Algebras of approximation sequences: Fredholmness

Steffen Roch

Abstract

In this paper, a Fredholm theory for approximation sequences is proposed. A sequence is called Fredholm if it is invertible modulo a certain ideal which plays the role of the ideal of the compact operators in the Fredholm theory of operators. With every Fredholm sequence, there are associated three integers which are the analogues of the nullity, the defiency and the index of a Fredholm operator. The nullity of a Fredholm sequence (A_{n}) is interpreted as a quantity which describes the asymptotic behaviour of the small singular values of the matrices A_{n} an n → ∞, and an identity is derived which allows the computation of this nullity in many situations. Several examples and applications are discussed.

Keywords Fredholm theory, approximation methods, asymptotic behaviour of singular values AMS classification 46N40, 65R20

1 Introduction

Approximation methods. Let H be a separable complex Hilbert space, L(H) the C^{∗}-algebra of the linear and bounded operators on H, and (P_{n}) a sequence of orthogonal projections on H which converges strongly (= pointwisely) to the identity operator I on H:

s-lim P_{n} = I ⇔ P_{n}x → x for all x ∈ H.

Assume that dim Im P_{n} = n; so Im P_{n} and L(Im P_{n}) can be identified with the linear space ℂ^{n} and with the algebra ℂ^{n×n} = L(ℂ^{n}), respectively. The n × n identity matrix will be denoted by I_{n}.

Let A ∈ L(H). An approximation method for A is a sequence (A_{n}) of matrices A_{n} ∈ ℂ^{n×n} such that A_{n}P_{n} → A and A_{n}^{∗}P_{n} → A^{∗} strongly as n → ∞. This method converges, or is applicable to A, if the equations

A_{n}x^{(n)} = P_{n}y  (1)

possess unique solutions x^{(n)} ∈ Im P_{n} for all sufficiently large n and all right hand sides y ∈ H, and if these solutions converge in the norm of H to a solution of the equation

Ax = y.  (2)

By the Banach-Steinhaus theorem, the method (A_{n}) for A is applicable if and only if the sequence (A_{n}) is stable in the sense that the matrices A_{n} are invertible for all sufficiently large n and that

sup ∥A_{n}^{−1}∥ < ∞.

For another characterization of stability, introduce the set F of all bounded sequences (A_{n}) of matrices A_{n} ∈ ℂ^{n×n}. Provided with elementwise operations and the supremum norm, this set becomes a C^{∗}-algebra with identity element (I_{n}), and the set G of all sequences (A_{n}) ∈ F with lim ∥A_{n}∥ = 0 forms a closed twosided ideal in F. A Neumann series argument shows that the sequence (A_{n}) ∈ F is stable if and only if its coset (A_{n})+G is invertible in the quotient algebra F/G.

Standard algebras of approximation methods. One way to study the stability properties of a class of approximation methods is to describe the subalgebra of F/G which is generated by the approximation sequences under consideration. For several concrete approximation methods such descriptions were given, e.g., in [3, 5, 10, 11]. To mention at least one example, we recall the results pertaining the polynomial collocation method for singular integral operators on the unit circle obtained in [6] and [10].

Consider the operator of singular integration against the unit circle 𝕋,

(Su)(t) := \frac{1}{πi} ∫_{T} \frac{u(s)}{s − t} ds, t ∈ 𝕋,

and let a and b be piecewise continuous functions on 𝕋 (i.e. a and b possess finite one-sided limits at each point of 𝕋). The operator aI + bS is bounded on the Lebesgue space L^{2}(𝕋). Let Π_{n} stand for the subspace of L^{2}(𝕋) spanned by the polynomials z^{j} with −n ≤ j ≤ n. For Riemann integrable and bounded right hand sides f, we seek approximate solutions u^{(n)} ∈ Π_{n} of the singular integral equation

(aI + bS)u = f  (3)

by replacing (3) by the linear system

a(z_{j})u(z_{j}) + b(z_{j})(Su^{(n)})(z_{j}) = f(z_{j}), j = −n, …, n,  (4)

where z_{j} := exp(2πij/(2n + 1)). Introducing the orthogonal projection P_{n} from L^{2}(𝕋) onto Π_{n} as well as the interpolation projection L_{n} which associates with every Riemann integrable function f the function L_{n}f in Π_{n} such that f(z_{j}) = (L_{n}f)(z_{j}) for j = −n, …, n, one can rewrite (4) as

L_{n}(aI + bS)P_{n}u^{(n)} = L_{n}f.

Thus, what is crucial is the stability of the sequence (L_{n}(aI + bS)P_{n}). Define operators R_{n} : L^{2}(𝕋) → Π_{n} by

∑ c_{k}z^{k} ↦ c_{−1}z^{−n} + … + c_{−n}z^{−1} + c_{0}z^{n} + … c^{n}z^{0},

and let A stand for the smallest closed subalgebra of the algebra F which contains all sequences (L_{n}(aI +bS)P_{n}) with piecewise continuous functions a and b as well as all sequences (P_{n}KP_{n} + R_{n}LR_{n} + G_{n}) with K and L compact and with (G_{n}) tending to zero in the norm. Actually, the algebra F consists of sequences (A_{n}) of (2n + 1) × (2n + 1) matrices A_{n}, but this difference to the above definition of F doesn’t matter. The algebra A is a C^{∗}-algebra, and there are two natural ∗-homomorphisms W and W̃ from A into L(L^{2}(𝕋)):

W : (A_{n}) ↦ s-lim A_{n}P_{n} and W̃ : (A_{n}) ↦ s-lim R_{n}A_{n}R_{n}.

In particular, W (L_{n}(aI + bS)P_{n}) = aI + bS and W̃ (L_{n}(aI + bS)P_{n}) = ãI + b̃S with ã(t) := a(1/t). The following stability criterion is derived in [6, 10].

Theorem 1 A sequence (A_{n}) ∈ A is stable if and only if both associated operators W (A_{n}) and W̃ (A_{n}) are invertible.

Similar results hold for other approximation methods. The common structure of the C^{∗}-algebras A ⊆ F associated with the approximation methods considered in the above cited papers can be summarized as follows.

1. There is a (possibly infinite) set T, and for every t ∈ T, there is a Hilbert space H^{t} and a sequence (E_{n}^{t}) of partial isometries E_{n}^{t} : H^{t} → H such that - the initial projections (E_{n}^{t})^{∗}E_{n}^{t} converge strongly to the identity on H^{t},

- the range projections E_{n}^{t}(E_{n}^{t})^{∗} coincide with P_{n}, and

- the separation condition holds:

(E_{n}^{s})^{∗}E_{n}^{t} → 0 weakly for all s ≠ t.

2. For every t ∈ T and every (A_{n}) ∈ A, there exists the strong limit

s-lim (E_{n}^{t})^{∗}A_{n}E_{n}^{t} =: W^{t}(A_{n}).

3. A is unital and contains all sequences (E_{n}^{t}K(E_{n}^{t})^{∗}) with K compact on H^{t} as well as all sequences (G_{n}) ∈ F tending to zero in the norm. The closed linear span of all of these sequences forms a closed ideal J of A.

4. A sequence (A_{n}) ∈ A is stable if and only if all associated operators W^{t}(A_{n}) are invertible on H^{t}.

An algebra A which satisfies these axioms will be called a standard algebra in what follows. The algebra of the polynomial collocation method for singular integral operators is a standard algebra: choose T = {0, 1}, and set E_{n}^{0} = P_{n} and E_{n}^{1} = R_{n}.

Fredholm theory in standard algebras. With every sequence (A_{n}) in a standard algebra A, we associate the function t ↦ W^{t}(A_{n}) which is defined on T and takes a value in L(H^{t}) at t ∈ T. A remarkable consequence of the C^{∗}-property of A and of axiom 4 of a standard algebra is that the mapping

(A_{n}) + G ↦ (t ↦ W^{t}(A_{n}))

is an isometry from A/G into the algebra of all operator-valued functions on T:

∥(A_{n}) + G∥ = sup_{t∈T} ∥W^{t}(A_{n})∥.

This isometry has been employed in [12] in order to establish a Fredholm theory for approximation sequences in standard algebras. A sequence (A_{n}) ∈ A is Fredholm in this sense if every operator W^{t}(A_{n}) is a Fredholm operator. It turns out that, for every Fredholm sequence, the quantities

α(A_{n}) := ∑_{t∈T} dim Ker W^{t}(A_{n}) and β(A_{n}) := ∑_{t∈T} dim Coker W^{t}(A_{n}) (5)

are finite; thus we are suggested to define the index of a Fredholm sequence by

ind (A_{n}) := α(A_{n}) − β(A_{n}).

The so-defined Fredholm sequences and the functionals α, β and ind obviously satisfy all the properties which are known from common Fredholm theory for operators: stability under small perturbations, stability under perturbations belonging to the ideal J (which serves as a substitute for the ideal of the compact operators), symmetry, and multiplicativity. Moreover, if (A_{n}) is a Fredholm sequence and if both numbers α(A_{n}) and β(A_{n}) are zero, then the sequence (A_{n}) is stable (which is a consequence of the 4th axiom of a standard algebra).

Furthermore, in [12] there is derived a characterization of the α-number of a Fredholm sequence in terms of the asymptotic behaviour of the singular values of A_{n}. Let 0 ≤ λ_{1}^{(n)} ≤ … ≤ λ_{n}^{(n)} denote the eigenvalues of A_{n}^{∗}A_{n}. If (A_{n}) ∈ A is a Fredholm sequence, and if α(A_{n}) = k, then

lim λ_{k}^{(n)} = 0 but lim inf λ_{k+1}^{(n)} > 0.  (6)

Contents of this paper. The theory of Fredholm sequences as sketched above has interesting consequences and applications (see [12, 13] for a first discussion of applications to the regularization of ill-posed approximation sequences and to the asymptotic behaviour of the singular values of Cauchy-Toeplitz matrices), but it is still unsatisfactory. The main point is that, so far, Fredholmness is defined only for sequences in a standard algebra. Thus, at least formally, the Fredholmness of a sequence (A_{n}) depends on the algebra as an element of which (A_{n}) is regarded. Of course, the characterization (6) reveals that, actually, the quantities α(A_{n}) and β(A_{n}) do not depend on the embedding of (A_{n}) into a standard algebra. But, for example, the identity ind (A_{n}B_{n}) = ind (A_{n}) + ind (B_{n}) can only be guaranteed if both (A_{n}) and (B_{n}) are elements of one and the same standard algebra.

