Algebras of approximation sequences: Fractality

Steffen Roch

This paper deals with a fundamental property of an approximation sequence which is responsible for the uniformity of certain limiting processes: its fractality. Roughly speaking, a sequence is fractal if the knowledge of any of its infinite subsequences allows to reconstruct the whole sequence up to a sequence tending to zero in the norm. Typical features of a fractal sequence (A_{n}) are the existence of the limit of the condition numbers of the matrices A_{n} and the existence of the limit in the sense of the Hausdorff metric of the spectra, the pseudospectra, and the numerical ranges of the A_{n}. It will be moreover shown that every approximation sequence possesses a fractal subsequence.

1. Introduction

Fractality is a property of an approximation method which makes certain limiting processes uniform. To motivate and illustrate this notion we start with a simple and transparent example.

Stable approximation methods and condition numbers. 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 for a moment that dim Im P_{n} = n. Then 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.

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 → ∞. As a consequence of the Banach-Steinhaus theorem, every approximation method is bounded in the sense that

∥(A_{n})∥ := sup ∥A_{n}∥ < ∞. (1.1)

The method (A_{n}) converges, or is applicable to A, if the equations

A_{n}x^{(n)} = P_{n}y (1.2) 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. (1.3)

By the Banach-Steinhaus theorem again, 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}∥ < ∞. (1.4)

If the sequence (A_{n}) is stable, then the condition numbers cond A_{n} := ∥A_{n}∥ ∥A_{n}^{−1}∥ are well-defined for n large enough. Without further information, all one can say in the moment is that the sequence (cond A_{n}) is bounded (due to (1.1) and (1.4)).

Fredholm integral equations of second kind. An obvious example of an approximation method where even convergence of the sequence (cond A_{n}) can be verified, is the Galerkin method for a Fredholm integral equation of the second kind: Ax = (I − K)x = y, y ∈ H, (1.5)

where K is a compact operator on H. For the Galerkin method, choose an orthonormal basis (e_{i})_{i≥1} of H, specify P_{n} as the orthogonal projection from H onto the span of {e_{1}, …, e_{n}}, and replace the equation (1.5) by the sequence of the approximation equations

A_{n}x^{(n)} := (I − P_{n}KP_{n})x^{(n)} = P_{n}y, n ≥ 1. (1.6)

Since ∥(I − P_{n}KP_{n}) − (I − K)∥ → 0 as n → ∞, (1.7)

a standard Neumann series argument yields that the sequence (I − P_{n}KP_{n}) is stable if and only if the operator I − K is invertible, and that

∥(I − P_{n}KP_{n})^{−1} − (I − K)^{−1}∥ → 0 as n → ∞ (1.8)

in this case. From (1.7) and (1.8) one immediately gets

Proposition 1.1. If K ∈ L(H) is compact and I − K is invertible, then the sequence (cond (I − P_{n}KP_{n}))_{n≥1} of the condition numbers of the Galerkin method is convergent, and its limit is ∥I −K∥ ∥(I −K)^{−1}∥, the condition number of I −K.

Thanks to the norm convergence (1.7), projection methods for Fredholm integral equations of second kind play an exceptional role. A similar norm convergence cannot hold if one wishes to approximate non-compact operators by n × n matrices. In these instances, one has to deal with strong (= pointwise) convergence, and the above arguments to prove Proposition 1.1 do not apply. Nevertheless, one can verify the convergence of the condition numbers also for many strongly convergent approximation methods, see [2, 14]. To illustrate this, and to motivate the subsequent introduction of the notion of a fractal algebra, we proceed with an alternative proof of Proposition 1.1.

Alternative approach. This approach requires two ingredients. The first one is a formula for the limes superior of the norms ∥A_{n}∥ which is valid for an arbitrary bounded approximation sequence (A_{n}). For its formulation, let F stand for the C^{∗}-algebra of all bounded sequences (A_{n}) of operators A_{n} ∈ L(H), provided with elementwise operations and the supremum norm, and let G refer to the closed ideal of F consisting of all sequences (A_{n}) with lim ∥A_{n}∥ = 0. The quotient algebra F/G plays an outstanding role for several problems in numerical analysis; for example the sequence (A_{n}) ∈ F is stable if and only if its coset (A_{n}) + G is invertible in F/G. The following identity, due to Böttcher [2], relates the norms ∥A_{n}∥ to the norm of the coset (A_{n}) + G.

Proposition 1.2. For every sequence (A_{n}) ∈ F,

lim sup ∥A_{n}∥ = ∥(A_{n}) + G∥_{F/G}.

The second ingredient is a characteristic property of the Galerkin method for Fredholm integral equations which, of course, is a consequence of the norm convergence (1.7), but which also belongs to many other (not necessarily normconvergent) approximation sequences.

Let A stand for the C^{∗}-subalgebra of F consisting of all sequences (A_{n}) which converge in the norm. For example, the sequence (I − P_{n}KP_{n}) of the Galerkin method is an element of A. Further let (A_{n}) ∈ A, let η be an infinite subsequence of ℕ, and suppose that we only know the subsequence (A_{η(n)}) of (A_{n}). Then we know lim A_{η(n)}, hence we know lim A_{n}, and consequently we know the coset (A_{n}) + G. Thus, knowledge of an arbitrary subsequence of (A_{n}) allows us to reconstruct the complete sequence (A_{n}) up to a sequence tending to zero in the norm. Precisely this property means that the mapping

π_{η} : (A_{η(n)}) ↦ (A_{n}) + G

is a continuous ^{∗}-homomorphism with norm 1.

Now we can prove Proposition 1.1 without (directly) invoking the norm convergence. What we have to show is that the limes superior in Proposition 1.2 is actually a limes if the sequence (A_{n}) belongs to A. Assume it is not. Then there exists an infinite subsequence η of ℕ such that

sup ∥A_{η(n)}∥ < lim sup ∥A_{n}∥.

Since ∥π_{η}∥ = 1 and by Proposition 1.2, we further have

lim sup ∥A_{n}∥ = ∥(A_{n}) + G∥ = ∥π_{η}(A_{η(n)})∥ ≤ ∥(A_{η(n)})∥ = sup ∥A_{η(n)}∥,

which is a contradiction.

Contents of this paper. In the following section, the notions of fractal homomorphisms, fractal algebras and fractal approximation sequences are defined, and some of their properties are discussed. In Section 3, these notions are applied to certain spectral approximation processes which will yield the convergence in the Hausdorff sense of the sets of the eigenvalues, the sets of the pseudo-eigenvalues, and of the numerical ranges of A_{n} for several approximation sequences (A_{n}). The concluding section is devoted to some special features of self-adjoint fractal sequences, and to the question whether every approximation sequence possesses a fractal subsequence.

The notion of a fractal homomorphism has been introduced in [14]. In that paper, algebras of approximation sequences were studied for which there exists a sufficient number of fractal homomorphisms of a special kind (related with socalled lifting theorems). For these algebras, analogues of Theorem 2.11, Corollary 2.12 and Propositions 3.7 and 3.11 (see below) were derived.

2. Fractal sequences

Definitions. It is not important in what follows that the elements of the sequences under consideration are matrices or operators. So we will use slighly generalized definitions of F and G. Given unital C^{∗}-algebras C_{n} (n = 0, 1, 2, ...) with identity elements e_{n}, let F stand for the set of all sequences (c_{0}, c_{1}, c_{2}, ...) with c_{n} ∈ C_{n} which are bounded in the sense that

∥(c_{n})∥_{F} := sup ∥c_{n}∥_{C_{n}} < ∞,

and write G for the set of all sequences (c_{0}, c_{1}, c_{2}, ...) in F with ∥c_{n}∥ → 0 as n → ∞. Provided with elementwise operations, an elementwise involution and the supremum norm, F becomes a unital C^{∗}-algebra and G a closed ideal of F.

For every strongly monotonically increasing subsequence η of ℕ, define F_{η} and G_{η} analogously as the C^{∗}-algebra consisting of all bounded sequences and the ideal of all zero sequences (c_{η(0)}, c_{η(1)}, c_{η(2)}, ...) with c_{η(n)} ∈ C_{η(n)}, and let R_{η} stand for the restriction mapping

R_{η} : F → F_{η}, (a_{n}) ↦ (a_{η(n)}).

The mapping R_{η} is a ^{∗}-homomorphism from F onto F_{η} which maps G onto G_{η}. Further, given a C^{∗}-subalgebra A of F, let A_{η} refer to the image of A under R_{η}. Clearly, A_{η} is a C^{∗}-subalgebra of F_{η}.

Definition 2.1. Let A be a C^{∗}-subalgebra of 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.

(c) A sequence (a_{n}) ∈ F is fractal if the smallest C^{∗}-subalgebra of F which contains (a_{n}) 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 its subsequence modulo sequences in A ∩ G. This assumption is quite natural for sequences arising from discretization procedures. For example, the algebra A examined in the introduction is fractal, and W : A → L(H), (A_{n}) ↦ lim A_{n} is a fractal homomorphism.

Fractal homomorphisms. We proceed with two lemmas on fractal homomorphisms.

Lemma 2.2. Let A be a C^{∗}-subalgebra of F, and let B be a C^{∗}-algebra. Then A ∩ G lies in the kernel of every fractal ^{∗}-homomorphism W : A → B.

