Algebras of approximation sequences: Finite sections of band-dominated operators

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

Abstract

We develop the stability theory for the finite section method for general band-dominated operators on l^{p} spaces over ℤ^{k}. The main result says that this method is stable if and only if each member of a whole family of operators – the so-called limit operators of the method – is invertible and if the norms of these inverses are uniformly bounded.

1 Introduction

Let K and N be positive integers and p ≥ 1 be a real number, and let ∥.∥_{C}^{N} be an arbitrary norm on ℂ^{N}. By l^{p}(ℤ^{K}, ℂ^{N}) resp. l^{∞}(ℤ^{K}, ℂ^{N}) we denote the Banach spaces of all functions f : ℤ^{K} → ℂ^{N} such that

∥f∥_{l^{p}}^{p} := ∑_{x∈ Z Z^{K}} ∥f(x)∥_{C^{N}}^{p} < ∞

resp.

∥f∥_{l^{∞}} := sup_{x∈ Z Z^{K}} ∥f(x)∥_{C}^{N} < ∞,

and l^{0}(ℤ^{K}, ℂ^{N}) is the closed subspace of l^{∞}(ℤ^{K}, ℂ^{N}) which consists of all functions f with

lim_{x→∞} ∥f(x)∥_{C}^{N} = 0.

Moreover, l_{N×N}^{∞} := l^{∞}(ℤ^{K}, ℂ^{N×N}) stands for the Banach algebra of all functions f : ℤ^{K} → ℂ^{N×N} such that

∥f∥_{l_{N×N}^{∞}} := sup_{x∈ Z Z^{K}} ∥f(x)∥_{C}^{N×N} < ∞

where ∥.∥_{C}^{N×N} is the matrix norm on ℂ^{N×N} which is induced by the norm chosen on ℂ^{N}. Finally, let E^{K} refer to one of the spaces l^{p}(ℤ^{K}, ℂ^{N}) with 1 ≤ p < ∞ or p = 0. The Banach algebra of all linear bounded operators on E^{K} will be denoted by L(E^{K}).

∗Supported by the German Research Foundation under Grant Nr. 436RUS17/67/98.

The basic objects of the present paper – band and band-dominated operators – are constituted by shift and multiplication operators. For m ∈ ℤ^{K}, the shift operator V_{m} on E^{K} is defined by

(V_{m}f)(x) = f(x − m), x ∈ ℤ^{K}.

Clearly, V_{m} ∈ L(E^{K}) and ∥V_{m}∥ = 1. Further, every function a ∈ l_{N×N}^{∞} gives rise to a multiplication operator aI acting on E^{K} via

(af)(x) = a(x)f(x), x ∈ ℤ^{K}.

Evidently, aI ∈ L(E^{K}) and ∥aI∥ = ∥a∥_{l_{N×N}^{∞}}.

Definition 1 A band operator on E^{K} is a finite sum of the form ∑ a_{m}V_{m} where m ∈ ℤ^{K} and a_{m} ∈ l_{N×N}^{∞}. A band-dominated operator on E^{K} is the uniform limit of a sequence of band operators.

To justify this notation observe that, in case K = N = 1 and with respect to the standard basis of E^{1}, band operators are nothing but usual band matrices. Observe also that the class of band operators is independent of the concrete space E^{K}, whereas the class of band-dominated operators depends on p essentially. It is not hard to check that the band operators form an algebra which implies that class of all band-dominated operators constitutes a closed subalgebra of L(E^{K}).

Let Ω be a subset of ℝ^{K} which has the point 0 ∈ ℝ^{K} in its interior. The characteristic function of Ω is the function χ_{Ω} which is 1 on x ∈ Ω and 0 if x ∉ Ω. We denote by P_{n} the operator of multiplication by the restriction of the function χ_{nΩ} onto ℤ^{K}. Clearly, P_{n} is a projection operator on E^{K}, and the condition 0 ∈ int Ω guarantees that the operators P_{n} converge strongly on E^{K} to the identity operator.

For the approximate solution of equations Ax = y with A a band-dominated operator on E^{K} by the finite section method, we replace the equation Ax = y by the sequence of the approximate equations

P_{n}AP_{n}x_{n} = P_{n}y,  n ≥ 1.  (1)

The crucial question is whether the sequence (P_{n}AP_{n}) is stable, i.e. whether the operators P_{n}AP_{n} : Im P_{n} → Im P_{n} are invertible for sufficiently large n and whether the norms of their inverses are uniformly bounded. The stability of (P_{n}AP_{n}) together with the strong convergence of the P_{n} and with the invertibility of A imply the applicability of the finite section method in the sense that, for every y ∈ E and n large enough, the equations (1) possess unique solutions x_{n} which converge to a solution of Ax = y as n → ∞.

It is often convenient to write the equations (1) in the form

(P_{n}AP_{n} + Q_{n})X_{n} = y where Q_{n} := I − P_{n}.  (2)

Clearly, the solutions X_{n} are related to the solutions x_{n} of (1) by x_{n} = P_{n}X_{n} and X_{n} := x_{n} + Q_{n}y. It is also evident that the sequence (P_{n}AP_{n}) is stable if and only if the sequence (P_{n}AP_{n} + Q_{n}) has this property. Moreover, on defining P_{n} := 0 for n < 0, the stability of (P_{n}AP_{n} + Q_{n})_{n≥0} becomes equivalent to the stability of (P_{n}AP_{n} + Q_{n})_{n∈ Z Z}.

There is a vast literature dealing with the finite section method for various concrete classes of band-dominated operators, see [2, 5, 6, 8]. The only paper (we know) which deals with the finite section method of general band-dominated operators is Gohberg/Kaashoek [3] where the case K = 1 is considered. In this paper, the problem of the stability of the sequence (P_{n}AP_{n}) is reduced to the stability problem for two related simpler sequences.