The goal of the present paper is to propose a general Fredholm theory which principally applies to every approximation sequence (A_{n}) ∈ F, and which reduces to the above sketched theory in case of sequences in a standard algebra. In particular, the identities (5) (which are no longer definitions but consequences of the theory) will be generalized to a much larger class of algebras which includes standard algebras. Moreover, it will be pointed out how the Fredholm theory of approximation sequences is related to the theory of Fredholm elements in Banach and C^{∗}-algebras as described, e.g., in [1]. And finally, a few new insights into the structure of algebras of approximation sequences (i.e. of subalgebras of F) will be derived.

2 Centrally compact and Fredholm sequences

Compact elements in C^{∗}-algebras. Let B be a C^{∗}-algebra. An element k ∈ B is of rank one if, for every b ∈ B, there is a complex number μ(b) such that kbk = μ(b)k. An element of B is of finite rank if it is the sum of a finite number of elements of rank one, and it is compact if it lies in the closure of the set of all finite rank elements. We denote the set of all compact elements in B by G(B). It is easy to check that both the elements of finite rank and the compact elements form two-sided ideals in B. In case B = L(H), an element b ∈ B is of rank one, of finite rank, or compact if and only if the operator b has range dimension less than or equal to one, finite range dimension, or is compact, respectively.

Proposition 1 Let A be a C^{∗}-subalgebra of F which contains the ideal G. Then G(A) = G.

Proof. Let (A_{n}) ≠ 0 be a rank one element of F. Then A_{k} ≠ 0 for a certain k. Let A_{k}^{+} denote the Moore-Penrose inverse of A_{k}, and consider the sequence

B := (0, …, 0, A_{k}^{+}, 0, …) ∈ G

with the A_{k}^{+} standing at the kth position. By assumption, there is a μ(B) ∈ ℂ such that

(A_{n})B(A_{n}) = μ(B)(A_{n}),

whence μ(B) = 1 and

A_{1} = … = A_{k−1} = A_{k+1} = A_{k+2} = … = 0.

Thus, every rank one sequence in F is necessarily of the form

(0, …, 0, A_{k}, 0, …)  (7)

with some A_{k} ∈ ℂ^{k×k}. Further, A_{k} must be a rank one element in ℂ^{k×k}, that is, it is zero or has one-dimensional range. It is clear that, conversely, all sequences (7) with dim Im A_{k} ≤ 1 are elements of rank one in F. Now the assertion follows immediately from the definitions.

Centrally compact elements. One might call a sequence (A_{n}) ∈ F Fredholm if it is invertible modulo the ideal G(F) = G. This indeed yields a reasonable Fredholm theory (see the following section), but it doesn’t give the desired notion of Fredholmness, since Fredholmness of a sequence in this sense simply means stability of that sequence. Here is a modified notion of compactness which fits exactly to our purposes. Recall that the center of an algebra is the set of all elements which commute with every element of the algebra.

Definition 1 Let B be a unital C^{∗}-algebra. An element k ∈ B is of central rank one if, for every b ∈ B, there is an element μ(b) belonging to the center of B, such that kbk = μ(b)k. An element of B is of finite central rank if it is the sum of a finite number of elements of central rank one, and it is centrally compact if it lies in the closure of the set of all elements of finite central rank.

We denote the set of all centrally compact elements in B by J(B). It is again easy to check that both the elements of finite central rank and the centrally compact elements form two-sided ideals in B.

In case B = L(H), the rank one, finite rank, and compact elements coincide with their central analogues, since the center of L(H) consists of the scalar multiples of the identity operator only. On the other hand, the center of the algebra F coincides with l^{∞} (where the number sequence (a_{n}) ∈ l^{∞} is identified with the matrix sequence (a_{n}I_{n}); see Lemma 1 below). Hence, the ideal J(F) should be much larger than the ideal G(F) = G of the zero sequences.

Proposition 2 A sequence (A_{n}) ∈ F is centrally compact if and only if, for every ε > 0, there is a sequence (K_{n}) ∈ F such that

sup_{n} ∥A_{n} − K_{n}∥ < ε and sup_{n} dim Im K_{n} < ∞.

Proof. If (A_{n}) is of central rank one in F, then every matrix A_{n} is of rank one in ℂ^{n×n}, hence dim Im A_{n} ≤ 1.

Conversely, let (A_{n}) ∈ F be a sequence of matrices with dim Im A_{n} ≤ 1 for every n, and let (B_{n}) ∈ F arbitrarily. Then there are numbers μ_{n} such that

A_{n}B_{n}A_{n} = μ_{n}A_{n}.  (8)

The numbers μ_{n} are uniquely determined if A_{n} ≠ 0; in case A_{n} = 0 we choose μ_{n} = 0. The so-defined sequence (μ_{n}) is bounded. Indeed, (8) implies |μ_{n}| ∥A_{n}∥ ≤ ∥A_{n}∥^{2}∥B_{n}∥ for every n whence

|μ_{n}| ≤ ∥A_{n}∥ ∥B_{n}∥ for every n with A_{n} ≠ 0.

This observation identifies the elements of central rank one. It is clear now that the elements of finite central rank are just the sequences (A_{n}) ∈ F with

sup dim Im A_{n} < ∞

which yields the assertion.

Observe that J(F) is a proper ideal of F. Indeed, suppose for contrary, that there is a sequence (K_{n}) ∈ F such that

sup ∥I_{n} − K_{n}∥ < 1/2 and sup dim Im K_{n} < ∞.

Then every matrix K_{n} is invertible, hence, dim Im K_{n} = n, which contradicts the second condition of the choice of (K_{n}).

Fredholm sequences. Based on the ideal of the centrally compact sequences in F one can introduce an appropriate class of Fredholm sequences.

Definition 2 A sequence (A_{n}) ∈ F is a Fredholm sequence if it is invertible modulo the ideal J(F) of the centrally compact sequences.

The following properties of Fredholm sequences are obvious.

– Every stable sequence is Fredholm.

– The adjoint of a Fredholm sequence is Fredholm.

– The product of Fredholm sequences is Fredholm.

– If (A_{n}) is Fredholm and (K_{n}) ∈ J(F), then (A_{n} + K_{n}) is Fredholm.

– The set of the Fredholm sequences is open in F. For another characterization of Fredholm sequences, let 0 ≤ λ_{1}^{(n)} ≤ … ≤ λ_{n}^{(n)} denote the eigenvalues of A_{n}^{∗}A_{n} and write σ_{i}^{(n)} := (λ_{i}^{(n)})^{1/2} ≥ 0 for the singular values of A_{n}.

Theorem 2 Each of the following conditions is equivalent to the Fredholmness of a sequence (A_{n}) ∈ F:

(a) There is a sequence (B_{n}) ∈ F and a sequence (J_{n}) ∈ J(F) of central finite rank such that

B_{n}A_{n}^{∗}A_{n} = I_{n} + J_{n}.  (9)

(b) There is a k such that

lim inf_{n→∞} σ_{k+1}^{(n)} > 0.  (10)

Proof. Let (A_{n}) be a Fredholm sequence. Then (A_{n}^{∗}A_{n}) is a Fredholm sequence and, by definition, there are sequences (B_{n}) ∈ F and (J_{n}) ∈ J(F) such that

(B_{n})(A_{n}^{∗}A_{n}) = (I_{n}) + (J_{n}).

One can assume that sup dim Im J_{n} < ∞. Indeed, by Proposition 2, there exists a sequence (K_{n}) ∈ F with ∥(J_{n})−(K_{n})∥ < 1/2 and sup dim Im K_{n} < ∞. Writing (9) as

(B_{n})(A_{n}^{∗}A_{n}) = (I_{n}) + (J_{n} − K_{n}) + (K_{n})

and taking into account the invertibility of (I_{n}) + (J_{n} − K_{n}) in F, one gets

(I_{n} + J_{n} − K_{n})^{−1}(B_{n})(A_{n}^{∗}A_{n}) = (I_{n}) + (I_{n} + J_{n} − K_{n})^{−1}(K_{n})

with

dim Im (I_{n} + J_{n} − K_{n})^{−1}K_{n} ≤ dim Im K_{n}.

Denoting (I_{n} +J_{n} −K_{n})^{−1}(B_{n}) and (I_{n} +J_{n} −K_{n})^{−1}(K_{n}) by (B_{n}) and (J_{n}) again, we arrive at (9).

Let now the sequences (B_{n}) and (J_{n}) be as in (9) and let

A_{n}^{∗}A_{n} = U_{n}^{∗}Λ_{n}U_{n} with Λ := diag (λ_{1}^{(n)}, …, λ_{n}^{(n)})

and with unitary matrices U_{n} refer to the diagonalization of A_{n}^{∗}A_{n}. After multiplication by U_{n} and U_{n}^{∗}, the identity (9) becomes

(U_{n}B_{n}U_{n}^{∗})(Λ_{n}) = (I_{n}) + (U_{n}J_{n}U_{n}^{∗}).

Abbreviating C_{n} := U_{n}B_{n}U_{n}^{∗} and F_{n} := U_{n}J_{n}U_{n}^{∗} we get

C_{n}Λ_{n} = C_{n}diag (λ_{1}^{(n)}, …, λ_{n}^{(n)}) = I_{n} + F_{n} for all n  (11)

where still sup dim Im F_{n} < ∞. Set

k := lim sup_{n→∞} dim Im F_{n}.

We claim that lim inf λ_{k+1}^{(n)} > 0. Assume this is wrong. Then there is an infinite subsequence (n_{l})_{l≥1} of ℕ such that lim_{l→∞} λ_{k+1}^{(n_{l})} = 0. Multiplying (11) from both sides by P_{k+1} = diag (1, …, 1, 0, …, 0) (with the 1 occurring k + 1 times) we get

P_{k+1}C_{n_{l}}Λ_{n_{l}}P_{k+1} = P_{k+1} + P_{k+1}F_{n_{l}}P_{k+1},

and, since

∥Λ_{n_{l}}P_{k+1}∥ = ∥diag (λ_{1}^{(n_{l})}, …, λ_{k+1}^{(n_{l})}, 0, …, 0)∥ = λ_{k+1}^{(n_{l})} → 0,

lim_{l→∞} ∥P_{k+1} + P_{k+1}F_{n_{l}}P_{k+1}∥ = 0.

Thus, the matrices P_{k+1}F_{n_{l}}P_{k+1} ∈ ℂ^{(k+1)×(k+1)} are invertible for all sufficiently large n_{l}. This is impossible because of

dim Im P_{k+1}F_{n_{l}}P_{k+1} ≤ dim Im F_{n_{l}} ≤ k < k + 1

which proves the claim (b).

