the zero sequences (whereas the corresponding ideal in [11] can be much larger in general). Observe also that the constant C appearing in [11] is equal to 1 because of ∥P_{t}∥ = 1.

Assertion (b) of the theorem is a general consequence of the KMS-concept (and in fact the main goal of this concept); proofs can be found in [3] or [11], Theorem 1.13.

Thus, the proof of Theorem 4 will follow once we have shown that

∥Φ_{s}(A_{t})∥ = ∥S_{s}(A_{t})∥ if s ∈ {0, 1}  (9)

and

∥Φ_{s}(A_{t})∥ ≤ max {∥Φ_{0}(A_{t})∥, ∥Φ_{1}(A_{t})∥} if s ∈ Ṙ \ {0, 1}  (10)

for all cosets (A_{t}) + G ∈ A. Since the cosets (A_{t}) + G with A_{t} := ∑_{i} ∏_{j} W_{t}(a_{ij}) + α(I−P_{t}) lie densely in A, and since all occurring mappings are continuous algebra homomorphisms, it is sufficient to check (9) and (10) for these special cosets.

Proposition 4 Let (A_{t}) + G ∈ A. Then (9) holds.

Proof. Consider the case s = 1. Let (A_{t}) be an arbitrary sequence such that (A_{t}) + G ∈ A, and let (G_{t}) be a sequence in G. Then, by the Banach-Steinhaus theorem,

∥S_{1}(A_{t})∥ ≤ lim inf_{t→∞} ∥U_{−t}A_{t}U_{t}∥ = lim inf_{t→∞} ∥U_{−t}(A_{t} + G_{t})U_{t}∥
≤ sup_{t} ∥A_{t} + G_{t}∥ = ∥(A_{t}) + (G_{t})∥_{F},

and taking the infimum over all (G_{t}) ∈ G yields

∥S_{1}(A_{t})∥ ≤ ∥(A_{t}) + G∥_{F/G}.  (11)

Let now (J_{t}) + G be a coset which belongs to the local ideal J_{1}. By the definition of local ideals, we can approximate (J_{t})+G as closely as desired by a sum of cosets of the form ((B_{t}) + G)((f_{t}I) + G) where (B_{t}) + G ∈ B and where f is a function which is continuous on Ṙ and has a zero at 1. It is easy to check that then S_{1}(f_{t}I) = 0. Hence, S_{1}((B_{t})(f_{t}I) + G) = 0 and, consequently, S_{1}((J_{t}) + G) = 0 for all cosets in J_{1}. Together with (11) we obtain

∥S_{1}(A_{t})∥ = ∥S_{1}(A_{t} + J_{t})∥ ≤ ∥(A_{t}) + (J_{t}) + G∥,

and taking the infimum over all cosets (J_{t}) + G ∈ J_{1} gives

∥S_{1}(A_{t})∥ ≤ ∥Φ_{1}(A_{t})∥.  (12)

The reverse inequality is obvious: Indeed, from (7) we infer that Φ_{1}(A_{t}) = Φ_{1}(U_{t}S_{1}(A_{t})U_{−1}), whence follows that

∥Φ_{1}(A_{t})∥ ≤ ∥(U_{t}S_{1}(A_{t})U_{−1})∥_{F} ≤ ∥S_{1}(A_{t})∥.

The proof for s = 0 is similar.

Proposition 5 Let (A_{t}) + G ∈ A. Then (10) holds.

Proof. First let s ∈ (0, 1), and let S be the homomorphism defined by (8). Then, as we have seen in the proof of Proposition 2,

Φ_{s}(A_{t}) = Φ_{s}(S(A)) = Φ_{s}(S(S_{0}(A_{t})))

whence follows that

∥Φ_{s}(A_{t})∥ ≤ ∥S_{0}(A_{t})∥ = ∥Φ_{0}(A_{t})∥

for all operators A_{t} and A as above.

For s ∈ Ṙ \ [0, 1] we have Φ_{s}(A_{t}) = αΦ_{s}(I) and, thus, ∥Φ_{s}(A_{t})∥ ≤ α, and this cannot be larger than

∥Φ_{0}(A_{t})∥ = ∥S_{0}(A_{t})∥ = ∥ ∑_{i} ∏_{j} W (a_{ij}) + α(I − P)∥

what finishes the proof of both Proposition 5 and Theorem 4.

3 Spectral approximation

Now we are in position to confirm the convergence of the ε-pseudospectra as well as of the numerical ranges.