The formulation of our stability criterion is in Section 2. Its proof is surprisingly simple and rests on an application of the results of [8] on the Fredholmness of band-dominated operators. In Section 3, we will consider some special choices of Ω and make the stability conditions more explicite. The concluding forth section is devoted to a discussion of the fractality properties of the finite section method for band-dominated operators. Fractality is a property of an approximation sequence which guarantees that certain limiting processes behave more uniform than expected (see [9, 10]). For example, the condition numbers of a stable approximation method (A_{n}) are bounded:

lim sup ∥A_{n}∥ ∥A_{n}^{−1}∥ < ∞,

but if the sequence (A_{n}) is in addition fractal, then this limes superior is actually a limes. It turns out that the finite section method for band-dominated operators is non-fractal in an essential manner. So, the finite section method seems to be the first instance of an non-fractal approximation method where complete necessary and sufficient stability criteria are available. In general, the non-fractality of an approximation method raises serious difficulties (some concrete examples are mentioned in Section 4). The success of the approach developed here gives some hope that these techniques (in particular the concept of the limit operators) might be useful also for analyzing other, in the moment still untractable approximation sequences.

Let us finally mention that, if the strong convergence is replaced by a weaker one (cp. [7], Sections 4.32 - 4.43) then analogous results can be derived also for the finite sections of band dominated operators on l^{∞}(ℤ^{K}, ℂ^{N}).

2 Stability of the finite section method

We agree upon writing every vector x ∈ ℤ^{K+1} as x = (x′, x_{K+1}) ∈ ℤ^{K} × ℤ. The Banach space E^{K+1} can be viewed as the direct sum over all m ∈ ℤ of its closed subspaces

E_{m}^{K+1} := {f ∈ E^{K+1} : f(x) = 0 if x_{K+1} ≠ m}.

Further, for every m ∈ ℤ, consider the operator of restriction

R_{m} : E^{K+1} → E^{K}, (R_{m}f)(x′) := f(x′, m)

as well as the operator of embedding

S_{m} : E^{K} → E^{K+1}, (S_{m})(x′, n) := \{ \begin{matrix}f(x′) & if n = m \\ 0 & if n ≠ m. \end{matrix}

The operator T_{m} := S_{m}R_{m} is the projection (in case p = 2 the orthogonal projection) of E^{K+1} onto E_{m}^{K+1} parallel to the direct sum of all E_{r}^{K+1} with r ≠ m, whereas R_{m}S_{m} acts on E^{K} as the identity operator. In particular, each of the spaces E_{m}^{K+1} is isometrically isomorphic to E^{K}.

We say that an operator A ∈ L(E^{K+1}) is reduced by the family (T_{m})_{m∈ℤ} if

AT_{m} = T_{m}A for all m ∈ ℤ.

The class of all of these operators will be denoted by L_{red}(E^{K+1}). Clearly, this class is a closed, unital, and inverse closed subalgebra of L(E^{K+1}).

After these preparations, we turn our attention to approximation sequences for operators on E^{K}. By F we denote the set of all bounded sequences (A_{n})_{n∈ Z Z} of operators A_{n} ∈ L(E^{K}) . To every sequence (A_{n}) ∈ F, there corresponds an operator Op(A_{n}) on E^{K+1} via

Op(A_{n})f := ∑_{m∈ Z Z}S_{m}A_{m}R_{m}f.

Lemma 1 The mapping Op is an isometrical algebra isomorphism from F onto L_{red}(E^{K+1}).

The proof is rather simple. From

Op(A_{n})T_{m} = ∑ S_{n}A_{n}R_{n}T_{m}
= S_{m}A_{m}R_{m}T_{m} = T_{m}S_{m}A_{m}R_{m} = T_{m} ∑ S_{n}A_{n}R_{n} = Op(A_{n}),

it follows that (T_{m}) reduces every operator Op(A_{n}). Conversely, if an operator A is reduced by (T_{m}), then (A_{n}) with A_{n} := R_{n}AS_{n} is a sequence in F with

Op(A_{n}) = ∑ S_{n}A_{n}R_{n} = ∑ S_{n}R_{n}AS_{n}R_{n} = ∑ T_{n}AT_{n} = A.

The isometry of the mapping Op is evident.

The next lemma settles the connection between stability and Fredholmness for a class of special sequences in F which includes all finite section sequences (P_{n}AP_{n}+ Q_{n})_{n∈ Z Z} for compact sets Ω. Let F_{c} refer to the set of all sequences (A_{n}) ∈ F with the property that A_{m} − A_{n} is compact for all m, n.

Lemma 2 A sequence (A_{n}) ∈ F_{c} is stable if and only if the operator Op(A_{n}) is Fredholm.

Proof. If (A_{n}) ∈ F is a stable sequence, then there are sequences (B_{n}), (G_{n}) and (H_{n}) in F such that A_{n}B_{n} = I + G_{n} and B_{n}A_{n} = I + H_{n} for all n and that G_{n} = H_{n} = 0 for all sufficiently large n. In case (A_{n}) belongs to F_{c}, one can say more. Namely, if one of the operators A_{n} is invertible, then all of these operators are Fredholm operators (of index 0). Hence, the operators G_{n} and H_{n} can be chosen to be of finite rank. Then Op(G_{n}) and Op(H_{n}) are finite rank, hence compact, operators on E(ℤ^{K+1}, ℂ^{N}), whence the Fredholmness of Op(A_{n}).