Proof. Let (g_{n}) ∈ A ∩ G. Given ε > 0 there is an n_{0} such that ∥g_{n}∥ ≤ ε for n ≥ n_{0}. Consider the sequence η(n) := n + n_{0}. Since W is fractal,

W (g_{n}) = (W_{η}R_{η})(g_{n}) = W_{η}(g_{η(n)}) = W_{η}(g_{n+n_{0}}).

Consequently,

∥W (g_{n})∥ = ∥W_{η}(g_{n+n_{0}})∥ ≤ ∥W_{η}∥ ∥(g_{n+n_{0}})∥_{F_{η}} ≤ ε

(observe that ∥W_{η}∥ ≤ 1). Letting ε go to zero we get the assertion. □

Thus, if W : A → B is a fractal ^{∗}-homomorphism, then W (a_{n}) does only depend on the coset π(a_{n}) = (a_{n}) + A ∩ G.

Lemma 2.3. Let A be a fractal C^{∗}-subalgebra of F and B be a C^{∗}-algebra. A ∗-homomorphism W : A → B is fractal if and only if A ∩ G ⊆ Ker W.

Proof. If W is fractal, then A ∩ G ⊆ Ker W by Lemma 2.2. Let, conversely, A ∩ G ⊆ ker W, and let η be a monotonically increasing sequence. If W^{π} denotes the quotient mapping

W^{π} : A/(A ∩ G) → B, π(a_{n}) ↦ W (a_{n}), (2.1)

then, evidently, W = W^{π}π. Further, since π is fractal by assumption, there is a ∗-homomorphism π_{η} such that π = π_{η}R_{η}. Thus,

W = W^{π}π = W^{π}(π_{η}R_{η}) = (W^{π}π_{η})R_{η}

which shows fractality of W. □

Criteria for fractality of algebras. The following two theorems provide equivalent characterizations of fractal subalgebras of F.

Theorem 2.4. 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.

Proof. Let A be fractal, and assume there is a strongly monotonically increasing sequence η as well as an element (a_{n}) ∈ A with R_{η}(a_{n}) ∈ G_{η} for which (a_{n}) is not in A ∩ G, i.e.

∥π(a_{n})∥ = ∥(a_{n}) + A ∩ G∥ =: C > 0.

Since R_{η}(a_{n}) ∈ G_{η}, there is an n_{0} such that ∥a_{η(n)}∥ ≤ C/2 for n ≥ n_{0}. Define a sequence μ by μ(n) := η(n + n_{0}). Then ∥R_{μ}(a_{n})∥ ≤ C/2. On the other hand, the fractality of A implies that

C ≤ ∥π_{μ}R_{μ}(a_{n})∥ ≤ ∥π_{μ}∥ ∥R_{μ}(a_{n})∥ = ∥R_{μ}(a_{n})∥. (2.2)

for every strongly monotonically increasing sequence μ, which is a contradiction. Hence, (a_{n}) belongs to A ∩ G.

For the reverse direction, let (a_{n}), (b_{n}) ∈ A be sequences with R_{η}(a_{n}) = R_{η}(b_{n}). Then, by hypotheses, (a_{n}) − (b_{n}) is in A ∩ G, and so it is correct to define a ∗-homomorphism π_{η} via

π_{η} : A_{η} → A/(A ∩ G), R_{η}(a_{n}) ↦ (a_{n}) + A ∩ G.

Evidently, π_{η}R_{η} = π whence follows that π is a fractal homomorphism and A is a fractal algebra. □

Corollary 2.5. (a) Every C^{∗}-subalgebra of a fractal algebra A ⊆ F is fractal.

(b) A C^{∗}-subalgebra A of F is fractal if and only if each of its elements is fractal.

Proof. (a) Let B be a C^{∗}-subalgebra of A, and let (b_{n}) ∈ B be an element with R_{η}(b_{n}) ∈ G_{η}. Since A is fractal, Theorem 2.4 implies that (b_{n}) belongs to A ∩ G. Then, clearly, (b_{n}) lies in B ∩ G, and invoking Theorem 2.4 once more one gets the fractality of B.

(b) If A is a fractal C^{∗}-subalgebra of F, then each of its elements is fractal due to assertion (a). For the reverse assertion we claim that, whenever A is not fractal, then it contains a non-fractal sequence. Indeed, if A fails to be fractal, then, by Theorem 2.4, there is an element (a_{n}) ∈ A as well as a sequence η such that R_{η}(a_{n}) ∈ G_{η}, i.e. ∥a_{η(n)}∥ → 0 as n → ∞, but (a_{n}) ∉ A ∩ G, i.e. ∥a_{n}∥ ↛ 0 as n → ∞. Applying Theorem 2.4 once again this shows that the C^{∗}-subalgebra of A which is generated by this sequence (a_{n}) cannot be fractal, i. a. (a_{n}) is a non-fractal sequence. □

Corollary 2.6. If A is a fractal C^{∗}-subalgebra of F, then A_{η} ∩ G_{η} = (A ∩ G)_{η}.

The following criterion proves to be useful for verifying the fractality of several concrete algebras of approximation methods.

Theorem 2.7. Let A be a unital C^{∗}-subalgebra of F. The algebra A is fractal if and only if there exists a family {W_{t}}_{t∈T} of unital and fractal ^{∗}-homomorphisms W_{t} from A into unital C^{∗}-algebras B_{t} such that the following equivalence holds for every sequence (a_{n}) ∈ A: The coset (a_{n}) + A ∩ G is invertible in A/(A ∩ G) if and only if W_{t}(a_{n}) is invertible in B_{t} for every t ∈ T.

Proof. If A is fractal, then the canonical homomorphism π : A → A/(A ∩ G) yields the ’family’ {π} which obviously has the desired properties.

Let, conversely, {W_{t}}_{t∈T} be a family of unital and fractal ^{∗}-homomorphisms which is subject to the conditions of the theorem. Given a strongly monotonically increasing sequence η, define the operator T_{η} by

T_{η} : A/A ∩ G → A_{η}/(A ∩ G)_{η}, (a_{n}) + A ∩ G ↦ R_{η}(a_{n}) + (A ∩ G)_{η}.

One easily checks that this definition is correct, and that T_{η} is a ^{∗}-homomorphism from A/A ∩ G onto A_{η}/(A ∩ G)_{η}. We claim that T_{η} is an isomorphism.

For this goal, it is sufficient to show that T_{η} is a symbol mapping in the sense that, if (a_{n}) ∈ A, and if T_{η}((a_{n}) + A ∩ G) = R_{η}(a_{n}) + (A ∩ G)_{η} is invertible, then (a_{n}) + A ∩ G is invertible.

If the coset R_{η}(a_{n}) + (A ∩ G)_{η} is invertible, then there are a sequence (b_{n}) ∈ A as well as sequences (g_{n}), (h_{n}) ∈ A ∩ G such that

R_{η}(a_{n})R_{η}(b_{n}) = R_{η}(e_{n}) + R_{η}(g_{n}), R_{η}(b_{n})R_{η}(a_{n}) = R_{η}(e_{n}) + R_{η}(h_{n}). (2.3)

The homomorphisms W_{t} are fractal by assumption, i.e. there are homomorphisms W_{t,η} such that W_{t} = W_{t,η}R_{η}. Applying W_{t,η} to the equalities (2.3) we get

W_{t}(a_{n})W_{t}(b_{n}) = W_{t}(e_{n}) + W_{t}(g_{n}), W_{t}(b_{n})W_{t}(a_{n}) = W_{t}(e_{n}) + W_{t}(h_{n}).

Since W_{t}(e_{n}) is the identity element of B_{t} and W_{t}(g_{n}) = W_{t}(h_{n}) = 0 by Lemma 2.2, all elements W_{t}(a_{n}) are invertible in B_{t}. Thus, the coset (a_{n}) + A ∩ G is invertible, and T_{η} is indeed a symbol mapping and, thus, a ^{∗}-isomorphism between A/A ∩ G and A_{η}/(A ∩ G)_{η}.

Let finally Π_{η} denote the canonical homomorphism from A_{η} onto A_{η}/(A ∩ G)_{η}. Then, evidently, T_{η}^{−1}Π_{η}R_{η} = π, i.e. π is fractal (with π_{η} = T_{η}^{−1}Π_{η}). □

Example: Spline projection methods for singular integral equations. To mention at least one concrete class of fractal approximation sequences we consider spline projection methods for singular integrals on the real line. All details, as well as more general results (pertaining arbitrary spline spaces, singular integrals on arbitrary Lyapunov curves etc.) can be found in [6]. Here we restrict our attention to a situation without technical complications.

Let S_{n} stand for the smallest closed subspace of L^{2}(ℝ) which contains, for every k, the functions φ_{kn}(t) := χ_{[0,1)}(nt − k) where χ_{[0,1)} is the characteristic function of the interval [0, 1). Every spline space S_{n} is isometric to the Hilbert space l^{2}(ℤ), with the isometry given by

E_{n} : l^{2}(ℤ) → S_{n}, (x_{k})_{k∈ℤ} ↦ n^{1/2} ∑_{k∈ℤ} x_{k}φ_{kn}.