Finally, for the proof that (b) implies the Fredholmness of (A_{n}), let k be a number such that

lim inf σ_{k+1}^{(n)} > 0,

and let A_{n} = U_{n}Σ_{n}V_{n}^{∗} with Σ_{n} = diag (σ_{1}^{(n)}, …, σ_{n}^{(n)}) and with unitary matrices U_{n} and V_{n} refer to the singular value decomposition of A_{n}. The choice of k guarantees that the sequence (Σ_{n} + P_{k})_{n≥1} is stable. Then (A_{n} + U_{n}P_{k}V_{n}^{∗})_{n≥1} is a stable sequence, too. Thus, there are sequences (C_{n}) ∈ F and (G_{n}) ∈ G such that

(C_{n})(A_{n} + U_{n}P_{k}V_{n}^{∗}) = (I_{n}) + (G_{n})

or, equivalently,

(C_{n})(A_{n}) = (I_{n}) + (G_{n}) − (C_{n}U_{n}P_{k}V_{n}^{∗}).

The sequence (C_{n}U_{n}P_{k}V_{n}^{∗}) is finite central rank, hence, (G_{n}) − (C_{n}U_{n}P_{k}V_{n}^{∗}) is a centrally compact sequence. Thus, (A_{n}) is invertible modulo J(F) from the left hand side, and its invertibility from the right hand side follows analogously.

The preceding theorem suggests to introduce the α-number of a Fredholm sequence (A_{n}) (corresponding to the kernel dimension of a Fredholm operator) as the smallest number k for which (10) is true. Equivalently, α(A_{n}) is the smallest number for which there exists a sequence (B_{n}) ∈ F as well as a sequence (J_{n}) of finite central rank such that B_{n}A_{n}^{∗}A_{n} = I_{n} + J_{n} and lim sup_{n→∞} dim Im J_{n} = α(A_{n}). The index of a Fredholm sequence is the quantity

ind (A_{n}) := α(A_{n}) − α(A_{n}^{∗}).

Observe that, in the case at hand, this index is always zero. This is a consequence of the fact that the entries of the sequences under consideration are finite-dimensional operators and, hence, the matrices A_{n}^{∗}A_{n} and A_{n}A_{n}^{∗} have the same eigenvalues even with respect to their multiplicity. So the most interesting quantity associated with a Fredholm sequence of matrices seems to be its α-number. On the other hand, the vanishing of the index of (A_{n}) also has remarkable consequences, mainly due to the identities (5), as it will be pointed out in Section 5.

Let (A_{n}) be a Fredholm sequence and k := α(A_{n}). Is there an analogue of the splitting property (6) which holds for Fredholm sequences in standard algebras? The following simple example says that the answer is NO.

Example. Let (a_{n}) be an enumeration of the rational numbers in [0, 1], and set

A_{n} := P_{n}(a_{n}P_{1} + (I − P_{1}))P_{n} = diag (a_{n}, 1, …, 1).

Since

(P_{n})(A_{n}) = (a_{n})(P_{1}) + (P_{n}(I − P_{1})P_{n}) = (P_{n}) − (1 − a_{n})(P_{1}),

and since (1 − a_{n})(P_{1}) is a sequence of central rank one, the sequence (A_{n}) is Fredholm, but the smallest singular values σ_{1}^{(n)} of the matrices A_{n} lie dense in [0, 1].

Thus, one cannot expect that lim_{n→∞} σ_{k}^{(n)} = 0 if (A_{n}) is Fredholm sequence with k = α(A_{n}), but one obviously has lim inf σ_{k}^{(n)} = 0. Hence, every Fredholm sequence in F possesses an infinite subsequence which owns the splitting property (6).

Finally, let us agree upon the following. The phrase ’Fredholm sequence’ is reserved for sequences in F which are invertible modulo the ideal J(F). Occassionally, we also will have to deal with sequences or elements which are invertible modulo other ideals J of compact or centrally compact sequences or elements. To these kind of Fredholmness we will refer as J-Fredholmness.

3 Fredholmness modulo compact elements

In this section, a brief sketch of the Fredholm theory in a C^{∗}-algebra A modulo the ideal G(A) is given. Some of these results are well known (see [1]); they are recalled here for the reader’s convenience with their (as a rule, short) proofs. As mentioned before, a direct application of this Fredholm theory to the algebra F does not yield anything of interest. But, as will be pointed out in the forthcoming section, applying this Fredholm theory in case of a standard algebra A ⊆ F twice (namely in the algebra A/G(A) modulo the ideal G(A/G(A))), one will exactly obtain the Fredholm theory described in the introduction.

Ideals generated by elements of rank one. In what follows, H is a complex Hilbert space, L(H) the C^{∗}-algebra of the bounded linear operators on H, and K(H) the ideal of the compact linear operators on H. We start with a result on irreducible representations of the ideal J(A).

Theorem 3 Let A be a unital C^{∗}-algebra and π : A → L(H) an irreducible representation of A. Then π(J(A)) ⊆ K(H).

Proof. We will prove that, if k ∈ J(A) is of central rank one, then π(k) is an operator with range dimension at most one. This clearly implies the assertion of the theorem.

If π(k) = 0, then nothing is to prove. So let π(k) ≠ 0. For every a ∈ A, there exists an element μ in the center of A such that kak = μk. Then π(μ) is in the center of π(A), and the identity

π(k)π(a)π(k) = π(μ)π(k)  (12)

shows that π(k) is a central rank one element of π(A). Since π(μ) is in the center of π(A), the operator π(μ) is a scalar multiple of the identity operator due to the irreducibility of π (Schur’s lemma; see [7], Section 5.4). Hence, the π(μ) in (12) can be chosen as a complex number, and π(k) is a (common) rank one element of π(A).

Let now x, x̃ be vectors in Im π(k) with x ≠ 0, and choose vectors y, ỹ ∈ H such that x = π(k)y and x̃ = π(k)ỹ. Again due to the irreducibility, π(A)x = H. In particular, there is an a ∈ A such that π(a)π(k)y = π(a)x = ỹ. Multiplying this identity by π(k) we get π(k)π(a)π(k)y = π(k)ỹ which, together with (12), yields π(μ)π(k)y = π(k)ỹ or π(μ)x = x̃. Since π(μ) is a number, this shows that Im π(k) = span {x}. In particular, π(k) has range dimension one.

Since K(H) has no proper closed ideals besides the zero ideal, this result implies that π(J(A)) is either {0} or K(H).

Now we turn over to the ideal G(A) of the compact elements. For every nonzero rank one element k of A, we denote by I(k) the smallest closed ideal of A which contains this element. From Theorem 3 one immediately gets

Corollary 1 Let A be a unital C^{∗}-algebra. Then, for every irreducible representation π : A → L(H) and every rank one element k of A,

π(I(k)) ⊆ K(H).

Actually, much more can be shown.

Theorem 4 Let A be a unital C^{∗}-algebra and k a non-zero rank one element of A. Then there exists an irreducible representation π : A → L(H) such that

π(I(k)) = K(H) and Ker(π|_{I(k)}) = {0}.

In particular, every ideal I(k) is^{∗}-isomorphic to the ideal of the compact operators on a Hilbert space. We split the proof into several steps. The first partial result says that every ideal I(k) is generated by a rank one projection.

Proposition 3 Let A be a unital C^{∗}-algebra and let k ∈ A \ {0} be a non-zero rank one element. Then there exists a rank one projection p ∈ A such that I(k) = I(p).

Proof. Let k be a non-zero rank one element of A, i.e. given A ∈ A there is a complex number μ(a) such that kak = μ(a)k. Then the elements k^{∗}, kk^{∗} and k^{∗}k are rank one and non-zero, too. Indeed, for every a ∈ A,

k^{∗}ak^{∗} = \overline{μ(a^{∗})}k^{∗}, k^{∗}kak^{∗}k = μ(ak^{∗})k^{∗}k and kk^{∗}akk^{∗} = μ(k^{∗}a)kk^{∗}.

So, these elements are rank one, and moreover

0 ≠ ∥k∥^{2} = ∥k^{∗}∥^{2} = ∥kk^{∗}∥ = ∥k^{∗}k∥.

In the next step we verify that I(k) = I(k^{∗}k). The inclusion I(k^{∗}k) ⊆ I(k) is obvious. For the reverse inclusion, consider

kk^{∗}k = μ(k^{∗})k.  (13)

Since k ≠ 0, the number μ(k^{∗}) is uniquely determined. Assume that μ(k^{∗}) = 0. Then kk^{∗}k = 0 and, consequently, ∥k∥^{4} = ∥k^{∗}k∥^{2} = ∥k^{∗}kk^{∗}k∥ = 0 which contradicts k ≠ 0. Thus, μ(k^{∗}) ≠ 0, which implies

k = μ(k^{∗})^{−1}kk^{∗}k ∈ I(k^{∗}k)

and hence, I(k) ⊆ I(k^{∗}k). From (13) we further conclude

k^{∗}kk^{∗}k = μ(k^{∗})k^{∗}k.  (14)

Both sides of (14) are non-negative elements of A and μ(k^{∗}) ≠ 0. Thus μ(k^{∗}) > 0, and taking norms in (14) gives

∥k^{∗}kk^{∗}k∥ = ∥k^{∗}k∥^{2} = μ(k^{∗})∥k^{∗}k∥.

Since ∥k^{∗}k∥ = ∥k∥^{2} ≠ 0, this implies μ(k^{∗}) = ∥k^{∗}k∥. Now it is evident from (14) that p := ∥k^{∗}k∥^{−1}k^{∗}k is a projection in A which is rank one and that I(p) = I(k^{∗}k) = I(k).

Proposition 4 Let A be a C^{∗}-algebra with unit element e and let p ∈ A \ {0} be a non-zero rank one projection. Then the identity

pap = τ(a)p

defines uniquely a pure state τ of A.

Proof. The uniqueness follows from p ≠ 0. For a = e one gets τ(e)p = pep = p^{2} = p, hence τ(e) = 1. Since ∥p∥ = 1, one moreover has

|τ(a)| = ∥τ(a)p∥ = ∥pap∥ ≤ ∥a∥

for every a ∈ A, whence ∥τ∥ = 1. It is also clear that the functional τ is linear, hence τ is a state of A. It remains to show that this state is pure. Let

L_{τ} := {a ∈ A : τ(a^{∗}a) = 0}

denote the left kernel of τ. The state τ is a pure if and only if

Ker τ = L_{τ} + L_{τ}^{∗}  (15)

([7], Theorem 10.2.8). Since the inclusion L_{τ} + L_{τ}^{∗} ⊆ Ker τ holds for every state, it remains to check the reverse inclusion. Let a ∈ Ker τ, i.e. pap = τ(a)p = 0. Since pap = 0,

a = pa + qa = paq + qa with q = e − p.