Let, conversely, Op(A_{n}) be a Fredholm operator. Then there are an operator B ∈ L(E^{K+1}) as well as compact operators G and H on the same space such that B Op(A_{n}) = I + G and Op(A_{n})B = I + H. Multiplying these equalities by R_{n} from the left and by S_{n} from the right hand side and taking into account that Op(A_{n}) is reduced by (T_{n})_{n∈ Z Z}, we obtain the identities R_{n}BS_{n}A_{n} = I + R_{n}GS_{n} and A_{n}R_{n}BS_{n} = I + R_{n}HS_{n}. The sequences (R_{n}GS_{n}) and (R_{n}HS_{n}) converge to 0 in the operator norm as n → ±∞ which is a consequence of the compactness of G and H and of the strong convergence to 0 of the projections T_{n}. Thus, (A_{n}) is a stable sequence.

In case of band-dominated operators on l^{∞}(ℤ^{K}, ℂ^{N}) one has to modify the latter arguments appropriately. We let ⊕_{m∈ Z Z} l^{∞}(ℤ^{K}, ℂ^{N}) stand for the linear space of all bounded sequences (x_{m})_{m∈ Z Z}) of vectors x_{m} ∈ l^{∞}(ℤ^{K}, ℂ^{N}), provided with norm ∥(x_{m})∥ := ∑_{m} ∥x_{m}∥. The mapping

l^{∞}(ℤ^{K+1}, ℂ^{N}) → ⊕_{m∈ Z Z} l^{∞}(ℤ^{K}, ℂ^{N}), x ↦ (R_{m}x)_{m∈ Z Z}

is an isometry between these spaces. To every operator A on l^{∞}(ℤ^{K+1}, ℂ^{N}), we associate the operator S(A) ∈ L(⊕_{m∈ Z Z} l^{∞}(ℤ^{K}, ℂ^{N})) via

S(A)(x_{m}) := (R_{m}AS_{m}x_{m}).

It is clear that ∥S(A)∥ ≤ ∥A∥, but observe that S is not an algebra homomorphism.

Let now A ∈ L_{red}(l^{∞}(ℤ^{K+1}, ℂ^{N})) be a Fredholm operator. Thus, there exist operators B, C, K and L in L(l^{∞}(ℤ^{K+1}, ℂ^{N})) such that K and L are of finite rank and

BA = I + K, AC := I + L.

Then we have

S(B)S(A) = (R_{m}BS_{m}R_{m}AS_{m}) = (R_{m}BT_{m}AS_{m})
= (R_{m}BAS_{m}) = (R_{m}(I + K)S_{m}) = S(I) + S(K).

We claim that only a finite number of the operators R_{m}KS_{m} is not zero. Suppose that infinitely many of these operators is non-zero, say R_{1}KS_{1}, R_{2}KS_{1}2, . . ..

Then, for every m ≥ 1, there exist vectors y_{m} ∈ l^{∞}(ℤ^{K}, ℂ^{N})) such that y_{m} ≠ 0 and y_{m} ∈ Im R_{m}K. Accordingly, the vectors Y_{m} := S_{m}y_{m} are non-zero, and Y_{m} ∈ Im T_{m} ∩ Im K. Since T_{m}T_{n} = 0 whenever m ≠ n, the set of all Y_{m} is linearly independent. Hence, the linear span of all Y_{m} is an infinite-dimensional subspace of the range of K, which is impossible.

Repeating these considerations with L in place of K we get the existence of an n_{0} such that

R_{m}KS_{m} = R_{m}LS_{m} = 0 for all |m| ≥ n_{0}.

Define a new operator Â ∈ l^{∞}(ℤ^{K+1}, ℂ^{N})) by

Â := A + ∑_{|m|<n_{0}}(T_{m} − T_{m}AT_{m}),

and define B̂ and Ĉ analogously. Then S(B̂)S(Â) = S(Â)S(Ĉ) = S(I), i.e. S(Â) is an invertible operator. This clearly implies that all operators R_{m}ÂS_{m} are invertible and that the norms of their inverses are uniformly bounded. Consequently, the operators R_{m}AS_{m} are invertible for all |m| ≥ n_{0}, and the norms of their inverses are uniformly bounded. If, in particular, A = Op(A_{n}), this implies the stability of the sequence (A_{n}) because of R_{n}AS_{n} = A_{n}.

So we are left with the problem of the Fredholmness of the operator Op(A_{n}). It turns out that, if (A_{n}) is the sequence of the approximation operators in (2), i.e. if A_{n} = P_{n}AP_{n} +Q_{n} with a band-dominated operator A on E^{K}, then Op(A_{n}) is a band-dominated operator on E^{K+1}. More generally, let A stand for the smallest closed subalgebra of the algebra F which contains all constant sequences (A) with A being band-dominated on E^{K} and which contains the sequence (P_{n}).

Lemma 3 Let (A_{n}) ∈ A. Then (A_{n}) belongs to F_{c}, and Op(A_{n}) is a banddominated operator on E^{K+1}.

Proof. The algebra A can be characterized as the smallest closed subalgebra of F which contains all constant sequences (V_{m}) and (aI) with m ∈ ℤ^{K} and a ∈ l_{N×N}^{∞} as well as the sequence (P_{n}). It suffices to check both assertions of the lemma for these generating sequences in place of (A_{n}). The first assertion is evident. It is further clear that Op(aI) and Op(P_{n}) are multiplication operators on E^{K+1} and that Op(V_{m}) is the shift operator V_{(m,0)}. Hence, in any case, Op(A_{n}) is band-dominated.

Next we recall the criterion for the Fredholmness of band-dominated operators derived in [8].