The inverse of E_{n} : l^{2}(ℤ) → S_{n} will be denoted by E_{−n}. Further write L_{n} for the orthogonal projection from L^{2}(ℝ) onto S_{n}; these projections converge strongly to the identity operator as n tends to infinity.

On l^{2}(ℤ) we introduce the following operators. Given a piecewise continuous function a on the unit circle 𝕋 with kth Fourier coefficient a_{k}, let T^{0}(a) ∈ L(l^{2}(ℤ)) stand for the Laurent operator with matrix representation (a_{i−j})_{i,j∈ℤ}. The projection operator (x_{k}) ↦ (…, 0, 0, x_{0}, x_{1}, …) will be denoted by P. Finally, given n ∈ ℤ and t ∈ 𝕋, define operators V_{n} and Y_{t} by

V_{n} : l^{2}(ℤ) → l^{2}(ℤ), (x_{k}) ↦ (x_{k−n}), Y_{t} : l^{2}(ℤ) → l^{2}(ℤ), (x_{k}) ↦ (t^{−k}x_{k}).

Now let the C^{∗}-algebra F of all bounded approximation sequences be defined as above with the algebras C_{n} specified as L(S_{n}), and let A refer to the smallest closed subalgebra of F which contains all sequences of the form

(E_{n}V_{{sn}}P V_{−{sn}}E_{−n})_{n≥1} and (E_{n}T^{0}(a)E_{−n})_{n≥1}

with a running through the piecewise continuous functions on 𝕋, with s ∈ ℝ, and with {sn} referring to the smallest integer which is not smaller that sn. This definition of A seems to be artificial at the first glance, but one can show that this algebra indeed contains a bulk of interesting and concrete approximation methods. For example, the sequence (L_{n}(aI+bS)L_{n}) of the Galerkin method for the singular integral operator

((aI + bS)u)(t) := a(t)u(t) + \frac{b(t)}{πi} ∫_{−∞}^{∞} \frac{u(s)}{s − t} ds, t ∈ ℝ

belongs to A if a and b are piecewise continuous functions on the real line with jumps only at integer points and at infinity.

Proposition 2.8. Let (A_{n}) ∈ A.

(a) For every t ∈ 𝕋, the strong limit

W^{t}(A_{n}) := s-lim E_{n}Y_{t^{−1}}E_{−n}A_{n}E_{n}Y_{t}E_{−n}L_{n}

exists, and the mapping W^{t} is a ^{∗}-homomorphism from A into L(L^{2}(ℝ)).

(b) For every s ∈ ℝ, the strong limit

W_{s}(A_{n}) := s-lim V_{−{sn}}E_{−n}A_{n}E_{n}V_{{sn}}

exists, and the mapping W_{s} is a ^{∗}-homomorphism from A into L(l^{2}(ℤ)) .

(c) The cosets of the operators E_{−n}A_{n}E_{n} modulo the ideal K(l^{2}(ℤ)) of the compact operators on l^{2}(ℤ) are independent of n. If W_{∞}(A_{n}) denotes one of these cosets, then W_{∞} is a ^{∗}-homomorphism from A into L(l^{2}(ℤ))/K(l^{2}(ℤ)).

The stability criterion for sequences in A reads as follows (Theorem 3.3 in [6]):

Theorem 2.9. A sequence (A_{n}) ∈ A is stable if and only if the operators resp. cosets W^{t}(A_{n}) and W_{s}(A_{n}) are invertible for every t ∈ 𝕋 and s ∈ ℝ ∪ {∞}.

It is evident that the homomorphisms W^{t} and W_{s} are fractal for every t ∈ 𝕋 and s ∈ ℝ ∪ {∞}. Hence, Theorem 2.9 implies in combination with Theorem 2.7:

Corollary 2.10. The algebra A is fractal.

Fractal algebras and convergence of norms. Here we turn back once again the problem whether the limit lim ∥a_{n}∥ exists for a sequence (a_{n}) ∈ F. It will be pointed out that fractality of (a_{n}) is indeed enough to guarantee the existence of this limit.

Theorem 2.11. Let A be a fractal C^{∗}-subalgebra of F.

(a) If (a_{n}) ∈ A and η is a strongly monotonically increasing sequence, then

∥(a_{n}) + G∥_{F/G} = ∥(a_{η(n)}) + G_{η}∥_{F_{η}/G_{η}}.

(b) If (a_{n}) ∈ A, then the limit lim ∥a_{n}∥ exists and is equal to ∥(a_{n}) + G∥.

Proof. (a) The third isomorphy theorem for C^{∗}-algebras states that, if B is a C^{∗}-subalgebra of F and (b_{n}) ∈ B, then

∥(b_{n}) + G∥_{F/G} = ∥(b_{n}) + G∥_{(B+G)/G} = ∥(b_{n}) + B ∩ G∥_{B/(B∩G)}. (2.4)

Let now (a_{n}) ∈ A and (g_{n}) ∈ A ∩ G. Then

∥(a_{n}) + G∥_{F/G} = ∥(a_{n}) + A ∩ G∥_{A/(A∩G)} (by (2.4))
= ∥π(a_{n} + g_{n})∥_{A/(A∩G)}  (definition of π)
= ∥π_{η}R_{η}(a_{n} + g_{n})∥_{A/(A∩G)} (fractality of A)
≤ ∥(a_{η(n)} + g_{η(n)})_{n=0}^{∞}∥_{A_{η}}  (since ∥π_{η}∥ ≤ 1).

Taking the infimum over all (g_{n}) ∈ A ∩ G, and invoking Corollary 2.6, we obtain

∥(a_{n}) + G∥_{F/G} ≤ ∥(a_{η(n)})_{n=0}^{∞} + (A ∩ G)_{η}∥_{A_{η}/(A∩G)_{η}}
= ∥(a_{η(n)})_{n=0}^{∞} + A_{η} ∩ G_{η}∥_{A_{η}/(A_{η}∩G_{η})},

and a further application of (2.4) gives

∥(a_{n}) + G∥_{F/G} ≤ ∥(a_{η(n)})_{n=0}^{∞} + G_{η}∥_{F_{η}/G_{η}} . (2.5)

On the other hand, the lim sup-formula for the norm in F/G (i.e. Proposition 1.2, which also holds in the present, slightly more general situation) yields

∥(a_{η(n)}) + G_{η}∥_{F_{η}/G_{η}} = lim sup ∥a_{η(n)}∥ ≤ lim sup ∥a_{n}∥ = ∥(a_{n}) + G∥_{F/G}

which together with (2.5) proves the assertion.

(b) Assume there is a sequence (a_{n}) ∈ A such that lim inf ∥a_{n}∥ < lim sup ∥a_{n}∥. Then there is a strongly monotonically sequence η such that the limit lim ∥a_{η(n)}∥ exists and is equal to lim inf ∥a_{n}∥. By part (a) of this theorem and by the lim supformula again,

lim sup ∥a_{n}∥ = ∥(a_{n}) + G∥_{F/G} = ∥(a_{η(n)}) + G_{η}∥_{F_{η}/G_{η}}
= lim sup ∥a_{η(n)}∥ = lim ∥a_{η(n)}∥ = lim inf ∥a_{n}∥

which is a contradiction. □

Corollary 2.12. Let A be a fractal C^{∗}-subalgebra of F, and let (a_{n}) ∈ A be a stable sequence. Then the limit lim cond a_{n} := lim ∥a_{n}∥ ∥a_{n}^{−1}∥ exists and is equal to cond ((a_{n}) + G) := ∥(a_{n}) + G∥ ∥((a_{n}) + G)^{−1}∥.

As a particular consequence we obtain that the norms ∥A_{n}∥ as well as (in case of a stable sequence) the condition numbers cond A_{n} converge for every sequence (A_{n}) in the algebra A of the spline projection methods. Moreover, the limit lim ∥A_{n}∥ is equal to sup{∥W^{t}(A_{n})∥, ∥W_{s}(A_{n})∥} where the supremum is taken over all t ∈ 𝕋 and s ∈ ℝ∪{∞}. An analogous result holds for the limit of the condition numbers.

It is obvious that the decisive point in this approach is a precise knowledge on the stability of sequences in A: it is both important to have a criterion for the stability of an arbitrary sequence in A, and that this criterion has the ’right form’, i.e. via a fractal mappings. Appropriate stability criteria are available for many algebras of concrete approximation methods (see, e.g. [6] and the references therein).

3. Examples and applications

The theme of this section is limiting sets of eigenvalues of approximation operators, i.e. the question whether the spectrum of a given operator A can be determined approximately by computing the spectrum of certain approximations A_{n} of A. It turns out that this spectral approximation is in general possible if A and the A_{n} are self-adjoint, whereas it can fail drastically in case A is a nonself-adjoint operator. Thus, two alternative approaches to the approximation of spectra will be considered, namely via pseudospectra and via numerical ranges. Both pseudospectra and numerical ranges can be viewed as approximants of the usual spectra, and both exhibit a much better asymptotic behaviour than the latter. Further, in all cases we will observe that the approximation processes under consideration become more uniform in the presence of fractality.

Set sequences. We start with recalling some elementary facts on set sequences. A general reference is [7], Section 28. A set sequence is a sequence of subsets of the complex plane ℂ. If (A_{n}) is a sequence of operators, then the mapping n ↦ σ(A_{n}) which assigns with n ∈ ℕ the spectrum of A_{n} is a set sequence in this sense.