For b := paq one gets

τ(b^{∗}b)p = pb^{∗}bp = pqa^{∗}paqp = 0,

whence τ(b^{∗}b) = 0 and b ∈ L_{τ}. Analogously, for c := qa one finds

τ(cc^{∗})p = pcc^{∗}p = pqaa^{∗}qp = 0,

hence, τ(cc^{∗}) = 0 and c ∈ L_{τ}^{∗}. Consequently, a = b + c ∈ L_{τ} + L_{τ}^{∗}, and τ is a pure state.

Since τ(a^{∗}a)p = pa^{∗}ap = (ap)^{∗}(ap), it is τ(a^{∗}a) = 0 if and only if ap = 0. Thus,

L_{τ} = Aq = {aq : a ∈ A},

and the Hilbert space associated via the GNS construction with the pure state τ is H := A/L_{τ} = A/Aq with inner product

⟨a + Aq, b + Aq⟩ := τ(b^{∗}a)

(it is not necessary to take the completion since τ is pure, cp. [7], Theorem 10.2.3). The pureness of τ also guarantees that the representation

π : A → L(H), a ↦ (b + Aq ↦ ab + Aq)  (16)

is irreducible. The following proposition finishes the proof of Theorem 4.

Proposition 5 Let π as in (16). Then Ker (π|_{I(k)}) = {0}.

Proof. Let r ∈ I(k) = I(p) and π(r) = 0. Then, by (16),

rb + Aq = 0 resp. rb ∈ Aq resp. rbp = 0 for all b ∈ A.

This implies rbpc = 0 for all b, c ∈ A and, consequently, r ∑_{i} b_{i}pc_{i} = 0 for all b_{i}, c_{i} ∈ A. The elements ∑ b_{i}pc_{i} lie densely in I(p). Hence, rj = 0 for every j ∈ I(p). In particular, rr^{∗} = 0, i.e. r = 0.

Since K(H) has no proper closed subideals besides {0}, one has the following consequence of Theorem 4.

Corollary 2 Let k_{1}, k_{2} be non-zero rank one elements of the unital C^{∗}-algebra A. Then either I(k_{1}) = I(k_{2}) or I(k_{1}) ∩ I(k_{2}) = {0}.

Lifting theorems. The preceding results suggest to introduce an equivalence relation in the set of all non-zero rank one elements of a unital C^{∗}-algebra A by calling k_{1} and k_{2} equivalent if I(k_{1}) = I(k_{2}). Let T abbreviate the set of all equivalence classes and, given t ∈ T, choose a representative p_{t} of the coset t, abbreviate the ideal I(p_{t}) by I_{t}, and let π_{t} : A → L(H_{t}) stand for the associated irreducible representation (16). Thus, G(A) is generated by its minimal subideals I_{t} where t ∈ T. With these notations, the following so-called lifting theorem holds. The results of this subsection are taken from [11].

Theorem 5 (Lifting theorem, part 1.) Let A be a unital C^{∗}-algebra and T the set of the equivalence classes of the non-zero rank one elements of A. Then an element a ∈ A is invertible in A if and only if the operators π_{t}(a) are invertible in L(H_{t}) for every t ∈ T and if the coset a + G(A) is invertible in the quotient algebra A/G(A).

In other words: If a ∈ A is a G(A)-Fredholm element, then all operators π_{t}(a) are Fredholm, and the Fredholm element a is invertible if and only if all Fredholm operators π_{t}(a) are invertible.

Proof. If a is invertible, then coset a+G(A) and all operators π_{t}(a) are invertible. Conversely: If a+G(A) is invertible, then there are elements b ∈ A and k ∈ G(A) such that ba = e + k. Since G(A) is the smallest closed ideal which contains all ideals I_{t}, one finds an element j ∈ G(A) as well as finitely many elements j_{t_{i}} ∈ I_{t_{i}} such that j = j_{t_{1}} + … + j_{t_{m}} and ∥k − j∥ < 1/2. Multiplying the equation ba = e + k from the left hand side by (e + k − j)^{−1} and setting c := (e + k − j)^{−1}b and k_{t_{i}} := (e + k − j)^{−1}j_{t_{i}}, one arrives at

ca = e + k_{t_{1}} + … + k_{t_{m}} with k_{t_{i}} ∈ I_{t_{i}}.  (17)

Since I_{t} is isomorphic to π_{t}(I_{t}) = K(H_{t}), there are elements r_{t_{i}} ∈ I_{t_{i}} such that π_{t_{i}}(r_{t_{i}}) = π_{t_{i}}(k_{t_{i}})π_{t_{i}}(a)^{−1}. Set ĉ:= c − r_{t_{1}} − … − r_{t_{m}}. Then

ĉa = e + (k_{t_{1}} − r_{t_{1}}a) + … + (k_{t_{m}} − r_{t_{m}}a).

The elements k_{t_{i}} − r_{t_{i}}a belong to I_{t_{i}} and π_{t_{i}}(k_{t_{i}} − r_{t_{i}}a) = 0, hence k_{t_{i}} − r_{t_{i}}a = 0 and ĉa = e. The invertibility of a from the right hand side can be checked analogously.

Together with the following separation property, the lifting theorem can be essentially completed.

Proposition 6 Let the notations be as before. Then

(a) Let t_{1}, …, t_{m} ∈ T and t_{i} ≠ t_{j} for i ≠ j. Then (I_{t_{1}} + … + I_{t_{m−1}}) ∩ I_{t_{m}} = {0}.

(b) Let s, t ∈ T with s ≠ t. Then π_{s}(I_{t}) = {0}.

Proof. (a) Clearly, (I_{t_{1}} + … + I_{t_{m−1}}) ∩ I_{t_{m}} is an ideal in I_{t_{m}}. Since I_{t_{m}} is isomorphic to K(H_{t_{m}}), its only closed subideals are {0} and I_{t_{m}} itself. Thus, if the assertion would be wrong then, necessarily, I_{t_{m}} ⊆ I_{t_{1}} +…+I_{t_{m−1}}. In this case, let I_{t_{m}} = I(p) with a rank one projection p, and choose elements k_{t_{i}} ∈ I_{t_{i}} such that p = k_{t_{1}} + … + k_{t_{m−1}}. Multiplying this identity by p from both sides yields p = pk_{t_{1}}p + … + pk_{t_{m−1}}p. If pk_{t_{i}}p = 0 for every i, then p = 0 which is impossible. So pk_{t_{i}}p ≠ 0 for some i. Since p is rank one, there is a complex number μ such that μp = pk_{t_{i}}p ∈ I_{t_{i}}. Hence, p ∈ I_{t_{i}} which contradicts I_{t_{i}} ∩ I_{t_{m}} = {0}.

(b) Let r ∈ π_{s}(I_{t}), i.e. r = π_{s}(k_{t}) for a k_{t} ∈ I_{t}. By Theorem 3, r belongs to K(H_{s}), and by Theorem 4, there exists a k_{s} ∈ I_{s} such that π(k_{s}) = r^{∗}. Then π_{s}(k_{s}k_{t}) = r^{∗}r. On the other hand, since I_{s} ∩ I_{t} = {0}, one has k_{s}k_{t} = 0 which implies π_{s}(k_{s}k_{t}) = 0. Thus, r = 0.

Theorem 6 (Lifting theorem, part 2.) Let the situation be as in Theorem 5, and let a ∈ A be a G(A)-Fredholm element. Then all operators π_{t}(a) are Fredholm, and there are only finitely many t ∈ T for which π_{t}(a) is not invertible.

Proof. Apply the representation π_{t} to both sides of identity (17) to obtain the invertibility of π_{t}(a) from the left hand side for all t ∉ {t_{1}, …, t_{m}}.

The rank of an element. Let k ∈ G(A) be an non-zero element of finite rank. We say that k has rank r, if k is the sum of r elements of rank one, but not a sum of r −1 rank one elements. The rank of k will be denoted by rank k. Further define rank 0 = 0.

Proposition 7 Let k ∈ G(A) be of finite rank. Then, for every t ∈ T, there exist finite rank elements k_{t} ∈ I_{t} with k_{t} = 0 for all but a finite number of t such that k = ∑_{t∈T} k_{t}. The k_{t} are uniquely determined, and

rank k = ∑_{t∈T} rank k_{t}.

Proof. Let k ∈ G(A) be the sum of the rank one elements k_{1}, …, k_{r}. Every k_{j} belongs to exactly one of the ideals I_{t} (namely to I(k_{j})). So one gets a decomposition of k as a sum ∑_{t∈T} k_{t} with only finitely many non-vanishing elements k_{t} ∈ I_{t} of finite rank.

The uniqueness of this decomposition can be checked as follows: Let h_{1} + … + h_{m} = 0 for certain elements h_{i} ∈ I_{t_{i}} with t_{i} ≠ t_{j} for i ≠ j. Then h_{t_{1}} = −h_{t_{2}} − … − h_{t_{m}}, i.e. h_{t_{1}} ∈ I_{t_{1}} ∩ (I_{t_{2}} + … + I_{t_{m}}). By Proposition 6 (a), h_{t_{1}} = 0.

It remains to show the rank identity. Let k = ∑ k_{t} with k_{t} ∈ I_{t} and let rank k_{t} = r_{t}. Then every k_{t} is the sum of r_{t} rank one elements, hence, k is the sum of ∑ r_{t} rank one elements. Consequently,

rank k ≤ ∑ r_{t} = ∑ rank k_{t}.

Conversely, let k be the sum of r = rank k elements k_{1}, …, k_{r} of rank one. For every t ∈ T and every rank one element h, define n_{t}(h) to be 1 if h ∈ I_{t} and set n_{t}(h) = 0 if h ∉ I_{t}. Further let k_{t} := ∑_{i=1}^{r} n_{t}(k_{i})k_{i}. Then every k_{t} is the sum of r_{t} rank one elements where r_{t} is the number of the k_{i} which lie in I_{t}. Since every k_{i} belongs to exactly one of the ideals I_{t}, one has ∑ r_{t} = r. Thus,

∑ rank k_{t} ≤ ∑ r_{t} = r = rank k

which verifies the rank identity.

We proceed with relations between the rank of an element and the range dimension of its image under irreducible representations.

Proposition 8 Let k ∈ I_{t} be of finite rank. Then rank k = dim Im π_{t}(k).