Let A ∈ L(E^{K+1}), and let h = (h(n))_{n=0}^{∞} ⊆ ℤ^{K+1} be a sequence tending to infinity. The operator A_{h} ∈ L(E^{K+1}) is called a limit operator of A with respect to h if

lim_{n→∞} ∥(V_{−h(n)}AV_{h(n)} − A_{h})χI∥_{L(E^{K+1})} = 0

and

lim_{n→∞} ∥χ(V_{−h(m)}AV_{h(m)} − A_{h})∥_{L(E^{K+1})} = 0

for every function χ ∈ l_{N×N}^{∞} with finite support.

It is easy to see that an operator can possess only one limit operator with respect to a given sequence (which justifies the notation A_{h} for the limit operator). Observe also that, if g is an (infinite) subsequence of h and if A_{h} exists, then A_{g} exists, too, and A_{g} = A_{h}. The set of all limit operators of a given operator A will be denoted by σ_{op}(A).

Theorem 1 Let A be a band-dominated operator on E^{K+1}. Then A is a Fredholm operator if and only if all limit operators of A are invertible and if the norms of their inverses are uniformly bounded.

In particular, this result applies to the operators Op(A_{n}) with (A_{n}) ∈ A and, hence, yields a stability result for the sequence (A_{n}). Since this sequence consists of operators acting on E^{K}, one would prefer a criterion for the stability of (A_{n}) in terms of operators on E^{K}, not on E^{K+1}. The following lemma shows that the stability criterion for sequences in A can be formulated in the desired way.

Lemma 4 Let A ∈ L_{red}(E^{K+1}) and let A_{h} be a limit operator of A. Then A_{h} belongs to L_{red}(E^{K+1}), too. In particular, A_{h} is invertible if and only if all operators R_{m}A_{h}S_{m} with m ∈ ℤ are invertible in E^{K} and if the norms of their inverses are uniformly bounded.

Proof. The only thing which needs a proof is that A_{h} is reduced by the family (T_{m}) if A is so. Indeed, for every k ∈ ℤ and n ∈ ℕ, one has

T_{k}V_{−h(n)}AV_{h(n)} = T_{k}V_{−(h(n)′, m(n))}AV_{(h(n)′, m(n))}
= V_{−(h(n)′, m(n))}T_{k+m(n)}AV_{(h(n)′, m(n))}
= V_{−(h(n)′, m(n))}AT_{k+m(n)}V_{(h(n)′, m(n))}
= V_{−(h(n)′, m(n))}AV_{(h(n)′, m(n))}T_{k}

whence T_{k}A_{h} = A_{h}T_{k}.

Given (A_{n}) ∈ A, let σ_{stab}(A_{n}) stand for the set of all operators R_{m}A_{h}S_{m} where m ∈ ℤ and where A_{h} runs through the collection of all limit operators of Op(A_{n}). Summarizing the previous results, we arrive at the following stability criterion for (A_{n}).

Theorem 2 A sequence (A_{n}) ∈ A is stable if and only if all operators in σ_{stab}(A_{n}) are invertible and if the norms of their inverses are uniformly bounded.

3 The finite section method for band-dominated operators on ℤ^{1} and ℤ^{2}

In this section, we are going to compute the operator-valued spectrum σ_{stab}(A_{n}) for the finite section sequences (A_{n}) = (P_{n}AP_{n} + Q_{n}) with

P_{n} = \{ \begin{matrix}χ_{nΩ}I|_{Z Z}^{K} & if & n ≥ 0 \\ 0 & if & n < 0, \end{matrix} Q_{n} = I − P_{n}

in the cases when A is band-dominated on E^{1} and Ω = [−1, 1], and when A is band-dominated on E^{2} and Ω is a convex polygon with 0 as its interior point and with vertices in ℤ^{2}.

For this goal, recall the following result from [8]: For every band-dominated operator B on E^{K} and every sequence h ⊆ ℤ^{K} tending to infinity, there is a subsequence g of h such that the limit operator B_{g} exists. In particular, if ∑ ∏ A_{ij} is a finite sum of finite products of band-dominated operators A_{ij} on E^{K} and if the limit operator B_{h} with respect to a sequence h exists, then there is a subsequence g of h such that the limit operators (A_{ij})_{g} exist for all i, j, and in this case we have

B_{h} = B_{g} = ∑ ∏(A_{ij})_{g}.

Thus, on finite sums of finite products, the mapping B ↦ B_{h} can be thought of as a homomorphism after passing to a suitable subsequence of h.

We start with describing the limit operators of the operators Op(A) where A is band-dominated on E^{K} and K is arbitrary. Given a sequence h ⊆ ℤ^{K+1} = ℤ^{K} ×ℤ write h(m) = (s(m), t(m)) ⊆ ℤ^{K} × ℤ. Then

V_{−h(m)} Op(A) V_{h(m)}
= V_{−(s(m), 0)}V_{−(0, t(m))} Op(A) V_{(0, t(m))}V_{(s(m), 0)}
= V_{−(s(m), 0)} Op(A) V_{(s(m), 0)} = Op(V_{−s(m)}AV_{s(m)}).

Let now h be a sequence which tends to infinity and for which the limit operator Op(A)_{h} exists. Then the sequence s can be bounded or unbounded. If it is bounded, there is a subsequence η of ℕ such that s ∘ η is a constant sequence, (c), say. For the sequence h ∘ η one has

Op(A)_{h} = Op(A)_{h∘η} = Op(V_{−(s∘η)(m)}AV_{(s∘η)(m)}) = Op(V_{−c}AV_{c}).

If s is unbounded, there is a subsequence η of ℕ such that the limit operator A_{s∘η} exists, and consequently

Op(A)_{h} = Op(A)_{h∘η} = Op(A_{s∘η}).