3.1 Pseudospectra

Let ε > 0. The ε -pseudospectrum of an operator A ∈ L(L^{p}) is defined as the set of all complex numbers λ such that either A − λI is not invertible, or that this operator is invertible, but ∥(A − λI)^{−1}∥_{L(L^{p})} ≥ 1/ε. We denote the ε-pseudospectrum of A by σ^{ε}(A).

Probably, Henry Landau [12], [13] was the first to study pseudospectra of finite sections of Wiener-Hopf and Toeplitz operators. For p = 2 and continuous generating functions, the discrete analogue of Theorem 6 below is due to Reichel and Trefethen [15] and Böttcher [1], and the case 1 < p < ∞ with generating functions in C_{p} is subject of Böttcher, Grudsky, Silbermann [2].

Given a sequence (M_{t}) of subsets of the complex plane, we let lim_{t→∞} M_{t} stand for set of all complex numbers λ for which there is a strongly monotonically increasing sequence 0 < t_{0} < t_{1} < t_{2} < … as well as points λ_{n} ∈ M_{t_{n}} such that lim_{n→∞} λ_{n} = λ.

Theorem 6 Let (A_{t}) + G ∈ A and ε > 0. Then

lim_{t→∞} σ^{ε}(A_{t}) = σ^{ε}(S_{0}(A_{t}), S_{1}(A_{t})) = σ^{ε}(S_{0}(A_{t})) ∪ σ^{ε}(S_{1}(A_{t})).

Once Theorem 4 is established (which is the counterpart of Theorem 3.1 in [2]), the proof of Theorem 6 proceeds in complete analogy to that of Theorem 5.2 in [2]. The main ingredient of the proof is a generalization of a lemma due to Daniluk which goes back to Böttcher, Grudsky and Silbermann, and which states that, for operators A acting on L^{p}-spaces, the function λ ↦ (A − λI)^{−1} satisfies the maximum principle (Theorem 5.1 in [2]). The details will be omitted.

Specifying Theorem 6 to the case of the finite section method for Wiener-Hopf operators one gets:

Theorem 7 Let a ∈ AP_{p} + C_{p}. Then

lim σ^{ε}(W_{t}(a)|_{L^{p}(0,t)}) = σ^{ε}(W (a)|_{L^{p}(0,∞)}) ∪ σ^{ε}((I − P)C(a)(I − P)|_{L^{p}(−∞,0)}).

Proof. From Theorem 6 we know that, for every a ∈ AP_{p}+C_{p} and every complex number α,

lim σ^{ε}(W_{t}(a) + α(I − P_{t}))
= σ^{ε}(W (a) + α(I − P)) ∪ σ^{ε}((I − P)C(a)(I − P) + αP)

with all occuring operators acting on L^{p}(R). Since

σ^{ε}(QAQ + (I − Q)B(I − Q)) = σ^{ε}(QAQ|_{Im Q}) ∪ σ^{ε}((I − Q)B(I − Q)|_{Im (I−Q)})

for all operators A, B ∈ L(L^{p}(R)) and all operators Q of multiplication by the characteristic function of a measurable subset of the real line, and since the εpseudospectrum of αI is just the disk U_{ε}(α) with center α and radius ε, we conclude that

U_{ε}(α) ∪ lim σ^{ε}(W_{t}(a)|_{L^{p}(0,t)})
= U_{ε}(α) ∪ σ^{ε}(W (a)|_{L^{p}(0,∞)}) ∪ σ^{ε}((I − P)C(a)(I − P)|_{L^{p}(−∞,0)}).

This identity holds for arbitrary α. Considering this identity for two different numbers α_{1} and α_{2} with U_{ε}(α_{1}) ∩ U_{ε}(α_{2}) = ∅ one easily gets the assertion.

3.2 Numerical ranges

Given a complex Banach space X with dual X^{∗} define

Π(X) := {(f, x) ∈ X^{∗} × X : ∥f∥ = ∥x∥ = 1, f(x) = 1}.

For A a linear bounded operator on X, the spatial numerical range of A is the set

SN_{X}(A) := {f(Ax) : (f, x) ∈ Π(X)}.

In case X = H is a Hilbert space with inner product ⟨., .⟩, this definition reduces to the classical one:

SN_{H}(A) = {⟨Ax, x⟩ : ∥x∥ = 1}

because Π(H) can be identified with {(x, x) ∈ H × H : ∥x∥ = 1} in this case.

There is an alternative notion of a numerical range which works for elements of a Banach algebra which we call the algebraic numerical range for distinction. Given a Banach algebra L with identity e, the algebraic numerical range AN_{L}(a) of an element a ∈ L is defined as the set of all numbers φ(a) with φ running through the states of L, i.e. through the set of all linear functionals on L such that φ(e) = ∥φ∥ = 1.