Proof. Recall from the proof of Theorem 3 that the irreducible representation π_{t} of A maps elements of rank one onto operators with range dimension at most one. Hence, if k is the sum of r rank one elements, then π_{t}(k) is the sum of r operators of rank one, whence

dim Im π_{t}(k) ≤ r = rank k.

Conversely, suppose π_{t}(k) is a compact operator with range dimension r. Choose an orthonormal basis e_{1}, …, e_{r} in Im π_{t}(k), let P_{i} stand for the orthogonal projection from H_{t} onto ℂe_{i}, and let p_{i} denote the (uniquely determined) element in I_{t} such that π(p_{i}) = P_{i}. Since ∑ P_{i} is the orthogonal projection from H_{t} onto the range of π_{t}(k), one has

π_{t}(k) = ∑_{i=1}^{r} P_{i}π_{t}(k) = ∑_{i=1}^{r} π_{t}(p_{i}k) = π_{t}(∑_{i=1}^{r} p_{i}k).

Due to Proposition 5, this implies k = ∑ p_{i}k. One easily checks that every p_{i}k is a rank one element. Thus, k is the sum of r rank one elements whence the estimate rank k ≤ dim Im π_{t}(k).

Fredholmness modulo G(A). Now we will have a closer look at the Fredholm theory associated with the ideal G(A) of the compact elements. The remainder of this section is not needed in what follows. Recall that a ∈ A is G(A)-Fredholm if the coset a + G(A) is invertible in the quotient algebra A/G(A). If a is G(A)- Fredholm then a^{∗}a and aa^{∗} are G(A)-Fredholm, too, and there exist elements b, c ∈ A as well as elements k_{1}, k_{2} of finite rank such that

ba^{∗}a = e + k_{1} and aa^{∗}c = e + k_{2}.

Let α(a) stand for the smallest non-negative integer which owns the following property: there are a finite rank element k_{1} with rank k_{1} = α(a) and an element b ∈ A such that ba^{∗}a = e + k_{1}. Analogously, β(a) is defined as the smallest possible rank of k_{2}. Finally, define the index of a by ind a := α(a) − β(a). In case A = L(H), one has G(L(H)) = K(H) and

α(A) = dim Ker A and β(A) = dim Coker A

for every Fredholm operator. Here is a generalization of these results to arbitrary G(A)-Fredholm elements.

Theorem 7 Let a ∈ A be a Fredholm element modulo G(A). Then

α(a) = ∑_{t∈T} dim Ker π_{t}(a) and β(a) = ∑_{t∈T} dim Coker π_{t}(a).

Observe that the occurring sums are actually finite thanks to Theorem 6.

Proof. We will verify the first assertion only. Let a ∈ A be a Fredholm element, and let b ∈ A and k ∈ G(A) be elements such that

ba^{∗}a = e + k and rank k = α(a).

Write k as ∑ k_{t} with k_{t} ∈ I_{t} (which can be done uniquely). By Proposition 7,

ba^{∗}a = e + ∑ k_{t} and α(a) = ∑ rank k_{t}.

The separation property Proposition 6 (b) implies that

π_{t}(b)π_{t}(a^{∗})π_{t}(a) = e_{t} + π_{t}(k_{t})

with e_{t} referring to the identity operator on H_{t}. Hence,

dim Ker π_{t}(a) = α(π_{t}(a)) ≤ rank π_{t}(k_{t}),

whence

∑ dim Ker π_{t}(a) ≤ ∑ rank π_{t}(k_{t}) = ∑ rank k_{t} = rank k = α(a).

For the reverse inequality, let P_{t} stand for the orthogonal projection from H_{t} onto Ker π_{t}(a), and write p_{t} for the (uniquely determined) element in I_{t} such that π_{t}(p_{t}) = P_{t}. Consider the element â := a^{∗}a+∑ p_{t} (again this sum contains only a finite number of non-zero terms due to Theorem 6). Since a is a Fredholm element modulo G(A), also â is a G(A)-Fredholm element. Moreover, the operators

π_{t}(â) = π_{t}(a^{∗})π_{t}(a) + P_{t}

are invertible for every t ∈ T. The lifting theorem implies that â is an invertible element of A. In particular, there exists a b ∈ A such that bâ = e resp.

ba^{∗}a = e − b ∑ p_{t} = e − ∑ bp_{t}.

Hence,

α(a) ≤ rank ∑ bp_{t} ≤ ∑ rank bp_{t} ≤ ∑ rank p_{t}
= ∑ rank P_{t} = ∑ dim Im P_{t} = ∑ dim Ker π_{t}(a),

yielding finally the desired kernel dimension identity.

Let us mention a consequence of the kernel dimension identity. Clearly, the Fredholm element a determines its α-number uniquely, but the finite rank element k in ba^{∗}a = e + k is not determined uniquely by a (it depends on the choice of b also). Thus, it is a priori not evident whether a uniquely determines the ranks of the elements k_{t} (only their sum rank k is determined by a). The preceding theorem states that a also determines the ’local’ ranks rank k_{t} uniquely since rank k_{t} = dim Ker π_{t}(a).

Corollary 3 Let a ∈ A be a Fredholm element modulo G(A). Then

ind a = ∑_{t∈T} ind π_{t}(a).

With this corollary it becomes obvious that the introduced functionals α, β and ind for G(A)-Fredholm elements satisfy all the common properties one knows for the functionals dim Ker, dim Coker and ind in case of Fredholm operators.

4 Fredholm sequences in standard algebras

In this section we return to Fredholm sequences (i.e. sequences which are invertible modulo the ideal J(F) of the centrally compact sequences in F). The main goal is to single out a class of subalgebras A of F such that, for sequences belonging to A, their Fredholmness coincides with the invertibility modulo an ideal of compact elements as considered in the previous section. This makes the lifting theorem and its consequences (the dim Ker identity) available to the determination of the α-number of Fredholm sequences. Let us agree upon reserving the notation α(A_{n}) for the α-number of a Fredholm sequence (A_{n}) (i.e. modulo J(F)). If we want to consider α-numbers with respect to other ideals of compact or essentially compact sequences, we will mention this explicitely.

Algebras with center c. Let A be a unital C^{∗}-subalgebra of the algebra F, and throughout what follows suppose that the ideal G = G(F) belongs to A. Then G is a closed ideal of A, and it is easy to check that G(A) coincides with G.

Lemma 1 If G ⊆ A ⊆ F and A is unital, then the center of A is isomorphic to a subalgebra of l^{∞} which contains c.

Proof. Let I_{n} denote the n × n identity matrix, and let (a_{n}) ⊂ ℂ be a sequence in c with limit a. Then

(a_{n}I_{n}) = a(I_{n}) + ((a_{n} − a)I_{n}).

The sequence a(I_{n}) belongs to A since this algebra is unital, and the sequence ((a_{n} − a)I_{n}) tends to zero in the norm whence ((a_{n} − a)I_{n}) ∈ G ⊆ A. Clearly, (a_{n}I_{n}) belongs to the center of A.

Conversely, let the sequence (C_{n}) be in the center of A. Since G ⊆ A, every matrix C_{n} commutes with every other matrix in ℂ^{n×n}, hence, C_{n} = a_{n}I_{n} with a sequence (a_{n}) ∈ l^{∞}.

We say that the center of the algebra A ⊆ F is c if this center consists exactly of the sequences (c_{n}I_{n}) with (c_{n}) ∈ c. Here are two instances of algebras A with center c.

Suppose there is a separable infinite-dimensional Hilbert space H as well as a sequence (P_{n}) of orthogonal projections P_{n} from H onto an n-dimensional subspace of H such that (P_{n}) converges strongly to the identity operator on H. Assume further that all sequences (A_{n}) in A possess the following property: If the matrix A_{n} is identified with an operator on Im P_{n}, then the strong limit s-lim A_{n}P_{n} exists. If, in particular, A_{n} = c_{n}I_{n} with complex numbers c_{n}, then this strong convergence implies that (c_{n}) ∈ c.

Fractal algebras. Another class of algebras with center c is constituted by the fractal algebras. This class has been introduced and studied in [11, 9]. Here are the definition as well as an important property of fractal algebras as stated in [11, 9].

Given a strongly monotonically increasing sequence η : ℕ → ℕ, let F_{η} refer to the C^{∗}-algebra of all bounded sequences (A_{n}) with A_{n} ∈ ℂ^{η(n)×η(n)}, and write G_{η} for the ideal of all sequences (A_{n}) ∈ F_{η} which tend to zero in the norm. Further, let R_{η} stand for the restriction mapping R_{η} : F → F_{η}, (a_{n}) ↦ (a_{η(n)}). This mapping is a ^{∗}-homomorphism from F onto F_{η} which moreover maps G onto G_{η}. Given a C^{∗}-subalgebra A of F, let A_{η} denote the image of A under R_{η} which is a C^{∗}-algebra again.

Definition 3 Let A be a C^{∗}-subalgebra of the sequence algebra F.

(a) A ^{∗}-homomorphism W : A → B of A into a C^{∗}-algebra B is fractal if, for every strongly monotonically increasing sequence η, there is a ^{∗}-homomorphism W_{η} : A_{η} → B such that W = W_{η}R_{η}.

(b) The algebra A is fractal if the canonical homomorphism π : A → A/(A ∩ G) is fractal.

Thus, given a subsequence (a_{η(n)}) of a sequence (a_{n}) which belongs to a fractal algebra A, it is possible to reconstruct the original sequence (a_{n}) from this subsequence modulo sequences in A∩G. This assumption is very natural for sequences arising from discretization procedures. It is, for example, not hard to show that the algebra of the polynomial collocation method examined in the introduction is fractal. On the other hand, the algebra F of all bounded sequences fails to be fractal.

Theorem 8 A C^{∗}-subalgebra A of F is fractal if and only if the following implication holds for every element (a_{n}) ∈ A and every strongly monotonically increasing sequence η:

R_{η}(a_{n}) ∈ G_{η} ⇒ (a_{n}) ∈ A ∩ G.

This property also implies that the center of a fractal unital C^{∗}-subalgebra of F which contains the ideal G is c.

Fredholm inverse closed subalgebras. If A is a unital subalgebra of F which contains G and has center c, then every central rank one sequence in A is also a central rank one sequence in F, hence, J(A) ⊆ J(F).

Definition 4 The subalgebra A of F is called Fredholm inverse closed if

J(A) = A ∩ J(F).  (18)

Thus, if A is Fredholm inverse closed, and if (A_{n}) ∈ A is a Fredholm sequence (i.e. invertible modulo J(F)), then (A_{n}) is invertible modulo A ∩ J(F) (due to the common inverse closedness of (A+J(F))/J(F) ≅ A/(A∩J(F))) and, hence, (A_{n}) is invertible modulo J(A).