Definition 3.1. (a) Let (M_{n})_{n=1}^{∞} be a set sequence. The partial limiting set or limes superior lim sup M_{n} (resp. the uniform limiting set or limes inferior lim inf M_{n}) of the sequence (M_{n}) consists of all points m ∈ ℂ which are a partial limit (resp. the limit) of a sequence (m_{n}) of points m_{n} ∈ M_{n}.

Observe that the partial limiting set lim sup M_{n} is non-empty if infinitely many of the M_{n} are non-empty and if ∪_{n}M_{n} is bounded, whereas the uniform limiting set can be empty even under these restrictions as the trivial example M_{n} = {(−1)^{n}} shows. Both the partial and the uniform limiting set are closed.

Let ℂ^{C} denote the set of all non-empty and compact subsets of the complex plane. A criterion for the coincidence of the partial and uniform limiting sets of a set sequence taking values in ℂ^{C} can be given in terms of the convergence of that sequence with respect to the Hausdorff metric on ℂ^{C} where the distance of two sets A, B ⊆ ℂ^{C} is defined as

h(A, B) := max{max_{a∈A} dist(a, B), max_{b∈B} dist(b, A)}

with dist(a, B) = min_{b∈B} |a − b|. We denote the limit of a sequence (M_{n}) with respect to this metric by lim M_{n}.

Proposition 3.2. Let (M_{n}) be a sequence taking values in ℂ^{C}. Then lim sup M_{n} and lim inf M_{n} coincide if and only if the sequence (M_{n}) is convergent in sense of the Hausdorff metric. If one of these conditions is satisfied, then

lim sup M_{n} = lim inf M_{n} = lim M_{n}.

In Section 4, we will need the following compactness result saying that bounded subsets of ℂ^{C} are relatively compact with respect to h.

Theorem 3.3. Every bounded set sequence (M_{n}) in ℂ^{C} possesses a subsequence which converges with respect to the Hausdorff metric.

Spectra and their limiting sets. Again, let (C_{n}) be a family of C^{∗}-algebras with identity elements e_{n} and define the algebra F and its ideal G as above. The spectrum of an element a_{n} ∈ C_{n} will be denoted by σ(a_{n}). Our first goal is the asymptotic behaviour of the spectra σ(a_{n}) for bounded sequences (a_{n}) ∈ F.

Let (a_{n}) ∈ F. It turns out that the partial limiting set lim sup σ(a_{n}) is related with a new notion of stability which might be called ’spectral’ stability (in contrast to the ’normal’ stability defined in the introduction). A sequence (a_{n}) is spectrally stable if its entries a_{n} are invertible for all sufficiently large n, and if the spectral radii ρ(a_{n}^{−1}) of their inverses are uniformly bounded (whereas ’normal’ stability requires uniform boundedness of the norms ∥a_{n}^{−1}∥). Clearly, every normally stable sequence is also spectrally stable. The simple proof of the following result can be found in [13].

Proposition 3.4. Let (a_{n}) ∈ F. Then s ∈ ℂ belongs to the partial limiting set lim sup σ(a_{n}) if and only if the sequence (a_{n} − se_{n}) is not spectrally stable.

Thus, the determination of the partial limiting set lim sup σ(a_{n}) requires to investigate the spectral stability of the sequences (a_{n}−se_{n}) which can be is a rather involved problem in general. These difficulties disappear for normal sequences in which case the normal stability and the spectral stability coincide. So one has:

Corollary 3.5. Let (a_{n}) ∈ F be a sequence of normal elements. Then

lim sup σ(a_{n}) = σ_{F/G}((a_{n}) + G).

The fractality of the sequence (a_{n}) again guarantees that the limes superior in the latter spectral identity (which plays a similar role as the limsup identity for norms in Proposition 1.2) is a actually a Hausdorff limit.

Theorem 3.6. Let A be a fractal C^{∗}-subalgebra of F which contains the identity.

(a) A sequence (a_{n}) ∈ A is normally stable if and only if it possesses a normally stable subsequence.

(b) If (a_{n}) ∈ A is normal, then lim sup σ(a_{n}) = lim inf σ(a_{n}) = lim σ(a_{n}).

Proof. (a) Let the notations be as in Section 2, and let η be a strongly monotonically increasing sequence such that the sequence (a_{η(n)}) = R_{η}(a_{n}) is stable. Clearly, η can be chosen in such a way that all a_{η(n)} are invertible. By the inverse closedness of A_{η} in F_{η}, there is a sequence (b_{n}) ∈ A such that

R_{η}(a_{n}) R_{η}(b_{n}) = R_{η}(b_{n}) R_{η}(a_{n}) = R_{η}(e_{n}). (3.1)

The canonical homomorphism π : A → A/(A ∩ G) is fractal by hypothesis, i.e. π = π_{η}R_{η} with certain homomorphism π_{η}. Applying π_{η} to (3.1) one gets the invertibility of π(a_{n}) = (a_{n}) + A ∩ G in A/(A ∩ G), whence the stability of (a_{n}) follows.

(b) Assume, there is a t ∈ lim sup σ(a_{n})\lim inf σ(a_{n}). Then one can find a δ > 0 as well as a strongly monotonically increasing sequence η such that dist (t, σ(a_{η(n)})) ≥ δ for all n. The normality of the a_{η(n)} guarantees that (a_{η(n)} −te_{η(n)}) is a normally stable sequence (with the norms of the inverses being bounded above by 1/δ). Then, by part (a), the sequence (a_{n} − te_{n}) itself is normally stable. Hence, t ∉ σ((a_{n}) + G) = lim sup σ(a_{n}) by Corollary 3.5 which is a contradiction. □

As an example, consider once more the spline Galerkin method for singular integral operators, i.e. specify F, G and A as in the previous section. By Corollary 2.10, A is a fractal algebra.

Proposition 3.7. Let (A_{n}) ∈ A be a normal sequence. Then

lim sup σ(A_{n}) = lim inf σ(A_{n}) = ⋃_{s∈ℝ∪{∞}} σ(W_{s}(A_{n})) ∪ ⋃_{t∈𝕋} σ(W^{t}(A_{n})).

The proof follows immediately from the previous theorem, Theorem 2.9 and Corollary 3.5. This result applies, e.g., to the sequence (A_{n}), A_{n} = L_{n}(aI + bS + K)L_{n} where a and b are real-valued functions with a being piecewise continuous and continuous on all non-integer points and b continuous on ℝ ∪ {∞}, and where the compact operator K is chosen such that the operator aI + bS + K becomes self-adjoint.

Pseudospectra and their limiting sets. A computer working with finite accuracy cannot distinguish between a non-invertible matrix and an invertible matrix the inverse of which has a very large norm. This suggests the following definition of ε- invertibility which reflects finite accuracy. It is a remarkable observation mainly due to Landau, Reichel and Trefethen [8, 9, 15] that the related ε-pseudospectra also exhibit a much better convergence behaviour than the common spectra. Beautiful plots of limiting sets of pseudospectra can be found in [9, 15, 3].

Definition 3.8. Let B be a C^{∗}- algebra with identity e and let ε > 0. An element a ∈ B is ε-invertible if it is invertible and if ∥a^{−1}∥ < 1/ε. The ε-pseudospectrum σ^{(ε)}(a) of a consists of all λ ∈ ℂ for which a − λe is not ε-invertible.

Let the C^{∗}-algebras C_{n}, F and G as above. The following result identifies the partial limiting set lim sup σ^{(ε)}(a_{n}) of the ε-pseudospectra of a sequence (a_{n}) ∈ F with the ε-pseudospectra of the coset (a_{n}) + G in F/G. In contrast to the case ε = 0 (Corollary 3.5) this result holds for arbitrary sequences in F.

Theorem 3.9. Let (a_{n}) ∈ F and ε > 0. Then

lim sup σ_{C_{n}}^{(ε)}(a_{n}) = σ_{F/G}^{(ε)}((a_{n}) + G).

The proof of Theorem 3.9 follows essentially by adapting Böttcher’s proof for a special class of sequences in F which is related with the finite section method for Toeplitz operators [2]. A generalization of Theorem 3.9 to the case of pseudospectra of operator polynomials is derived in [12].

In the presence of fractality of the sequence (a_{n}), one will expect equality between the partial and the uniform limiting sets of the pseudospectra of the a_{n}. Indeed, this equality holds as we will see now. For, we agree upon calling a sequence (a_{n}) ∈ F ε-stable if the coset (a_{n}) + G is ε-invertible in F/G.

Theorem 3.10. Let A be a fractal C^{∗}-subalgebra of F which contains the identity of F, and let ε > 0.

(a) A sequence (a_{n}) ∈ A is ε-stable if and only if it possesses an ε-stable subsequence.

(b) If (a_{n}) ∈ A, then lim sup σ^{(ε)}(a_{n}) = lim inf σ^{(ε)}(a_{n}) = lim σ^{(ε)}(a_{n}).

Proof. (a) Suppose η is a strongly monotonically increasing sequence such that (a_{η(n)}) = R_{η}(a_{n}) is an ε-stable subsequence of (a_{n}). Then the sequence (a_{n}) is stable by Theorem 3.6(a), and Theorem 2.11 implies that

1/ε > ∥((a_{η(n)}) + G_{η})^{−1}∥_{F/G} = ∥((a_{n}) + G)^{−1}∥_{F/G},

i.e. (a_{n}) is ε-stable.