Thus, for an operator A ∈ L(L^{p}), there are at least two possibilities of introducing a numerical range: the spatial numerical range SN_{L^{p}}(A) where A is considered as an operator acting on the Banach space L^{p}, and the algebraic numerical range AN_{L(L^{p})}(A) where A is considered as an element of the Banach algebra L(L^{p}). The relation between these two notions is as follows.

Theorem 8 Let X be a Banach space and A ∈ L(X). Then

AN_{L(X)}(A) = clos conv SN_{X}(A).

Herein, clos convM refers to the closure of the convex hull of the set M. For a proof see Theorem 9.4 in [5].

In general, the algebraic numerical ranges give the most satisfactory results in the Banach space case. Particularly, AN_{L}(a) is a closed and convex subset of the complex plane which encloses the spectrum of a ∈ L. Moreover, Banach algebras are inverse closed with respect to algebraic numerical ranges in the sense that AN_{L}(a) = AN_{L′}(a) where L′ refers to the smallest closed subalgebra of L which contains both a and the identity element e of L. For these and more facts see the monograph [5].

The following result describes the asymptotic behaviour of the numerical ranges of the operators A_{t} in a sequence (A_{t}) + G ∈ A.

Theorem 9 Let (A_{t}) + G be a coset in the algebra A. Then

lim_{t→∞} AN_{L(L^{p}(R))}(A_{t}) = conv (AN_{L(L^{p}(R))}(S_{0}(A_{t})) ∪ AN_{L(L^{p}(R))}(S_{1}(A_{t}))).

In [17], the discrete analogue of this theorem is proved for the case of continuous generating functions. The proof given there is based heavily on the possibility of representing the algebra of all approximation methods under consideration (i.e. some analogue of our algebra A) as an algebra of ordered pairs of operators up to an isometrical isomorphism. Moreover, the proof in [17] works in all situations where a similar description is possible. Especially, and thanks to Theorem 4, it also works for cosets in the algebra A. So we refer to the proof of Theorem 3 in [17] for the details.

In case of the pure finite sections we can complete the result of Theorem 9 as follows:

Theorem 10 Let a ∈ AP_{p} + C_{p}. Then

lim SN_{L^{p}(0,t)}(W_{t}(a)) = SN_{L^{p}(0,∞)}(W (a)).

There is a quite simple proof of this result which makes use of the uniform convexity of the spaces L^{p} for 1 < p < ∞ (and which does not involve the general Theorem 9); for details see Section 4 in [17] again.

Concluding remarks

1. The sketched approach does also work in the case of matrix-valued generating functions with entries in AP_{p} + C_{p}, and the corresponding results of Theorems 6, 7 and 9 have exactly the same form as above.

2. The results in Theorems 6, 7, 9 and 10 can be sharpened as follows: Given a sequence (M_{t}) of subsets of the complex plane, let Lim_{t→∞}M_{t} stand for set of all complex numbers λ such that for every strongly monotonically increasing sequence 0 < t_{0} < t_{1} < t_{2} < … there exist points λ_{n} ∈ M_{t_{n}} with lim_{n→∞} λ_{n} = λ. Then the partial limiting sets lim M_{t} appearing in the formulations of the mentioned theorems can be replaced by the corresponding uniform limiting sets LimM_{t}. This can be seen by invoking the concept of fractality developed in [19] (observe that the homomorphisms S_{0} and S_{1} are fractal in the sense of [19]).

3. The case of the finite sections of Wiener-Hopf operators with piecewise continuous generating function on L^{p}-spaces is still open. In the Hilbert space case p = 2, all the results derived above can be shown also for Wiener-Hopf operators with piecewise generating function which has been done in [18] (where actually a much larger algebra of approximation methods has been investigated which also includes, for instance, the finite sections of Wiener-Hopf operators in combination with multiplication operators and the Carleman shift (Jf)(t) = f(−t)). But in case p ≠ 2, only a certain analogue of Theorem 2 is available (see [16]) which, however, is not sufficient for further considerations. The point is that, for piecewise continuous generating functions, we indeed can localize and apply the KMS principle as above, but this requires to factor out an essentially larger ideal than the ideal G of the zero sequences. For an idea of how this large ideal looks like compare [11], Section 3.2.6. So we actually can prove an isometrical representation in case of piecewise continuous generating functions too, but only for a certain quotient of the algebra we are really interested in, which does not seem very helpful.

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Author’s address:

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