It will be pointed out that, for Fredholm sequences (A_{n}) which belong to a fractal and Fredholm inverse closed subalgebra A of F, their α-number can be determined by an analogue of the kernel dimension identity in Theorem 7.

Sequences of essential rank one. Let A be a unital C^{∗}-subalgebra of F which contains the ideal G and has center c. A central rank one sequence of A is said to be of essential rank one if it does not belong to the ideal G. For every essential rank one sequence (K_{n}), let J(K_{n}) refer to the smallest closed ideal of A which contains the sequence (K_{n}) and the ideal G.

Since A has center c, it is obvious that, for every essential rank one sequence (K_{n}) of A, the coset (K_{n}) + G is a rank one element of the quotient algebra A/G and that

J(K_{n})/G = I((K_{n}) + G)

(with the notation I as in the previous section). This shows that Fredholmness of a sequence (A_{n}) ∈ A is closely related to the common Fredholm theory (as described in the previous section) in the algebra A/G. In particular, if T refers to the set of all equivalence classes of rank one elements in A/G, then there exists a t ∈ T such that

J(K_{n})/G = I_{t}.  (19)

Let T_{ess} stand for the set of all t ∈ T for which there is an essential rank one sequence (K_{n}) ∈ A such that (19) holds. Further, write G_{ess}(A/G) for the smallest closed ideal of A/G which contains all ideals I_{t} with T ∈ T_{ess}. It is easy to check that the lifting theorems (Theorems 5 and 6) and their consequences (in particular, Theorem 7) remain valid if the set T and the invertibility modulo the ideal G(A/G) are replaced by the set T_{ess} and by invertibility modulo G_{ess}(A/G). In particular, the coset (A_{n}) + G is invertible in A/G if and only if the operators π_{t}((A_{n}) + G) are invertible for all t ∈ T_{ess} and if (A_{n}) + G is invertible modulo G_{ess}(A/G). Moreover, if the coset (A_{n}) + G is invertible modulo G_{ess}(A/G), then

α_{G_{ess}(A/G)}((A_{n}) + G) = ∑_{t∈T_{ess}} dim Ker π_{t}((A_{n}) + G).

For t ∈ T_{ess}, abbreviate the homomorphism

A → L(H_{t}), (A_{n}) ↦ π_{t}((A_{n}) + G)

by W_{t}. The homomorphism W_{t} is an irreducible representation of the algebra A ([4], Proposition 2.11.2) which maps the ideal G to zero. With these notations and identifications we have

J(A)/G = G_{ess}(A/G),

and the following version of the lifting theorem holds.

Theorem 9 Let A be a unital and Fredholm inverse closed C^{∗}-subalgebra of F which contains the ideal G and has center c. A sequence (A_{n}) ∈ A is stable if and only if (A_{n}) is a Fredholm sequence and if the operators W_{t}(A_{n}) are invertible for all t ∈ T_{ess}.

Fredholm sequences in fractal and Fredholm inverse closed algebras. Now we are going to formulate and prove the main result of this section: a relation between the α-number of a Fredholm sequence (A_{n}) and the kernel dimensions of the operators W_{t}(A_{n}). This result is a generalization of the identity (5) which served in [12] as a basis to define Fredholm sequences and their α-number. In particular, it is not needed for this generalization that the underlying algebra is standard (cp. the introduction; the point is that the fourth axiom of the definition of a standard algebra is not required, and also the explicit form of the homomorphisms W_{t} as strong limits is not assumed in what follows).

Theorem 10 Let A be a unital, fractal and Fredholm inverse closed C^{∗}-subalge-bra of F which contains the ideal G, and let (A_{n}) ∈ A be a Fredholm sequence. Then

α(A_{n}) = ∑_{t∈T_{ess}} dim Ker W_{t}(A_{n}).  (20)

The remainder of this section is devoted to the proof of this result. Let (A_{n}) ∈ A be a Fredholm sequence. Since A is Fredholm inverse closed, (A_{n}) is invertible modulo J(A). Then, by Theorem 3, the operators W_{t}(A_{n}) are Fredholm for every t ∈ T_{ess} and, by Theorem 6, there are only finitely many operators W_{t}(A_{n}) which are not invertible. Let P_{Ker W_{t}(A_{n})} denote the orthogonal projection from H_{t} onto the kernel of W_{t}(A_{n}) (only a finite number of these projections are not zero). Decompose each of these projections into a sum of dim Ker W_{t}(A_{n}) orthogonal projections of range dimension one:

P_{Ker W_{t}(A_{n})} = ∑_{i=1}^{dim Ker W_{t}(A_{n})} P_{i,t}

such that P_{i,t}P_{j,t} = P_{j,t}P_{i,t} = 0 whenever i ≠ j (for example, by choosing an orthonormal basis in Ker W_{t}(A_{n}) and by defining P_{i,t} as the orthogonal projection onto the ith element of this basis). Since π_{t} is an isomorphism between I_{t} and K(H_{t}), every projection P_{i,t} corresponds uniquely to a coset p_{i,t} ∈ I_{t}. Clearly, p_{i,t}p_{j,t} = p_{j,t}p_{i,t} = 0 if i ≠ j. Since I_{s} ∩ I_{t} = {0} for all s, t ∈ T with s ≠ t, one has moreover

p_{i,t}p_{j,s} = p_{j,s}p_{i,t} = 0 whenever (i, t) ≠ (j, s).  (21)

The next results say that projections in A/G can be lifted.

Proposition 9 Let p ∈ F/G be a projection. Then there exists a sequence of projections (P_{n}) ∈ F such that (P_{n}) + G = p.

A proof of this well-known result is in [12].

In particular, Proposition 9 guarantees the existence of sequences (P_{n}^{i,t}) ∈ F of projections such that (P_{n}^{i,t}) belongs to the coset p_{i,t}. Observe that (P_{n}^{i,t}) ∈ A since A contains G. Moreover, since (P_{n}^{i,t}) + G belongs to I_{t} and t is in T_{ess}, every sequence (P_{n}^{i,t}) belongs to an ideal of the form J(K_{n}) with an essential rank one sequence (K_{n}). In particular, (P_{n}^{i,t}) ∈ J(A).

We claim that the lifting of the projections p_{i,t} to the sequences (P_{n}^{i,t}) can be done in such a way that all projections P_{n}^{i,t} have one-dimensional range and that

P_{n}^{i,t} P_{n}^{j,s} = P_{n}^{j,s} P_{n}^{i,t} = 0 whenever (i, t) ≠ (j, s)  (22)

for all sufficiently large n.

Proposition 10 Let A be as in Theorem 10 and (P_{n}^{i,t}) the lifting of a rank one projection p_{i,t} ∈ A/G. Then the projections P_{n}^{i,t} can be chosen in such a way that their range dimension is one.

Proof. Let (K_{n}) ∈ A be an essential rank one sequence such that (P_{n}^{i,t}) ∈ J(K_{n}). Then the operator K := W_{t}(K_{n}) has rank one (see the proof of Theorem 3), and P_{i,t} = W_{t}(P_{n}^{i,t}) is a projection operator with one-dimensional range by construction of (P_{n}^{i,t}). Choose compact operators E, F ∈ L(H_{t}) such that EKF = P_{i,t}, and let (E_{n}) and (F_{n}) be pre-images of E and F under the mapping W_{t} in J(K_{n}), respectively. Then

E_{n}K_{n}F_{n} = P_{n}^{i,t} + G_{n}

with a sequence (G_{n}) tending to zero in the norm (recall that the mapping W_{t} is a homomorphism from J(K_{n}) onto K(H_{t}) with kernel G due to its definition and Proposition 5). Multiplying this equality from both sides by P_{n}^{i,t} yields

P_{n}^{i,t}E_{n}K_{n}F_{n}P_{n}^{i,t} = P_{n}^{i,t} + P_{n}^{i,t}G_{n}P_{n}^{i,t}.  (23)

Consider the operators occuring in (23) as acting on Im P_{n}^{i,t}. Then the right hand side of (23) is invertible for n large enough (say, for n such that ∥G_{n}∥ < 1/2), whereas the left hand side of (23) has range dimension at most one for every n. In case dim Im P_{n}^{i,t} ≥ 2, this is a contradiction.

Assume that dim Im P_{n}^{i,t} = 0 for infinitely many n. Then the fractality of A implies via Theorem 8 that (P_{n}^{i,t}) belongs to G which contradicts the definition of p_{i,t} ≠ 0. Hence, dim Im P_{n}^{i,t} ≤ 1 for all n, and this range dimension is zero for at most finitely many n. Modifying the sequence (P_{n}^{i,t}) by adding a sequence which tends to zero one can obviously reach that all projections P_{n}^{i,t} have range dimension one.

Proposition 11 Let A be as in Theorem 10 and (P_{n}^{i,t}) the lifting of a rank one projection p_{i,t} ∈ A/G such that dim Im P_{n}^{i,t} = 1 for all n. Then the sequences (P_{n}^{i,t}) can be modified by adding sequences in G in such a way that the modified sequences still consist of rank one projections and that the orthogonality condition (22) is satisfied for all sufficiently large n.

Proof. We proceed by induction. For one lifted sequence there is nothing to prove. Assume that already k of the sequences (P_{n}^{i,t}) are modified such that (22) holds. Let (P_{n}) abbreviate the sum of these k sequences, and let (Q_{n}) be a further sequence of rank one projections Q_{n}. The orthogonality condition (22) implies that the operators P_{n} are projections for n large enough.

Consider the operators (I_{n} − P_{n})Q_{n}(I_{n} − P_{n}) =: Q̂_{n} with I_{n} referring to the n × n identity matrix again, and let p and q denote the cosets (P_{n}) + G and (Q_{n}) + G, respectively. Since dim Im Q_{n} = 1, the Q̂_{n} are operators with range dimension at most one, and

Q̂_{n}^{2} = (I_{n} − P_{n})Q_{n}(I_{n} − P_{n})Q_{n}(I_{n} − P_{n}) = μ_{n}(I_{n} − P_{n})Q_{n}(I_{n} − P_{n}) = μ_{n}Q̂_{n} (24)

with an l^{∞}-sequence (μ_{n}). Since the operators Q̂_{n} are self-adjoint and nonnegative, the numbers μ_{n} can be assumed to be real and non-negative. Further, since pq = qp = 0, one has (e − p)q(e − p) = q and, consequently,

∥(I_{n} − P_{n})Q_{n}(I_{n} − P_{n}) − Q_{n}∥ → 0 resp. ∥(I_{n} − P_{n})Q_{n}(I_{n} − P_{n})∥ → 1

as n → ∞. Together with (24), this shows that lim μ_{n} = 1. Hence, μ_{n} ≠ 0 for sufficiently large n, and the operators

\frac{1}{μ_{n}} Q̂_{n} = \frac{1}{μ_{n}} (I_{n} − P_{n})Q_{n}(I_{n} − P_{n})

are projections with rank one which are orthogonal to P_{n}. Modifying a finite number of entries of the sequence (\frac{1}{μ_{n}} Q̂_{n}) one gets a sequence all entries of which are projections.