(b) Assume t lies in lim sup σ^{(ε)}(a_{n}) \ lim inf σ^{(ε)}(a_{n}). Then there is a subsequence η of ℕ such that t ∉ lim sup σ^{(ε)}(a_{η(n)}). By Theorem 3.9, the sequence (a_{η(n)} − te_{η(n)}) is ε-stable, whence via assertion (a) the ε-stability of the complete sequence (a_{n} − te_{n}) follows. Again by Theorem 3.9, this implies t ∉ lim sup σ^{(ε)}(a_{n}) which contradicts the assumption. □

A consequence is the following result on spectral approximation of singular integral operators by a spline Galerkin method which, in contrast to Proposition 3.7, holds for arbitrary sequences (L_{n}(aI + bS + K)L_{n}) ∈ A.

Proposition 3.11. Let A be the algebra of spline approximation methods for singular integral operators, (A_{n}) ∈ A, and ε > 0. Then

lim sup σ^{(ε)}(A_{n}) = lim inf σ^{(ε)}(A_{n}) = ⋃_{s∈ℝ∪{∞}} σ^{(ε)}(W_{s}(A_{n})) ∪ ⋃_{t∈𝕋} σ^{(ε)}(W^{t}(A_{n})).

The proof of the latter identity follows from the fact that the function, which is defined on the disjoint union of ℝ ∪ {∞} with 𝕋 and which takes the value ∥W_{s}(a_{n})∥ at s ∈ ℝ ∪ {∞} and the value ∥W^{t}(a_{n})∥ at t ∈ 𝕋, attains its maximum (see [14]).

Numerical ranges and their limiting sets The (several kinds of) numerical ranges provide further examples of (upper) spectral approximants which share with the ε-pseudospectra their good asymptotic behaviour for all approximation sequences.

Let H be a (complex) Hilbert space with inner product ⟨· , ·⟩, and let A be a linear bounded operator on H. The spatial numerical range SN_{H}(A) of A is the set

SN_{H}(A) := {⟨Ax, x⟩ : x ∈ H, ∥x∥ = 1}.

In what follows we want to consider the asymptotic behaviour of the spatial numerical ranges SN_{ℂ^{n}}(A_{n}) of a sequence (A_{n}) of approximation operators, and our goal is to identify the limiting set lim SN_{ℂ^{n}}(A_{n}) with a subset of ℂ which depends on the coset (A_{n})+G in F/G only. So we need an analogue of the spatial numerical range for elements of a C^{∗}-algebra. If B is a C^{∗}-algebra with identity element e, then the algebraic numerical range AN_{B}(b) of b ∈ B is the set

AN_{B}(b) := {φ(b) : φ ∈ B^{∗} with ∥φ∥ = φ(e) = 1}.

For a detailed account on numerical ranges see, e.g., [4] and [5]. Here we only mention that AN_{B}(b) is a convex and compact subset of the complex plane, that AN_{B}(c) = AN_{C}(c) for every C^{∗}-subalgebra C of B and for every element c ∈ C (inverse closedness), and that

AN_{B/J} (b + J) = ∩_{j∈J} AN_{B}(b + j) (3.2)

for every closed ideal J of B and every element b ∈ B. Further, for every bounded linear operator A on a Hilbert space H,

AN_{L(H)}(A) = clos SN_{H}(A). (3.3)

Again we start with a general result relating the partial limiting set of the numerical ranges of a sequence (a_{n}) to the numerical range of the coset of (a_{n}) modulo zero sequences.

Theorem 3.12. For every sequence (a_{n}) ∈ F one has

conv lim sup_{n→∞} AN_{C_{n}}(a_{n}) = AN_{F/G}((a_{n}) + G).

Observe that the partial limiting set of a sequence of convex sets needs not to be convex again which explains the conv operator on the left hand side of this equality. For the proof of Theorem 3.12 we need two auxiliary results: one describing a convexity property of sequences of convex sets, and one characterizing the algebraic numerical range of elements of F.

Lemma 3.13. Let (M_{k}) be a monotonically decreasing (i.e. M_{k} ⊇ M_{k+1} for all k) sequence of compact subsets of ℂ. Then conv ∩_{k} M_{k} = ∩_{k} conv M_{k}.

The elementary proof is omitted.

Theorem 3.14. For every sequence (a_{n}) ∈ F one has

AN_{F}((a_{n})) = conv clos ∪_{k} AN_{C_{k}}(a_{k}).

Proof. Let m ∈ AN_{C_{k}}(a_{k}) for some fixed k, and let ϕ be a state of C_{k} with ϕ(a_{k}) = m. Then

ψ_{ϕ} : F → ℂ, (b_{n})_{n=1}^{∞} ↦ ϕ(b_{k})

is a state of F, and ψ_{ϕ}((a_{n})) = ϕ(a_{k}) = m. Thus, m ∈ AN_{F}((a_{n})), which implies that ∪_{k}AN_{C_{k}}(a_{k}) ⊆ AN_{F}((a_{n})). Since AN_{F}((a_{n})) is a closed convex set we arrive at the inclusion conv clos ∪_{k} AN_{C_{k}}(a_{k}) ⊆ AN_{F}((a_{n})).

For the reverse inclusion, think of C_{k} as a C^{∗}-algebra of linear bounded operators on some Hilbert space H_{k}, and let ⊕H_{k} refer to the orthogonal sum of these Hilbert spaces. Evidently, the elements of F can be identified with diagonal operators acting on ⊕H_{k} in an obvious manner. For simplicity, we denote the diagonal operator which corresponds to the sequence (a_{k}) ∈ F by (a_{k}) again.

From the inverse closedness with respect to algebraic numerical ranges and from (3.3) we infer that

AN_{F}((a_{n})) = AN_{L(⊕H_{k})}((a_{n})) = clos SN_{⊕H_{k}}((a_{n})). (3.4)

Thus, given m ∈ AN_{F}((a_{n})) and ε > 0, there is a vector (x_{k}) ∈ ⊕H_{k} with norm 1 such that

|m − ⟨(a_{k})(x_{k}), (x_{k})⟩_{⊕H_{k}}| = |m − ∑_{k=1}^{∞} ⟨a_{k}x_{k}, x_{k}⟩_{H_{k}}| < ε.

Let 𝕄 ⊆ ℕ denote the set of all k with x_{k} ≠ 0, choose arbitrary elements y_{k} ∈ H_{k} with ∥y_{k}∥ = 1 for k ∉ 𝕄, and set

x′_{k} := \{ \begin{matrix}x_{k}/∥x_{k}∥ & if & k ∈ 𝕄 \\ y_{k} & if & k ∉ 𝕄. \end{matrix}

Then

∑_{k=1}^{∞} ⟨a_{k}x_{k}, x_{k}⟩ = ∑_{k∈𝕄} ∥x_{k}∥^{2} ⟨a_{k} \frac{x_{k}}{∥x_{k}∥}, \frac{x_{k}}{∥x_{k}∥}⟩ = ∑_{k=1}^{∞} ∥x_{k}∥^{2} ⟨a_{k}x′_{k}, x′_{k}⟩

and, consequently,

|m − ∑_{k=1}^{∞} ∥x_{k}∥^{2} ⟨a_{k}x′_{k}, x′_{k}⟩_{H_{k}} | < ε.

Because of ∥x_{k}∥^{2} ≥ 0 and ∑_{k=1}^{∞} ∥x_{k}∥^{2} = ∥(x_{k})∥_{⊕H_{k}}^{2} = 1, this shows that m can be approximated by convex linear combinations of points ⟨a_{k}x′_{k}, x′_{k}⟩ ∈ ∪_{k}SN_{H_{k}}(a_{k}) as closely as desired. Hence, and by (3.3),

m ∈ clos conv ∪_{k} SN_{H_{k}}(a_{k}) ⊆ clos conv ∪_{k} AN_{L(H_{k})}(a_{k}),

and employing inverse closedness once more we find

(3.5) m ∈ clos conv ∪_{k} AN_{C_{k}}(a_{k}).

Since ∪_{k}AN_{C_{k}}(a_{k}) is bounded (the radius of this set is not greater than ∥(a_{k})∥_{F}), and since clos conv M = conv clos M for every bounded subset M of the complex plane, (3.5) is just the assertion. □

We are now in position to prove Theorem 3.12.

Proof. Recall from (3.2) that

AN_{F/G}((a_{n}) + G) = ∩_{(g_{n})∈G} AN_{F}((a_{n}) + (g_{n}))

whence, in combination with Theorem 3.14,

(3.6) AN_{F/G}((a_{n}) + G) = ∩_{(g_{n})∈G} conv clos ∪_{n} AN_{C_{n}}(a_{n} + g_{n}).