Thus, the operators in σ_{stab}(A) are necessarily limit operators of A or operators of the form V_{−c}AV_{c} with c ∈ ℤ^{K}. Conversely, if s is a sequence such that A_{s} exists, then h(m) := (s(m), 0) is a sequence tending to infinity such that the limit operator of Op(A) with respect to h exists and Op(A)_{h} = Op(A_{s}). If c ∈ ℤ^{K}, then h(m) := (c, m) defines a sequence such that Op(A)_{h} exists and coincides with Op(V_{−c}AV_{c}). Thus,

σ_{stab}(A) = σ_{op}(A) ∪ {V_{−c}AV_{c} : c ∈ ℤ^{K}}.  (3)

It remains to compute the limit operators of the operator Op(P_{n}). In the first case under consideration, i.e. when Ω = [−1, 1] , this operator can be written as the product χ_{H_{1}}I|_{Z Z}^{2} χ_{H_{−1}}I|_{Z Z}^{2} where χ_{H_{1}} and χ_{H_{−1}} are the characteristic functions of the half planes

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

In [8], we determined the operator spectrum σ_{op}(χI) for the operator χI = χ_{R}_{+}^{2}I|_{Z Z}^{2} where χ_{R}_{+}^{2} is the characteristic function of the upper half plane

ℝ_{+}^{2} := {(x, y) : y ≥ 0}.

It is quite convenient to cite these results in terms of local operator spectra which are defined (for arbitrary K) as follows. By definition, a sequence h = (h_{m}) ⊆ ℤ^{K} tends to infinity if, for each R > 0, there is an m_{0} such that

h(m) ∈ {z ∈ ℤ^{K} : |z| > R} for all m ≥ m_{0}.

For the purposes at hand, it is desirable to have an information about the ‘direction’ into which h tends. To make this precise, let S^{K−1} denote the unit sphere {ξ ∈ ℝ^{K}, |ξ| = 1} and, given R > 0, ξ ∈ S^{K−1}, and a neighborhood U ⊆ S^{K−1} of ξ, define

W_{R,U} := {z ∈ ℤ^{K} : |z| > R and z/|z| ∈ U}.

Let h be a sequence tending to infinity, and ξ ∈ S^{K−1}. We say that h tends to ξ if, for each R > 0 and for each neighborhood U of ξ in S^{K−1}, there is an m_{0} such that

h_{m} ∈ W_{R,U} for all m ≥ m_{0}.

Given ξ ∈ S^{K−1} and C ∈ L(E^{K}), the local operator spectrum σ_{ξ}(C) of C at ξ is defined as the set of all limit operators C_{h} of C with respect to sequences h tending to ξ. It is not hard to see that

σ_{op}(C) = ∪_{ξ∈S^{K−1}}σ_{ξ}(C)

for every operator C ∈ L(E^{K}). Indeed, it is evident that every local operator spectrum is contained in the operator spectrum. For the reverse inclusion, let C_{h} ∈ σ_{op}(C) with respect to a certain sequence h tending to infinity. Taking into account the compactness of S^{K−1}, it is easy to see that there is a subsequence g of h which converges to a certain point ξ in S^{K−1}. Thus, C_{h} = C_{g} ∈ σ_{ξ}(C). In particular, if A is a band-dominated operator on E^{1} then σ_{op}(A) splits into σ_{−1}(A) ∪ σ_{1}(A) (observe that S^{0} = {−1, 1}).

To cite the mentioned result concerning the limit operators of χI := χ_{R}_{+}^{2}I|_{Z Z}^{2}, write ξ ∈ S^{1} as ξ = exp(iφ) with φ ∈ [0, 2π). Further set S_{+}^{1} := {exp(iφ) : φ ∈ (0, π)}, S_{−}^{1} := {exp(iφ) : φ ∈ (π, 2π)} and S_{0}^{1} := {exp(iφ) : φ ∈ {π, 2π} and, given m ∈ ℤ, let U_{m} stand for the shift operator V_{(0,m)}. Then we have

σ_{ξ}(χI) = \{ \begin{matrix}\{I\} & if & ξ ∈ S_{+}^{1} \\ \{0\} & if & ξ ∈ S_{−}^{1} \\ \{0, I, U_{−m}χU_{m} : m ∈ ℤ\} & if & ξ ∈ S_{0}^{1}. \end{matrix}  (4)

Having this result at our disposal, it is clear what the limit operators of the operators χ_{H_{1}}I|_{Z Z}^{2} and χ_{H_{−1}}I|_{Z Z}^{2} and, consequently, of Op(P_{n}) are:

σ_{ξ}(Op(P_{n})) = \{ \begin{matrix}\{0, I, U_{−m} χ_{H_{1}}I|_{Z Z}^{2} U_{m} : m ∈ ℤ\} & if & φ = π/4 \\ \{I\} & if & φ ∈ (π/4, 3π/4) \\ \{0, I, U_{−m} χ_{H_{−1}}I|_{Z Z}^{2} U_{m} : m ∈ ℤ\} & if & φ = 3π/4 \\ \{0\} & if & φ ∉ [π/4, 3π/4]. \end{matrix}

Let, finally, Π_{−1} : E^{1} → E^{1} stand for the projection operator which sends the sequence (…, x_{−1}, x_{0}, x_{1}, …) into (…, x_{−1}, 0, 0, …), and set Π_{1} := I − Π_{−1}. Then

χ_{H_{−1}}I|_{Z Z}^{2} = Op(V_{−n}Π_{−1}V_{n})_{n≥0} and χ_{H_{1}}I|_{Z Z}^{2} = Op(V_{−n}Π_{1}V_{n})_{n≥0}.