Now we can finish the proof of Theorem 10. By the preceding propositions, we can assume that all sequences (P_{n}^{i,t}) consist of projections with range dimension one and that (22) holds for all large n. Let (P_{n}) stand for the sum of all these sequences. The orthogonality (22) ensures that the operators P_{n} are projections and that

dim Im P_{n} = ∑_{t∈T_{ess}} dim Ker W_{t}(A_{n})

for all sufficiently large n (the term on the right hand side is just the number of the different sequences (P_{n}^{i,t})). Furthermore,

(A_{n})^{∗}(A_{n}) + (P_{n}) is a stable sequence, and (A_{n})^{∗}(A_{n})(P_{n}) ∈ G.  (25)

Indeed, the sequence (A_{n}) is Fredholm by assumption and the sequence (P_{n}) belongs to the ideal J(A) by construction. Thus, (A_{n})^{∗}(A_{n}) + (P_{n}) is a Fredholm sequence. Further, all operators

W_{t}((A_{n})^{∗}(A_{n}) + (P_{n})) = W_{t}(A_{n})^{∗}W_{t}(A_{n}) + P_{Ker W_{t}(A_{n})}

are invertible, and the lifting theorem (Theorem 9) implies stability of the sequence (A_{n})^{∗}(A_{n}) + (P_{n}). Similarly, the sequence (A_{n})^{∗}(A_{n})(P_{n}) belongs both to the ideal J(A) and to the kernels of all representations W_{t} with t ∈ T_{ess}. Again by the lifting theorem and due to the semi-simplicity of C^{∗}-algebras, the intersection of J(A) with all these kernels is the ideal G whence the second assertion of (25).

Sequences (A_{n}) and (P_{n}) as in (25) are called a Moore-Penrose sequence and the sequence of the associated Moore-Penrose projection in [12]. It is further shown in [12] that, for every Moore-Penrose sequence (in particular, for every Fredholm sequence), there is a sequence (Π_{n}) of projections such that every projection Π_{n} belongs to the C^{∗}-algebra generated by A_{n}^{∗}A_{n} and by the identity matrix I_{n} and that (25) holds with Π_{n} in place of P_{n}. Moreover, since Moore-Penrose projections are uniquely determined modulo the ideal G, ∥P_{n} −Π_{n}∥ → 0.

The latter convergence implies that dim Im P_{n} = dim Im Π_{n} for sufficiently large n, and it remains to verify that the range dimension of the Π_{n} for large n (which is independent on n and equal to the sum of the kernel dimensions of the operators W_{t}(A_{n}) as we have already checked) coincides with the α-number of the sequence (A_{n}). The property Π_{n} ∈ alg (A_{n}^{∗}A_{n}, I_{n}) ensures that the matrices A_{n}^{∗}A_{n} and Π_{n} can be diagonalized simultaneously:

U_{n}^{∗}A_{n}^{∗}A_{n}U_{n} = diag (a_{1}^{(n)}, a_{2}^{(n)}, …, a_{n}^{(n)}),
U_{n}^{∗}Π_{n}U_{n} = diag (p_{1}^{(n)}, p_{2}^{(n)}, …, p_{n}^{(n)})

with 0 ≤ a_{1}^{(n)} ≤ a_{2}^{(n)} ≤ … ≤ a_{n}^{(n)}. Let k = α(A_{n}), i.e. lim inf_{n→∞} a_{k}^{(n)} = 0 and lim inf_{n→∞} a_{k+1}^{(n)} > 0. The stability of the sequence (A_{n})^{∗}(A_{n}) + (Π_{n}) requires that

lim inf_{n→∞}(a_{r}^{(n)} + p_{r}^{(n)}) > 0 whence lim_{n→∞} p_{r}^{(n)} = 1 for r ≤ k,

and the condition ∥A_{n}^{∗}A_{n}Π_{n}∥ → 0 implies

lim_{n→∞} a_{r}^{(n)}p_{r}^{(n)} = 0 whence lim_{n→∞} p_{r}^{(n)} = 0 for r > k

since the numbers p_{r}^{(n)} can take the values 0 and 1 only. Here we also used the fact that the range dimension of the projections Π_{n} stabilizes as n → ∞. This observation finishes the proof of the dim Ker-formula (20) and of Theorem 10.

Remarks. 1. The following example shows that, without the hypothesis of fractality, one cannot expect that the ideals J(K_{n})/G are isomorphic to K(H) for essential rank one sequences (K_{n}). Let k ∈ F be the sequence (P_{1}, P_{2}, P_{3}, …) where every P_{n} is a projection from ℂ^{n} onto a one-dimensional subspace of ℂ^{n}. This sequence has essential rank one. The ideal J(k) contains the sequence

k′ := (P_{1}, 0, P_{3}, 0, P_{5}, …) = (P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, …) (I_{1}, 0, I_{3}, 0, I_{5}, …) which, on its hand, is essential rank one, too, and generates a proper ideal J(k′) of J(k) which is strictly larger that G. Since K(H) has no proper non-zero ideals, the quotient J(k)/G cannot be isomorphic to K(H) for some Hilbert space H.

2. It turns out that the Fredholm inverse closedness of the algebra A is also necessary for the dim Ker formula. To be more precise, let A ⊆ F be a unital algebra with center c which contains the ideal G. Suppose that, for every Fredholm sequence (A_{n}) ∈ A (i.e. for every sequence in A which is invertible modulo J(F)), the operators W_{t}(A_{n}) are Fredholm, only a finite number of these operators is not invertible, and the identity

α(A_{n}) = ∑_{t∈T_{ess}} dim Ker W_{t}(A_{n})

holds. Then, necessarily, A is Fredholm inverse closed. Indeed, let (A_{n}) ∈ A be a Fredholm sequence, let P_{KerW_{t}(A_{n})} denote the orthogonal projection onto the kernel of W_{t}(A_{n}), and choose sequences (P_{n}^{t}) in J(A) such that W_{t}(P_{n}^{t}) = P_{KerW_{t}(A_{n})} (which can be done due to Proposition 9). Then the sequence

(B_{n}) := (A_{n}^{∗}A_{n} + ∑_{t∈T_{ess}} P_{n}^{t})

is a Fredholm sequence, too (since (A_{n})^{∗}(A_{n}) is Fredholm and J(A) ⊆ J(F)). Furthermore,

dim Ker W_{t}(B_{n}) = 0 for all t ∈ T_{ess}.

From the dim Ker-formula we conclude that α(B_{n}) = 0, hence (B_{n}) is a stable sequence. This implies the invertibility of the sequence (A_{n})^{∗}(A_{n}) modulo J(A). Similarly, one gets the invertibility of (A_{n})(A_{n})^{∗} modulo this ideal. But then, the sequence (A_{n}) itself is invertible modulo J(A). Thus, every sequence in A which is invertible modulo J(F) in F (and, hence, modulo J(F) ∩ A in A), is also invertible modulo J(A). So, A is Fredholm inverse closed.

3. In case of a standard algebra A, its Fredholm theory reduces to the theory sketched in the introduction. Indeed, it is clear from the definition that A is unital and contains the ideal G. Further, all homomorphisms W^{t}(A_{n}) = s-lim (E_{n}^{t})^{∗}A_{n}E_{n}^{t} are fractal, hence, A is a fractal algebra (see [11]). Finally, the fourth axiom ensures the Fredholm inverse closedness of A (actually, this axiom guarantees much more: It requires that a sequence (A_{n}) is stable if all operators W^{t}(A_{n}) are invertible, whereas Fredholm inverse closedness essentially means that (A_{n}) is stable if all operators W^{t}(A_{n}) are invertible and if (A_{n}) is a Fredholm sequence).

It is also easy to identify the ideals I_{t} and J(A) in case of a standard algebra, namely

I_{t} = {(E_{n}^{t}K(E_{n}^{t})^{∗}) + G : K ∈ K(H^{t})}

and J(A) = J. The irreducible representations W_{t} are just the strong limit homomorphisms W^{t}, and the dim Ker formula (20) reduces exactly to the identity (5).

5 Examples and applications

We finish with a brief discussion of several examples and applications of Fredholm approximation sequences.

Splitting of the singular values of Toeplitz matrices. Given an L^{∞}(𝕋) function a with kth Fourier coefficient a_{k}, let T (a) denote the Toeplitz operator generated by a, i.e. the operator which is defined by its matrix representation (a_{i−j})_{i,j=1}^{∞} with respect to the standard basis of l^{2}. Toeplitz operators with bounded generating function are bounded. Further write P_{n} for the projection operator on l^{2} which maps the sequence (x_{i})_{i=1}^{∞} to (x_{1}, x_{2}, …, x_{n}, 0, 0, …), and consider the sequence (P_{n}T (a)P_{n})_{n≥1} of the finite sections of T (a). The operators P_{n}T (a)P_{n} can be identified with finite Toeplitz matrices acting on ℂ^{n}.

Let A stand for the smallest closed subalgebra of the algebra F which contains all sequences (P_{n}T (a)P_{n})_{n≥1} with a a piecewise continuous function on 𝕋. Finally, write R_{n} for the reflection operators on l^{2}, (x_{i}) ↦ (x_{n}, x_{n−1}, …, x_{1}, 0, 0, …). It has been shown in [2] that A is a standard algebra with the set T consisting of two elements and with associated strong limits

W (A_{n}) := s-lim A_{n} and W̃ := s-lim R_{n}A_{n}R_{n}.  (26)

Theorem 11 ([3]) A sequence (A_{n}) ∈ A is stable if and only if the operators W (A_{n}) and W̃ (A_{n}) are invertible.