For every k ∈ ℕ, choose an m_{k} ∈ AN_{C_{n}}(a_{n}) and define a sequence (G_{n}^{(k)}) ∈ G by

g_{n}^{(k)} = \{ \begin{matrix}−a_{n} + m_{k}e_{n} & if & n ≤ k − 1 \\ 0 & if & n ≥ k. \end{matrix}

Then

AN_{C_{n}}(a_{n} + g_{n}^{(k)}) = \{ \begin{matrix}\{m_{k}\} ⊆ AN_{C_{n}}(a_{n}) & if & n ≤ k − 1 \\ AN_{C_{n}}(a_{n}) & if & n ≥ k, \end{matrix}

whence

AN_{F/G}((a_{n}) + G) ⊆ ∩_{k=1}^{∞}conv clos ∪_{n≥k} AN_{C_{n}}(a_{n}).

Applying Lemma 3.13 to the sets M_{k} := clos ∪_{n≥k} AN_{C_{n}}(a_{n}) gives

AN_{F/G}((a_{n}) + G) ⊆ conv ∩_{k=1}^{∞} clos ∪_{n≥k} AN_{C_{n}}(a_{n}),

and it is not hard to check that the set on the right hand side coincides with conv lim sup AN_{C_{n}}(a_{n}) which proves one half of the theorem.

For the second half, let m ∈ lim sup AN_{C_{n}}(a_{n}). Then there exist a subsequence (n_{k}) ⊆ ℕ tending to infinity as k → ∞ as well as points m_{k} ∈ AN_{C_{n_{k}}} (a_{n_{k}}) which converge to m as k → ∞. Choose states ϕ_{k} of C_{n_{k}} with ϕ_{k}(a_{n_{k}}) = m_{k}, and let L stand for the linear subspace of F consisting of all sequences

(b_{n}) := α(a_{n}) + β(e_{n}) + (g_{n})

where e_{n} is again the identity of C_{n}, α, β ∈ ℂ, and (g_{n}) runs through the ideal G. If (b_{n}) ∈ L, then the limit lim_{k→∞} ϕ_{k}(b_{n_{k}}) exists. Indeed, it is sufficient to check the existence of this limit for the generating sequences of L, for which one has

ϕ_{k}(a_{n_{k}}) = m_{k} → m, ϕ_{k}(e_{n_{k}}) = 1 → 1 and |ϕ_{k}(g_{n_{k}})| ≤ ∥ϕ_{k}∥ · ∥g_{n_{k}}∥ = ∥g_{n_{k}}∥ → 0.

Thus, we get a linear functional ϕ on L defined by

ϕ((b_{n})) = lim_{k→∞} ϕ_{k}(b_{n_{k}})

which maps the sequences (a_{n}), (e_{n}) and (g_{n}) into m, 1 and 0, respectively, and which is continuous with norm 1:

|ϕ((b_{n}))| = | lim ϕ_{k}(b_{n_{k}})| ≤ sup_{k} |ϕ_{k}(b_{n_{k}})| ≤ sup_{k} ∥b_{n_{k}}∥ ≤ ∥(b_{n})∥_{F}.

Using the Hahn-Banach theorem, one can extend ϕ to a linear functional with norm 1 on all of F. We denote this extension by ϕ again. Since ∥ϕ∥ = ϕ((e_{n})) = 1, this functional is a state on F. Further, the ideal G lies in the kernel of ϕ by construction, so one can define a further functional ψ acting on F/G by

ψ : F/G → ℂ, (c_{n}) + G ↦ ϕ((c_{n})).

Clearly, ψ((e_{n}) + G) = 1 and ψ((a_{n}) + G) = m, and ψ is continuous with norm 1. Thus, ψ is a state of F/G which implies that m ∈ AN_{F/G}((a_{n}) + G) and, consequently,

lim sup AN_{C_{n}}(a_{n}) ⊆ AN_{F/G}((a_{n}) + G).

Finally, since algebraic numerical ranges are convex,

conv lim sup AN_{C_{n}}(a_{n}) ⊆ AN_{F/G}((a_{n}) + G)

which verifies the second half of the theorem. □

Our final goal in this section is consequences of the fractality of the sequence (a_{n}) for the asymptotic behaviour of the partial and uniform limiting sets of the numerical ranges of a_{n}.

Theorem 3.15. Let A be a fractal C^{∗}-subalgebra of F which contains the identity. Then, for every (a_{n}) ∈ A,

lim sup AN_{C_{n}}(a_{n}) = lim inf AN_{C_{n}}(a_{n}).

Proof. The inclusion lim inf AN_{C_{n}}(a_{n}) ⊆ lim sup AN_{C_{n}}(a_{n}) is obvious. For the reverse inclusion recall that the numerical ranges AN_{C_{n}}(a_{n}) are convex, and that the uniform limiting set of convex sets is convex again. Thus, the inclusion lim sup AN_{C_{n}}(a_{n}) ⊆ lim inf AN_{C_{n}}(a_{n}) holds if and only if

conv lim sup AN_{C_{n}}(a_{n}) ⊆ lim inf AN_{C_{n}}(a_{n}).

So, by Theorem 3.12, what one has to prove is that

AN_{F/G}((a_{n}) + G) ⊆ lim inf AN_{C_{n}}(a_{n}) for every (a_{n}) ∈ A.

From the inverse closedness with respect to algebraic numerical ranges we conclude that AN_{F/G}((a_{n}) + G) = AN_{(A+G)/G}((a_{n}) + G), and the third isomorphy theorem implies that

AN_{F/G}((a_{n}) + G) = AN_{A/(A∩G)}((a_{n}) + A ∩ G).

Let m ∈ AN_{A/(A∩G)}((a_{n}) + A ∩ G), and let ϕ be a state of A/(A ∩ G) such that m = ϕ((a_{n}) + A ∩ G) = ϕ(π(a_{n})) where we use the notations of Section 2. Since π is fractal, m = ϕ(π_{η}R_{η}(a_{n})) for every monotonically increasing sequence η. It is obvious that ϕ ∘ π_{η} is a state on A_{η} = R_{η}A. Hence,

m ∈ AN_{A_{η}}(R_{η}(a_{n})) = AN_{F_{η}}(R_{η}(a_{n})),

whence, by Theorem 3.14,

(3.7) m ∈ conv clos ∪_{n} AN_{C_{η(n)}}(a_{η(n)}) for every η.

Now assume there exists an m ∈ AN_{F/G}((a_{n}) + G) which does not belong to the uniform limiting set of the AN_{C_{n}}(a_{n}). Then there is a strongly monotonically increasing sequence η^{∗} ⊆ ℕ such that dist (m, AN_{C_{η^{∗}(n)}}(a_{η^{∗}(n)})) ≥ d > 0 for all n.

Due to the compactness and convexity of AN_{C_{η^{∗}(n)}}(a_{η^{∗}(n)}), there exist points m_{η^{∗}(n)} in AN_{C_{η^{∗}(n)}}(a_{η^{∗}(n)}) such that

|m − m_{η^{∗}(n)}| = dist (m, AN_{C_{η^{∗}(n)}}(a_{η^{∗}(n)})),

and these points m_{η^{∗}(n)} are unique for every n. It is further clear that all numbers m_{η^{∗}(n)} lie in the disk around the origin with radius r := sup ∥a_{n}∥ = ∥(a_{n})∥_{F}. Hence, there is at least one cluster point m^{∗} of the sequence (m_{η^{∗}(n)})_{n∈ℕ}. Given ε > 0, consider those η^{∗}(n) for which the m_{η^{∗}(n)} belong to the ε-neighbourhood U of m^{∗}. These η^{∗}(n) single out an subsequence η_{ε} of η^{∗}. A little thought reveals that then (3.8) dist (m, conv clos ∪_{η_{ε}(n)} AN_{C_{ηε(n)}}(a_{η_{ε}(n)})) ≥ d/2 if only ε is small enough. Since (3.7) holds for every sequence (in particular for the sequence η_{ε}), (3.8) contradicts (3.7). □

Uniform limiting sets of convex sets are convex again. Hence, Theorems 3.12 and 3.15 yield

Corollary 3.16. Let A be a fractal C^{∗}-subalgebra of F which contains the identity. Then, for every sequence (a_{n}) ∈ A,

lim sup AN_{C_{n}}(a_{n}) = lim inf AN_{C_{n}}(a_{n}) = AN_{F/G}((a_{n}) + G).

To illlustrate these results, we consider once again spline Galerkin methods.

Proposition 3.17. Let A be the algebra of spline approximation methods for singular integral operators and (A_{n}) ∈ A. Then

lim sup AN_{ℂ^{n×n}}(A_{n}) = lim sup SN_{ℂ^{n}}(A_{n})

= lim inf AN_{ℂ^{n×n}}(A_{n}) = lim inf SN_{ℂ^{n}}(A_{n})

= AN_{F/G}((A_{n}) + G)

= conv ( ⋃_{s∈ℝ∪{∞}} AN_{L(l^{2})}(W_{s}(A_{n})) ∪ ⋃_{t∈𝕋} AN_{L(l^{2})}(W^{t}(A))) .

Proof. The identities in the first lines are consequences of the preceding corollary and of (3.3). The last identity can be checked by repeating the arguments of the proof of Theorem 3.14, where an analogous result for sequences instead of pairs of operators is derived, and by taking into account the convexity of algebraic numerical ranges. □

4. On non-fractal sequences

The main goal of this concluding section is to prove that every approximation sequence possesses a fractal subsequence. For, we need some results on fractality of self-adjoint sequences.