Theorem 3 Let A be a band-dominated operator on E^{1} and Ω = [−1, 1]. Then the finite section method (P_{n}AP_{n} + Q_{n}) is stable if and only if the operator A is invertible, if the operators

Π_{−1}A_{s}Π_{−1} + Π_{1}  (5)

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

Π_{1}A_{s}Π_{1} + Π_{−1}  (6)

are invertible for all limit operators A_{s} ∈ σ_{1}(A), and if the norms of the inverses of all operators in (5) and (6) are uniformly bounded.

Proof. The necessity of the invertiblity of A is a simple consequence of the stability of P_{n}AP_{n} + Q_{n} which holds for every operator A. Let us check all operators in (5) are necessarily invertible. The proof for (6) is analogous. Let s ⊂ ℤ be a sequence tending to −1 ∈ S^{0} such that the limit operator A_{s} exists, and let m ∈ ℤ. Then h(n) := (s(n), −s(n)) defines a sequence in ℤ^{2} which tends to exp(iφ) with φ = 3π/4, and the limit operators Op(A)_{h} and Op(P_{n})_{h} exist and are equal to

Op(A_{s}) and χ_{H_{−1}}I|_{Z Z}^{2} = Op(V_{−n}Π_{−1}V_{n})_{n≥0},

respectively. Thus, σ_{stab}(P_{n}AP_{n} + Q_{n}) contains all operators

V_{−n}Π_{−1}V_{n}A_{s}V_{−n}Π_{−1}V_{n} + V_{−n}Π_{1}V_{n}

with n ≥ 0. In particular, it contains the operator Π_{−1}A_{s}Π_{−1} + Π_{1} and, thus, all operators as in (5). The proof for the operators in (6) is analogous, and the uniform boundedness of the inverses of all of these operators is a consequence of Theorem 2.

Now suppose that the operator A as well as all operators in (5) and (6) are invertible and that the norms of their inverses are uniformly bounded. We have to show that then all operators in σ_{stab}(P_{n}AP_{n} + Q_{n}) are invertible and that the norms of their inverses are uniformly bounded (Theorem 2 again).

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

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

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

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

with a sequence s ∈ ℤ tending to infinity or with a constant c ∈ ℤ (compare (3)). In both cases, the invertibility of Op(P_{n}AP_{n} + Q_{n})_{g} follows from the invertibility of A, and it is also clear that

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

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

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

= U_{−m} χ_{H_{−1}}I|_{Z Z}^{2} U_{m}Op(A_{s})U_{−m} χ_{H_{−1}}I|_{Z Z}^{2} U_{m} + U_{−m} (1 − χ_{H_{−1}})I|_{Z Z}^{2} U_{m}
= U_{−m}Op(V_{−n}Π_{−1}V_{n})U_{m}Op(A_{s})U_{−m}Op(V_{−n}Π_{−1}V_{n})U_{m} +
+ U_{−m}Op(V_{−n}Π_{1}V_{n})U_{m}
= Op(V_{−n−m}Π_{−1}V_{n+m})Op(A_{s})Op(V_{−n−m}Π_{−1}V_{n+m}) + Op(V_{−n−m}Π_{1}V_{n+m})
= Op(V_{−n−m}(Π_{−1}V_{n+m}A_{s}V_{−n−m}Π_{−1} + Π_{1})V_{n+m}).

Clearly, this operator on E^{2} is invertible if and only if all operators

Π_{−1}V_{n+m}A_{s}V_{−n−m}Π_{−1} + Π_{1} with m, n ∈ ℤ

are invertible and if the norms of their inverses are uniformly bounded. But this invertibility is an obvious consequence of the invertibility of all operators in (5): Indeed, if A_{s} is a limit operator of A with respect to a sequence s and if k ∈ ℤ, then V_{−k}A_{s}V_{k} is a limit operator of A with respect to the sequence s + k (which tends to the same point of S^{0} as the sequence s). In other words, local operator spectra are shift invariant, which proves our claim.

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

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

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

Thus, as in case K = 1, the operator Op(P_{n}) is the product of characteristic functions of half-spaces which arise from a rotation of the characteristic function of the upper half-space, χI := χ_{R}_{+}^{3}I|_{Z Z}^{3}, where ℝ_{+}^{3} := {(x, y, z) : z > 0}. If S_{+}^{2}, S_{−}^{2} and S_{0}^{2} stand for the north and south hemisphere and for the equator of the sphere S^{2} of the infinite points of ℝ^{3}, respectively, and if U_{m} denotes the shift operator V_{(0, 0, m)} for m ∈ ℤ, then one gets similarily to (4)

σ_{ξ}(χI) = \{ \begin{matrix}\{I\} & if & ξ ∈ S_{+}^{2} \\ \{0\} & if & ξ ∈ S_{−}^{2} \\ \{0, I, U_{−m}χU_{m} : m ∈ ℤ\} & if & ξ ∈ S_{0}^{2}. \end{matrix}  (7)

Next we have to determine the local operator spectra of Op(P_{n}). Given a point x ∈ Ω, the ray {(nx, n) : n ≥ 0} ⊆ ℝ^{3} points to an element of the sphere S^{2} which we denote by x^{(2)}. As a consequence of (7) one gets:

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

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

• If x ∈ (u_{j}, u_{j+1}), then σ_{x^{(2)}}(Op(P_{n})) consists of the operators 0, I and of the characteristic functions of the shifted half-planes U_{−m}χ_{F_{j}}I|_{Z Z}^{3}U_{m} with m ∈ ℤ.