In particular, Theorem 10 (and also its predecessor (5)) implies that

α(A_{n}) = dim Ker W (A_{n}) + dim Ker W̃ (A_{n})  (27)

for every Fredholm sequence (A_{n}) ∈ A. In Figures 2 and 4, there are plotted the singular values of the Toeplitz matrices P_{n}T (a)P_{n} and P_{n}T (b)P_{n} with n between 1 and 150 for

a(t) = 5t^{−3} + t^{−2} + 3t^{−1} + 1 + 4t + 7t^{2} + t^{3}

and

b(t) = 0.7t + t^{5}

respectively. The generating functions a and b have winding numbers 1 and 4 (Figures 1 and 3), and Figures 2 and 4 show exactly the predicted splitting of the singular values. These computations were done by F. Meyer using standard matlab.

Figure 1: Image of the unit circle under the generating function a. Figure 2: Singular values of P_{n}T (a)P_{n} for n between 1 and 150.

Figure 3: Image of the unit circle under the generating function b. Figure 4: Singular values of P_{n}T (b)P_{n} for n between 1 and 150.

Thus, the singular value splitting is an effect which can be observed numerically. On the other hand, the generating functions a and b are polynomials which might be the reason for the excellent convergence in figures 2 and 4. There is an example (due to Tyrtyshnikov; [14]) of a piecewise continuous function c such that the smallest singular value of P_{n}T (c)P_{n} decays to zero as (ln n)^{−1}.

Numerical determination of the kernel dimension. In case of a Fredholm Toeplitz operator T (a) with continuous generating function, the kernel dimension of T (a) is simply the maximum of 0 and of the negative winding number of the curve a(𝕋) around the origin (where this curve is provided with the orientation which is naturally inherited from the counterclockwise orientation of the unit circle). A similar simple geometric argument applies to the determination of dim Ker T (a) if a is piecewise continuous. The geometric interpretation of the kernel dimension of Toeplitz operators is mainly a consequence of Coburn’s theorem (see [3], Theorem 1.10).

In contrast to this, the determination of the kernel dimension of a compactly perturbed Toeplitz operator T (a) + K can prove to be a serious problem even in case of a nice generating function a. An application of identity (27) to the sequence (A_{n}) where A_{n} = P_{n}(T (a) + K)P_{n} with a piecewise continuous and K compact yields

α(A_{n}) = dim Ker (T (a) + K) + dim Ker T (ã)

where again ã(t) := a(1/t). The kernel dimension of T (ã) can be determined via the winding number. Thus, if one is able to observe the α-number of (A_{n}) numerically (by consideration of the singular values of A_{n} for large n as in Figures 2 and 4 above), then this identity yields the desired kernel dimension of T (a)+K.

Sufficient stability conditions. Let T stand for the smallest closed subalgebra of L(l^{2}) which contains all Toeplitz operators T (a) with piecewise continuous generating function a. Again we consider the finite section method (P_{n}AP_{n}), but now for operators A in T . Accordingly, let B refer to the smallest closed subalgebra of F which contains all sequences (P_{n}AP_{n}) with A ∈ T . It is not hard to prove that the strong limits W (A_{n}) and W̃ (A_{n}) (defined as in (26)) exist for every sequence (A_{n}) ∈ B. Thus, the invertibility of the operators W (A_{n}) and W̃ (A_{n}) is a necessary condition for the stability of the sequence (A_{n}). Assume the invertibility of these operators is also sufficient for the stability of (A_{n}). Then (and under the preliminary assumption that B is a standard algebra) the index identity

0 = ind (A_{n}) = ind W (A_{n}) + ind W̃ (A_{n})

should hold for every Fredholm sequence (A_{n}) ∈ B, i.e. for every sequence (A_{n}) for which W (A_{n}) and W̃ (A_{n}) are Fredholm operators.

There are simple examples showing that this identity cannot be true for arbitrary Fredholm sequences in B. Indeed, let

a(e^{ix}) := \{ \begin{matrix}1 & if & x ∈ (0, 2π/3) \\ e^{5iπ/6} & if & x ∈ (2π/3, 4π/3) \\ e^{7iπ/6} & if & x ∈ (4π/3, 2π). \end{matrix}

Then

ã(e^{ix})^{2} := \{ \begin{matrix}e^{iπ/3} & if & x ∈ (0, 2π/3) \\ e^{−iπ/3} & if & x ∈ (2π/3, 4π/3) \\ 1 & if & x ∈ (4π/3, 2π), \end{matrix}

hence, for A_{n} = P_{n}T (a)^{2}P_{n}, the operator W (A_{n}) = T (a)^{2} has index −2 whereas the index of W̃ (A_{n}) = T (ã^{2}) is 0.

Thus, the index identity predicts that the invertibility of the operators W (A_{n}) and W̃ (A_{n}) cannot be sufficient for the stability of a sequence (A_{n}) ∈ B in general. (In a similar way, the kernel dimension identity implies that the invertibility of the operator W (A_{n}) is not sufficient for the stability of a sequence (A_{n}) ∈ A in general, although it is sufficient for sequences of the form (P_{n}T (a)P_{n}).)

A detailed analysis (performed by Werbitzky, Rathsfeld, Böttcher, Silbermann and others) yields the following stability result for sequences in B where, besides the invertibility of W (A_{n}) and W̃ (A_{n}), certain local stability conditions occur. Let S_{n} refer to the subspace of the Hilbert space L^{2}([0, 1]) which is spanned by the functions

φ_{k,n}(x) := \{ \begin{matrix}1 & if & x ∈ (k/n, (k + 1)/n) \\ 0 & if & x ∈ (0, k/n) ∪ ((k + 1)/n, 1) \end{matrix}

with k = 0, …, n − 1, write E_{n} for the operators

E_{n} : Im P_{n} → S_{n}, (x_{1}, …, x_{n}, 0, …) ↦ ∑_{k=1}^{n} x_{k}φ_{k,n},

and define E_{−n} : S_{n} → Im P_{n} by E_{−n} := (E_{n})^{−1}. Finally, for τ ∈ 𝕋, let Y_{τ} stand for the operator

Y_{τ} : l^{2} → l^{2}, (x_{k})_{k=1}^{∞} ↦ (τ^{−k}x_{k})_{k=1}^{∞}.

One can show that, for every sequence (A_{n}) ∈ B and for every τ ∈ 𝕋, the strong limit

W^{τ}(A_{n}) := s-lim_{n→∞}E_{n}Y_{τ^{−1}}A_{n}Y_{τ}E_{−n}

exists and that it defines a bounded linear operator W^{τ}(A_{n}) on L^{2}([0, 1]).

Theorem 12 A sequence (A_{n}) ∈ B is stable if and only if the operators W (A_{n}), W̃ (A_{n}) and W^{τ}(A_{n}) are invertible for every τ ∈ 𝕋.

For a proof see, e.g., [5], Theorem 4.1. This proof also shows that B is a standard algebra.

Global vs. local stability conditions. For a more refined version of Theorem 12, let X stand for a closed subset of the unit circle 𝕋 and denote by P C_{X} the C^{∗}-algebra of all piecewise continuous functions on 𝕋 which are continuous at the points of 𝕋 \ X. Accordingly, let T_{X} stand for the smallest closed subalgebra of L(l^{2}) which contains all Toeplitz operators T (a) with generating function a ∈ P C_{X}, and let B_{X} refer to the smallest closed subalgebra of F which contains all sequences (P_{n}AP_{n}) with A ∈ T_{X}. A closer look at the proof of Theorem 12 reveals the following.

Theorem 13 A sequence (A_{n}) ∈ B_{X} is stable if and only if the operators W (A_{n}), W̃ (A_{n}) and W^{τ}(A_{n}) are invertible for every τ ∈ X ⊆ 𝕋.

Of particular interest is the case when X is a singleton, say X = {1}. For A ∈ T_{{1}}, Theorem 13 says that the sequence (P_{n}AP_{n}) is stable if and only if the three operators W (P_{n}AP_{n}) = A, W̃ (P_{n}AP_{n}) and W^{1}(P_{n}AP_{n}) are invertible. Actually, the invertibility of W^{1}(P_{n}AP_{n}) proves to be redundant in this special setting which is again a consequence of the index identity.

Theorem 14 Let A ∈ T_{{1}}. Then the sequence (P_{n}AP_{n}) is stable if and only if the operators W (P_{n}AP_{n}) = A and W̃ (P_{n}AP_{n}) are invertible.

Proof. The index identity, specified to the setting of Theorem 13 with X = {1}, yields

ind W (A_{n}) + ind W̃ (A_{n}) + ind W^{1}(A_{n}) = 0

for every Fredholm sequence (A_{n}) ∈ T_{{1}}. Let, in particular, A_{n} = P_{n}AP_{n} with A ∈ T_{{1}}, and suppose W (P_{n}AP_{n}) and W̃ (P_{n}AP_{n}) are invertible. It is not hard to check (using the Gohberg/Krupnik symbol calculus) that then W^{1}(P_{n}AP_{n}) is a Fredholm operator. Hence, (P_{n}AP_{n}) is a Fredholm sequence, and the index identity yields ind W^{1}(P_{n}AP_{n}) = 0. Further, the special form of the sequence (A_{n}) = (P_{n}AP_{n}) implies that W^{1}(P_{n}AP_{n}) is a Mellin operator, which is subject to Coburn’s theorem. Hence, W^{1}(P_{n}AP_{n}) is invertible.

Regularization of non-stable sequences. Let us finally mention a further potential application of Fredholm sequences to the regularization of non-stable approximation sequences which has been discussed in detail in [12].

Let (A_{n}) be an approximation sequence for an invertible operator A (i.e. the A_{n} converge to A in the ^{∗}-strong operator topology). Assume that this sequence is not stable but Fredholm with α-number k (which can be determined numerically or via the dim Ker formula in case of a standard algebra, say). Finally, let A_{n} = U_{n}Σ_{n}V_{n}^{∗} refer to the singular value decomposition of A_{n} with Σ_{n} := diag (σ_{1}^{(n)}, …, σ_{n}^{(n)}) and with 0 ≤ σ_{1}^{(n)} ≤ … ≤ σ_{n}^{(n)} denoting the singular values of A_{n}. For n > k, set Σ′_{n} := diag (0, …, 0, σ_{k+1}^{(n)}, …, σ_{n}^{(n)}) and A′_{n} := U_{n}Σ′_{n}V_{n}^{∗}. Then (A′_{n}) is also an approximation sequence for A, and this sequence is Moore-Penrose stable in the sense that the Moore-Penrose inverses of the matrices A′_{n} are uniformly bounded. Consequently, the Moore-Penrose inverses of the A′_{n} converge strongly to the inverse of A. In other words, the least square solutions of the equations A′_{n}x_{n} = P_{n}y converge to the solution of Ax = y.

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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