Fractality of self-adjoint sequences. If (A_{n}) is a self-adjoint fractal approximation sequence, then

(4.1) lim sup σ(A_{n}) = lim inf σ(A_{n})

as we know from Theorem 3.6(b). We will see now that, conversely, (4.1) is the only obstruction for a self-adjoint bounded sequence to be fractal.

Theorem 4.1. Let F be as above and let (a_{n}) ∈ F be a self-adjoint sequence. Then (a_{n}) is fractal if and only if the equality (4.1) holds.

Proof. The ’only if’-part is Theorem 3.6(b). For the reverse conclusion suppose that (4.1) holds. Let A denote the smallest closed subalgebra of F which contains the sequence (a_{n}) and the identity sequence (e_{n}) of F. Further, given a monotonically increasing sequence η, write A_{η} for the algebra R_{η}A. We claim that (4.2) A/(A ∩ G) is isomorphic to A_{η}/(A_{η} ∩ G_{η})

with the isomorphism given by

(b_{n}) + (A ∩ G) ↦ (b_{η(n)}) + (A_{η} ∩ G_{η}). (4.3)

To get the claim recall that A/(A∩G) and A_{η}/(A_{η}∩G_{η}) are isomorphic to (A+G)/G and (A_{η} + G_{η})/G_{η}, respectively. The latter algebras are singly generated by their elements (a_{n})+G and (a_{η(n)})+G_{η}, and the spectra of these cosets are lim sup σ(a_{n}) and lim sup σ(a_{η(n)}) due to Corollary 3.5, respectively. The assumption (4.1) guarantees that these spectra coincide; hence, by the Gelfand-Naimark theorem for singly generated C^{∗}-algebras, the isomorphy (4.2) follows.

Let now π stand for the canonical homomorphism from A onto A/(A∩G) and let η be a monotonically increasing sequence. Then, evidently, π = π_{η}R_{η} = φ_{η}ψ_{η}R_{η} where ψ_{η} is the canonical homomorphism from A_{η} onto A_{η}/(A_{η} ∩ G_{η}) and where φ_{η} if the inverse of the isomorphism (4.3). Hence, π is fractal. □

As a first application of the previous theorem we derive a fractality result for the sequence of the finite sections of a self-adjoint operator. Observe in this connection that the spectrum of a self-adjoint operator A for which the finite section method (P_{n}AP_{n}) is fractal can be as complicated as possible: Given an arbitrary compact subset K of the real line, choose a dense subsequence (k_{i})_{i=1}^{∞} in K, and consider the operator A = diag (k_{1}, k_{2}, k_{3}, …). Then

lim sup σ(P_{n}AP_{n}) = lim inf σ(P_{n}AP_{n}) = lim σ(P_{n}AP_{n}) = K.

Theorem 4.2. If A ∈ L(H) is a self-adjoint operator with connected spectrum, then the sequence (P_{n}AP_{n}) is fractal.

Proof. Let λ ∈ ℝ. If the operator A − λI is invertible, then it is either positively or negatively definite. Clearly, definiteness implies the stability of the sequence (P_{n}(A − λI)P_{n}). Conversely, if the sequence (P_{n}(A − λI)P_{n}) is stable, then the operator A − λI is invertible due to Polski’s theorem ([6], Theorem 1.2). Thus,

σ(A) = σ_{F/G}((P_{n}AP_{n}) + G). (4.4)

Further we know from Corollary 3.5 that

σ_{F/G}((P_{n}AP_{n}) + G) = lim sup σ(P_{n}AP_{n}). (4.5)

Finally, we claim that σ(A) ⊆ lim inf σ(P_{n}AP_{n}). (4.6)

Once (4.6) is verified, (4.4) - (4.6) give lim sup σ(P_{n}AP_{n}) = lim inf σ(P_{n}AP_{n}) which is equivalent to the fractality of (P_{n}AP_{n}) by Theorem 4.1.

It remains to check (4.6) where we follow arguments from [1]. Suppose λ is a real number which is not in lim inf σ(P_{n}AP_{n}). We claim that then A − λI is invertible.

Let H_{n} denote the range of P_{n}. Since λ ∈ ℝ\lim inf σ(P_{n}AP_{n}), there is an ε > 0 as well as an infinite subsequence η of ℕ such that

σ(P_{n}AP_{n}) ∩ (λ − ε, λ + ε) = ∅ for all n ∈ η.

Thus, the distance of σ(P_{n}AP_{n}) to λ is at least ε, which shows that the operators P_{n}AP_{n} − λI|_{H_{n}} are invertible and that their inverses are uniformly bounded:

sup_{n∈η} ∥(P_{n}AP_{n} − λI|_{H_{n}})^{−1}∥ ≤ 1/ε. (4.7)

The strong convergence of (P_{n}AP_{n} − λI) to A − λI together with the uniform boundedness (4.7) readily implies the strong convergence of the sequence of the inverses ((P_{n}AP_{n} − λI)^{−1}). Write B for the strong limit of this sequence. Letting n go to infinity in

(P_{n}AP_{n} − λI|_{H_{n}})^{−1} (P_{n}AP_{n} − λI|_{H_{n}}) = (I|_{H_{n}})

yields B(A − λI) = I. Analogously, (A − λI)B = I whence the invertibility of A − λI. □

Existence of fractal subsequences. There are simple examples such as

A = diag (( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , …) ∈ L(l^{2}) (4.8)

which show that the finite section method for self-adjoint operators is not necessarily fractal: For A as in (4.8) one has

σ(P_{2n}AP_{2n}) = {−1, 1} ≠ σ(P_{2n+1}AP_{2n+1}) = {−1, 1, 0}

for all n. In the case at hand, it turns out that the (non-fractal) sequence (P_{n}AP_{n}) possesses a fractal subsequence (formed by the matrices of even order). So one might ask whether every (self-adjoint or not) approximation sequence possesses a fractal subsequence. For a long time I conjectured that the answer is no and, motivated by Theorem 4.2, I moreover conjectured that if (A_{n}) is a completely non-fractal sequence (i.e. if no infinite subsequence of (A_{n}) is fractal), then σ(A) is a set of Cantor type. Then T. Ehrhardt draw my attention to the connection between the coincidence of the partial and the uniform limiting set on the one hand and the convergence with respect to the Hausdorff metric on the other hand, as well as to Theorem 3.3, which in combination with Theorem 4.1 gave a surprisingly simple proof of the fact that the converse of my conjecture is true.

Theorem 4.3. Let F be as above. Every self-adjoint sequence (a_{n}) ∈ F possesses a fractal subsequence.

Proof. Consider the sets M_{n} := σ(a_{n}). By Theorem 3.3, there exists a subsequence (M_{η(n)})_{n≥1} of (M_{n}) which converges with respect to the Hausdorff metric. Hence,

lim sup_{n→∞} (M_{η(n)}) = lim inf_{n→∞} (M_{η(n)}),

and the sequence (a_{η(n)})_{n≥1} is fractal due to Theorem 4.1. □

Moreover, based on this result, it is now possible to verify the existence of a fractal subsequence for every (not necessarily self-adjoint) sequence. Actually, a little bit more can be shown.

Theorem 4.4. Let F be as above, and let A be a separable C^{∗}-subalgebra of F. Then there exists a sequence η ⊆ ℕ such that the algebra A_{η} = R_{η}A ⊆ F_{η} is fractal.

Since every finitely generated C^{∗}-algebra is separable, this result immediately implies:

Theorem 4.5. Every sequence in F possesses a fractal subsequence.

One cannot expect that Theorem 4.4 holds for arbitrary C^{∗}-subalgebras of F; for example it is certainly not valid for l^{∞}. On the other hand, there exist nonseparable but fractal algebras; the algebra A related with spline Galerkin methods for singular integral operators can serve as an example.

The proof of Theorem 4.4 will make use of the following equivalent characterization of fractal algebras which is a simple consequence of the third isomorphy theorem.

Lemma 4.6. A C^{∗}-algebra A ⊆ F is fractal if and only if the restriction of the canonical homomorphism π : F → F/G onto A is fractal.

Here is the proof of Theorem 4.4.

Proof. Let A be a separable C^{∗}-subalgebra of F with a countable dense subset ((a_{n}^{(k)})_{n≥1})_{k≥1}. Denote by b_{n}^{(2k)} and b_{n}^{(2k+1)} the real and the imaginary part of a_{n}^{(k)}, respectively, and write B ⊆ A for the set of all sequences (b_{n}^{(k)})_{n≥1} with k ≥ 1 and D ⊆ A for the set of all difference sequences (b_{n}^{(k)})_{n≥1} − (b_{n}^{(l)})_{n≥1} with k, l ≥ 1. The set B ∪ D is countable, and each of its elements is self-adjoint.

Let ((d_{n}^{(k)})_{n≥1})_{k≥1} be any numeration of the elements of B ∪ D. By Theorem 4.3, every sequence (d_{n}^{(k)})_{n≥1} possesses a fractal subsequence. By employing a standard diagonalization process, we will construct a sequence η ∈ ℕ such that the sequence (d_{η(n)}^{(k)})_{n≥1} is fractal for every k ≥ 1.