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

U_{−k}χ_{F_{j−1}}I|_{Z Z}^{3}U_{k} and U_{−m}χ_{F_{j}}I|_{Z Z}^{3}U_{m} with k, m ∈ ℤ,

and of the characteristic functions of the shifted dihedrals

U_{−k}χ_{F_{j−1}}I|_{Z Z}^{3}U_{k}U_{−m}χ_{F_{j}}I|_{Z Z}^{3}U_{m} with k, m ∈ ℤ.

Observe further that

χ_{F_{j}}I|_{Z Z}^{3} = Op(V_{−n}χ_{H_{j}^{0}}I|_{Z Z}^{2}V_{n})_{n≥0}.

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

Theorem 4 Let A be a band-dominated operator on E^{2} and Ω a convex polygon with 0 as its interior point and with vertices in ℤ^{2}. Then the finite section method (P_{n}AP_{n} + Q_{n}) is stable if and only if the operator A is invertible, if the operators

Π_{j}A_{s}Π_{j} + (I − Π_{j})  (8)

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

Π_{j}Π_{j−1}A_{s}Π_{j−1}Π_{j} + (I − Π_{j}Π_{j−1})  (9)

are invertible for all limit operators A_{s} ∈ σ_{x^{(1)}}(A) and for all x = u_{j}, and if the norms of the inverses of all operators in (8) and (9) are uniformly bounded.

Proof. Most parts of the proof run completely parallel to the proof of Theorem 3. The only essential difference is that now, at points x = u_{j}, we have (besides the trivial operators 0 and I) two kinds of limit operators of Op(P_{n}), namely shifted half-plane projections and shifted dihedral projections. So what we have to verify for every fixed vertex x = u_{j} of Ω is, for instance, that the invertibility of all operators in (9) implies the invertibility of the operators

U_{−m}χ_{F_{j}}I|_{Z Z}^{3}U_{m}Op(A_{s})U_{−m}χ_{F_{j}}I|_{Z Z}^{3}U_{m} + U_{−m}(1 − χ_{F_{j}})I|_{Z Z}^{3}U_{m}

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

Since χ_{F_{j}}I|_{Z Z}^{3} = Op(V_{−n}Π_{j}V_{n}), this is equivalent to the invertibility of all operators

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

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

Π_{j}A_{s}Π_{j} + (I − Π_{j}).  (10)

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

V_{−h(n)}Π_{j−1}V_{h(n)} → I strongly as n → ∞,

but

V_{−h(n)}Π_{j}V_{h(n)} = Π_{j} for all n.

Since every shifted limit operator of A is a limit operator of A again, we conclude from the invertibility of all operators in (9) that the operators

Π_{j}Π_{j−1}V_{h(n)}A_{s}V_{−h(n)}Π_{j−1}Π_{j} + (I − Π_{j}Π_{j−1})

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

Π_{j}V_{−h(n)}Π_{j−1}V_{h(n)}A_{s}V_{−h(n)}Π_{j−1}V_{h(n)}Π_{j} + (I − Π_{j}V_{−h(n)}Π_{j−1}V_{h(n)})  (11)

are invertible for all n and that the norms of their inverses are uniformly bounded. Letting n go to infinity, these operators converge strongly to the operator (10). Since also the adjoints of the operators (11) converge strongly, the invertibility of the operator (10) follows from standard arguments. Finally, the Banach-Steinhaus theorem entails that ∥(Π_{j}A_{s}Π_{j} + (I − Π_{j}))^{−1}∥ is bounded from above by the sameq constant as the norms of the inverses of the operators (11) are.

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

4 On fractal approximation sequences

Here we want to make the discussion of fractality from the introduction more explicit. For simplicity, we restrict our attention to the Hilbert space case p = 2 and to one-sided infinite approximation sequences. Thus, let now F stand for the C^{∗}-algebra of all bounded sequences (A_{n})_{n≥0} of operators A_{n} ∈ L(E^{K}), and denote by G the closed ideal of F consisting of all sequences (A_{n}) with lim ∥A_{n}∥ = 0. Given a 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 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).

(c) A sequence (A_{n}) ∈ F is fractal if the smallest C^{∗}-subalgebra of F which contains (A_{n}) is fractal.

There are several criteria for fractality. A simple and for our purposes sufficient one can be stated in terms of the Hausdorff upper and lower limit of certain set sequences. If (A_{n}) is a self-adjoint fractal approximation sequence, then

lim sup σ(A_{n}) = lim inf σ(A_{n})  (12)

where σ(A_{n}) refers to the spectrum of A_{n}. One can show that, conversely, (12) is the only obstruction for a self-adjoint bounded sequence to be fractal.

Theorem 5 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 (12) holds.

A proof of (12) as well as of the theorem in in [10].

It seems that the non-fractality of certain approximation sequences is the reason for their momentary intractability. Let us mention a few examples.

Example 1: Spline Galerkin methods for singular integral equations on intervals. Consider a singular integral equation (aI + bS)u = f on the Hilbert space L^{2}(ℝ) with piecewise continouous coefficients a and b and with the operator S of singular integration along the real line defined by

(Su)(t) := \frac{1}{πi} ∫_{−∞}^{∞} \frac{u(s)}{s − t} ds where t ∈ ℝ.

For the approximate solution of this equation let S_{n} stand for the smallest closed subspace of L^{2}(ℝ) which contains all functions which are constant on each of the intervals (k/n, (k + 1)/n) with k ∈ ℤ, and let L_{n} stand for the orthogonal projection from L^{2}(ℝ) onto the spline space S_{n}. It is not too hard to analyse the stability of the spline Galerkin sequence (L_{n}(aI + bS)|_{S_{n}}) in case a and b are piecewise continuous functions which are continuous on ℝ \ ℤ (see, e.g., [4] for an approach using Banach algebra techniques and for a stability criterion). But, as far as we know, nobody succeeded in deriving a general necessary and sufficient criterion if the coefficients a and b are merel supposed to be piecewise continuous on ℝ (without further assumptions on the location of the their discontinuities). We believe that the non-fractality of the sequence (L_{n}(aI + bS)|_{S_{n}}) is a main reason for this (observe that this sequence is fractal if the only discontinuities of a and b are at the integers).

The non-fractality of spline Galerkin methods for singular integrals with piecewise continuous coefficients can be seen most easily for the special self-adjoint sequence (L_{n}χI|_{S_{n}}) where χ is the characteristic function of the interval (\sqrt{2}, ∞). Indeed, the functions φ_{kn}, k ∈ ℤ, which are defined as \sqrt{n} times the characteristic function of the interval [k/n, (k + 1)/n), form an orthonormal basis of S_{n}, and the matrix representation of L_{n}χI|_{S_{n}} with respect to this basis is the diagonal matrix diag_{k} (α_{k}) where

α_{k} := \{ \begin{matrix}0 & if & k < \{\sqrt{2}n\} − 1 \\ \{\sqrt{2}n\} − \sqrt{2}n & if & k = \{\sqrt{2}n\} − 1 \\ 1 & if & k > \{\sqrt{2}n\} − 1, \end{matrix}

with {x} referring to the smallest integer which is not smaller than x ∈ ℝ. Since the numbers \{\sqrt{2}n\} − \sqrt{2}n form a dense subset of [0, 1] (the Jacobi-Kronecker theorem), we conclude that

lim sup σ(L_{n}χI|_{S_{n}}) = lim sup\{0, 1, \{\sqrt{2}n\} − \sqrt{2}n\} = [0, 1],

whereas

lim inf σ(L_{n}χI|_{S_{n}}) = lim inf{0, 1} = {0, 1}.

Due to Theorem 5, the sequence (L_{n}χI|_{S_{n}}) is not fractal. It is interesting to observe that certain collocation methods for singular integral equations with arbitrary piecewise continuous coefficients are fractal (see again [4]).

Example 2: Spline Galerkin methods for singular integral equations on domains. The same difficulties as in Example 1 arise for spline Galerkin methods for equations on bounded domains. A typical example is the spline Galerkin method for the singular integral equation (aI + bS_{Ω})u = f on L^{2}(Ω) with Ω a convex domain in ℝ^{2} with C^{2}-boundary (as considered in [1]). Here, S_{Ω} is the operator χ_{Ω}S_{R}^{2}χ_{Ω}I, and S_{R}^{2} is the singular integral operator

(S_{R}^{2}u)(x) := −\frac{1}{π} ∫_{R_{2}} \frac{u(y)}{(x − y)^{2}} dy.

A direct application of a spline Galerkin projection (based on a spline space over a suitable lattice of ℝ^{2}) to this equation leads necessarily to a non-fractal behaviour in the neighborhood of the boundary of Ω. In [1] it is tried to manage these effects by modifying the spline space by an appropriate cutting off technique.

Example 3: The finite section method for band-dominated operators. 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})

which show that the finite section method for band-dominated operators is not fractal in general: Indeed, for A one has

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

for all n. Of course, for this special sequence, one would simply turn to its subsequence (P_{2n}AP_{2n}) in order to get a fractal sequence (and this is what is actually done when talking about the finite section method for ”block Toeplitz operators”, say). But the algebra A of the finite section method for general band-dominated operators contains sequences of this kind with arbitrary block size at once and, thus, one cannot force fractality by figuring out a certain fixed subsequence of the integers.

References

[1] T. Finck, Splinaapproximationsverfahren für singuläre Integralgleichungen auf ebenen, glattberandeten Gebieten. – Ph.D. theses, TU Chemnitz, 1998.

[2] I. Gohberg, I. Feldman, Convolution equations and projection methods for their solution. – Nauka, Moskva 1971 (Russian; English transl.: Amer. Math. Soc. Transl. of Math. Monographs 41, Providence, R. I., 1974).

[3] I. Gohberg, M. A. Kaashoek, Projection method for Block Toeplitz operators with operator-valued symbols. – In: Toeplitz Operators and Related Topics, Operator Theory: Advances and Applications 71, Birkhäuser Verlag, Basel, Boston, Berlin 1994, 79 – 104.

[4] R. Hagen, S. Roch, B. Silbermann, Spectral Theory of Approximation Methods for Convolution Equations. – Birkhäuser Verlag, Basel, Boston, Berlin 1995.

[5] A. V. Kozak, A local principle in the theory of projection methods. – Dokl. AN SSSR 212(1973), 6, 1287 – 1289 (Russian).

[6] A. V. Kozak, I. B. Simonenko, Projection methods for solving discrete convolution equations. – Sib. Mat. J. 20(1979), 6, 1249 – 1260 (Russian).

[7] S. Prössdorf, B. Silbermann, Numerical Analysis for Integral and Related Operator Equations. – Akademie-Verlag, Berlin, 1991, and Birkhäuser Verlag, Basel, Boston, Stuttgart 1991.

[8] V. S. Rabinovich, S. Roch, B. Silbermann, Fredholm theory and finite section method for band-dominated operators. – Integral Equations Operator Theory 30(1998), 4, 452 – 495.

[9] S. Roch, B. Silbermann, C^{∗}-algebra techniques in numerical analysis. – Journal of Operator Theory 35(1996), 2, 241 – 280.

[10] S. Roch, Algebras of approximation sequences: Fractality. – Preprint TU Darmstadt, submitted to the Proceedings of the 11. TMP Conference, Chemnitz 1999.

Author’s address:

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

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

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