Let η_{1} ⊆ ℕ be a sequence such that (d_{η_{1}(n)}^{(1)})_{n≥1} is a fractal sequence and, for

every k ≥ 2, choose a subsequence η_{k} of η_{k−1} such that (d_{η_{k}(n)}^{(k)})_{n≥1} is a fractal sequence. Then define η by η(n) := η_{n}(n). The sequence η coincides (with possible exception of at most finitely many entries) with a subsequence of η_{k} for every k. Hence, every sequence in

D_{η} := R_{η}D = {(d_{η(n)}^{(k)})_{n≥1}, k ≥ 1}

is fractal.

We claim that the algebra A_{η} := R_{η}A is fractal. What we have to verify is that, given a subsequence μ of η, there is a homomorphism π̂_{μ} such that

π̂|_{A_{η}} = π̂_{μ}R_{μ}|_{A_{η}}

where π̂ is the canonical homomorphism from F_{η} onto F_{η}/G_{η} (Lemma 4.6). Observe that the set of all sequences

(a_{η(n)}^{(k)})_{n≥1} = (b_{η(n)}^{(2k)})_{n≥1} + i (b_{η(n)}^{(2k+1)})_{n≥1}

with k = 1, 2, … is dense in A_{η}. So, without loss of generality, we can assume η = ℕ in what follows. That is, we will have to deal with the following situation:

A is a C^{∗}-subalgebra of F with countable dense subset ((a_{n}^{(k)})_{n≥1})_{k≥1} such that each of the self-adjoint sequences (b_{n}^{(k)})_{n≥1} and (b_{n}^{(k)})_{n≥1} − (b_{n}^{(l)})_{n≥1} with k, l ≥ 1 is fractal. We have to show that A is fractal, i.e. given a subsequence μ of ℕ, we have to define a homomorphism π_{μ} such that π|_{A} = π_{μ}R_{μ}|_{A}, π referring to the canonical homomorphism from F onto F/G.

Write B and D in place of B_{η} and D_{η}. Let μ be a subsequence of ℕ. We start with defining the mapping π_{μ} on the set of the self-adjoint elements of A_{μ}. So let (a_{n}) ∈ A and assume (a_{μ(n)})_{n≥1} is self-adjoint.

Claim 1. There is a sequence ((c_{n}^{(k)})_{n≥1})_{k≥1} ⊆ B such that

∥(a_{μ(n)})_{n≥1} − (c_{μ(n)}^{(k)})_{n≥1}∥_{F_{μ}} → 0 as k → ∞. (4.9)

Indeed, write a_{n} as Re a_{n} + iIm a_{n}. Since ((a_{n}^{(k)})_{n≥1})_{k≥1} is a dense subset of A, we can approximate the sequence (Re a_{n})_{n≥1} as closely as desired by sequences of the form (Re a_{n}^{(k)})_{n≥1} = (b_{n}^{(2k)})_{n≥1}. Then, clearly, the sequence (Re a_{μ(n)})_{n≥1} can be approximated as closely as desired by sequences of the form (b_{μ(n)}^{(2k)})_{n≥1} ∈ B_{μ}, and since Re a_{μ(n)} = a_{μ(n)} by hypothesis, this gives the claim.

Now, given a self-adjoint sequence (a_{μ(n)}) ∈ A_{μ}, choose and fix a sequence ((c_{μ(n)}^{(k)})_{n≥1})_{k≥1} ⊆ B_{μ} with property (4.9) and, for every k, choose a sequence

(ĉ_{n}^{(k)}) ∈ B with R_{μ}(ĉ_{n}^{(k)}) = (ĉ_{μ(n)}^{(k)}).

Let A^{(k)} refer to the smallest C^{∗}-subalgebra of A which contains the sequence (ĉ_{n}^{(k)})_{n≥1}. The algebras A^{(k)} are fractal by construction, hence (Lemma 4.6), there are homomorphisms π_{μ}^{(k)} such that

π|_{A^{(k)}} = π_{μ}^{(k)}R_{μ}|_{A^{(k)}} for every k.

In particular,

π_{μ}^{(k)}(ĉ_{μ(n)}^{(k)}) = (ĉ_{n}^{(k)}) + G.

Moreover, the coset (ĉ_{n}^{(k)}) + G turns out to be independent of the choice of the ’representative’ (ĉ_{n}^{(k)}) of the sequence (ĉ_{μ(n)}^{(k)}):

Claim 2. Let (c_{n}), (d_{n}) ∈ B be sequences with c_{μ(n)} = d_{μ(n)} for every n. Then (c_{n}) + G = (d_{n}) + G.

Indeed, the sequence (c_{n} − d_{n}) belongs to D and is, thus, fractal by construction. By Theorem 2.11 (b), the limit lim ∥c_{n} −d_{n}∥ exists and is equal to ∥(c_{n} −d_{n})+G∥. On the other hand, this limit is zero since infinitely many of the differences c_{n} −d_{n} are zero by assumption. Hence, (c_{n} − d_{n}) ∈ G, proving our claim.

Thus, knowing only the subsequences (c_{μ(n)}^{(k)})_{n≥1}, one can rediscover the cosets

(c_{n}^{(k)}) + G uniquely.

Claim 3. As k → ∞, the cosets (c_{n}^{(k)}) + G converge in F/G.

This follows from

∥(c_{n}^{(k)}) + G − (c_{n}^{(l)}) + G∥_{F/G} = ∥(c_{μ(n)}^{(k)}) + G_{μ} − (c_{μ(n)}^{(l)}) + G_{μ}∥_{F_{μ}/Gμ}
(4.10)  ≤ ∥(c_{μ(n)}^{(k)}) − (c_{μ(n)}^{(l)})∥_{F_{μ}},

where the equality is a consequence of the fractality of the sequence (c_{n}^{(k)} − c_{n}^{(l)}) ∈ D, and from the convergence of the sequences (c_{μ(n)}^{(k)})_{n≥1} to (a_{μ(n)})_{n≥1}. Hence

((c_{n}^{(k)}) + G)_{k≥1} is a Cauchy sequence and thus convergent, which verifies Claim 3.

Let (c_{n}) + G denote the limit of the sequence ((c_{n}^{(k)}) + G)_{k≥1}.

Claim 4. The coset (c_{n}) + G does not depend on the choice of the sequence ((c_{μ(n)}^{(k)})_{n≥1})_{k≥1} which approximates (a_{μ(n)}).

Indeed, consider besides ((c_{μ(n)}^{(k)})_{n≥1})_{k≥1} another sequence ((d_{μ(n)}^{(k)})_{n≥1})_{k≥1} ⊆ B_{μ} which also converges to (a_{μ(n)}) and which generates (in the same way as the sequence ((c_{μ(n)}^{(k)})) does) a coset (d_{n}) + G. Then we have as in (4.10)

∥(c_{n}) + G − (d_{n}) + G∥_{F/G} = lim_{k→∞} ∥(c_{n}^{(k)}) + G − (d_{n}^{(k)}) + G∥_{F/G}
(4.11)  ≤ lim sup_{k→∞} ∥(c_{μ(n)}^{(k)}) − (d_{μ(n)}^{(k)})∥_{F_{μ}}

(recall that the sequence (c_{n}^{(k)} − d_{n}^{(k)}) is fractal by construction). Since both sequences ((c_{μ(n)}^{(k)})_{n≥1})_{k≥1} and ((d_{μ(n)}^{(k)})_{n≥1})_{k≥1} have the same limit as k → ∞, the right hand side of (4.11) tends to zero which gives the claim.

Thus, every self-adjoint sequence (a_{μ(n)}) ∈ A_{μ} corresponds to a uniquely determined coset (c_{n}) + G which we denote by π_{μ}(a_{μ(n)}).

Claim 5. For every self-adjoint sequence (a_{n}) ∈ A, and every sequence μ,

π_{μ}(a_{μ(n)})_{n≥1} = (a_{n}) + G. (4.12)

Indeed, possibly there are several sequences ((c_{μ(n)}^{(k)})_{n≥1})_{k≥1} which converge to (a_{μ(n)}). But among these sequences there is, by assumption, at least one such that (c_{n}^{(k)}) → (a_{n}) in F. For this special sequence, one has

(c_{n}^{(k)}) + G → (a_{n}) + G in F/G.

The limit lim_{k→∞}((c_{n}^{(k)}) + G) is, as we have seen in Claim 4, independent of the choice of (c_{n}^{(k)}). Hence,

π_{μ}(a_{μ(n)}) = (a_{n}) + G,

which settles the construction of π_{μ} on the set of the self-adjoint sequences of A_{μ}.

If (a_{μ(n)})_{n≥1} is an arbitrary (not necessarily self-adjoint) sequence in A_{μ}, then define

π_{μ}(a_{μ(n)}) := π_{μ}(Re a_{μ(n)}) + i π_{μ}(Im a_{μ(n)}).

Due to (4.12),

π_{μ}(a_{μ(n)}) = (Re a_{n}) + G + i ((Im a_{n}) + G) = (a_{n}) + G,

whence π_{μ}R_{μ}|_{A} = π|_{A} as desired. □

Acknowledgements

Dedicated to the memory of Siegfried Prössdorf.

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Technische Universität Darmstadt Fachbereich Mathematik Schlossgartenstrasse 7 D 64289 Darmstadt

1991 Mathematics Subject Classification: Primary 65R20; Secondary 47A10, 47A12

Submitted: Date inserted by the Editor