Algebras of approximation sequences: Fredholm
theory in fractal algebrasSubmitted to Studia Mathematica.
Abstract:
The present paper is a continuation of our previous papers where a Fredholm theory for approximation sequences is proposed and some of its properties and consequences are studied. Here this theory is specified to the class of fractal approximation methods. The main result is a formula for the so-called alpha-number of an approximation sequence (A_n) which is the analogue of the kernel dimension of a Fredholm operator.
Collocation methods for systems of Cauchy
singular integral equations on an interval
with P. Junghanns and B. Silbermann, to appear in Computational
Technologies.
Abstract:
Necessary and sufficient conditions for the stability of collocation methods with respect to Chebyshev nodes of first and second kind for Cauchy singular integral equations with piecewise continuous coefficients, given on an interval, in the scalar and in the system case are established. Moreover, the behaviour of the singular values of the matrices of the discrete equations is considered.
Algebras of approximation sequences: Structure
of fractal algebras
Submitted to the Proceedings of the IWOTA 2000 in Faro/Portugal.
Abstract:
A basic idea to study the stability of an approximation sequence is to translate the stability problem into an invertibility problem in a suitably defined C^*-algebra which offers the applicability of C^*-tools to problems in numerical analysis. This approach associates with every class L of operators and with every discretization procedure D for the operators in L a concrete C^*-algebra A(L, D). The surprising result of an analysis of these algebras is that, for many apparently quite different choices of L and D, they show a common structure which might be formally summarized in the so-called standard model. The goal of this paper is to analyse how this formal structure arises from intrinsic properties of the algebras A. The basic of these properties are the fractality of the algebra and a certain maximality condition for an ideal in A which is intimately related with the Fredholm theory in A. Here fractality of A means that every sequence in A can be completely reconstructed from each of its subsequences modulo a sequence tending to zero in the norm, whereas the Fredholm property of a sequence (A_n) is closely related to the asymptotic behaviour of the smallest singular values of the matrices A_n.
Band-dominated operators with operator-valued
coefficients, their Fredholm properties and finite sections
with V. Rabinovich and B. Silbermann, submitted to Integral Eq. Oper.
Th.
Abstract:
The central theme of the present paper are band and band-dominated operators, i.e. norm limits of band operators. In the first part, we generalize the results from our previous papers concerning the Fredholm properties of band-dominated operators and the applicability of the finite section method to the case of operators with operator-valued coefficients. We characterize these properties in terms of the limit operators of the given band-dominated operator. The main objective of the second part is to apply these results to pseudodifferential operators on cones in R^n which is possible after a suitable discretization.
Algebras of approximation sequences: Finite
sections of band-dominated operators
with V. Rabinovich and B. Silbermann, submitted to Acta Appl. Math.
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.
Algebras of approximation sequences: Fredholmness
Submitted to J. Oper. Theory.
Abstract:
In this paper, a Fredholm theory for approximation sequences is proposed. A sequence is called Fredholm if it is invertible modulo a certain ideal which plays the role of the ideal of the compact operators in the Fredholm theory of operators. With every Fredholm sequence, there are associated three integers which are the analogues of the nullity, the defiency and the index of a Fredholm operator. The nullity of a Fredholm sequence (A_n) is interpreted as a quantity which describes the asymptotic behaviour of the small singular values of the matrices A_n as n tends to infinity, and an identity is derived which allows the computation of this nullity in many situations. Several examples and applications are discussed.
Algebras of approximation sequences: Fractality
Operator Theory: Advances and Applications 121, Birkhäuser Verlag,
Basel 2001, 471 - 497.
Abstract:
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.
Some peculiarities of approximation methods
for singular integral equations with conjugation
with V. Didenko and B. Silbermann; to appear in Methods and Appl. of
Analysis.
Abstract:
Approximation methods for singular integral operators with continuous coefficients and conjugation on curves with corners are investigated with respect to their stability. Particular emphasis is devoted to index constraints for the local stability conditions. It turns out that, if an associated local operator is Fredholm, then the absolute value of its Fredholm index is necessarily bounded by 2, and this maximal value is attained in some instances (whereas it is known that in case of pure singular integral equations Fredholmness of the associated local operators already implies vanishing index).
Pseudospectra of operator polynomials
To appear in the Proceedings of the IWOTA 1998, Groningen, July 1998.
Abstract:
The recent interest in epsilon-pseudospectra of operators results from their (in comparison with usual spectra) excellent continuity properties. The goal of the present paper is to introduce and to examine epsilon-pseudospectra of operator polynomials with main emphasis on the continuity aspect.
Continuity of generalized inverses in Banach algebras
with B. Silbermann; Studia Mathematica 136(1999) (3), 197 - 227.
Abstract:
The main topic of the paper is the continuity of several kinds of generalized inversion of elements in a Banach algebra with identity. We introduce the notion of asymptotic generalized invertibility and characterize sequences of elements with this property completely. Based on this result, we derive continuity criteria which generalize the well known criteria from operator theory.
Spectral approximation of Wiener-Hopf operators
with almost periodic generating function
Numerical Functional Anal. Optimiz. 21(2000), 1 - 2, 241 - 253.
Abstract:
The paper deals with spectral approximation of Wiener-Hopf operators acting on L^p-spaces by their finite sections. The generating function of the Wiener-Hopf operators are supposed to be continuous plus almost periodic. Whereas the usual spectra of the finite sections drastically fail to converge to the spectrum of the Wiener-Hopf operator, it turns out that other spectral approximants, viz. the pseudospectra and the numerical ranges, do converge perfectly. The proof requires a modified approach to the finite section method for Wiener-Hopf operators which will be developed first. This note generalizes results obtained by Böttcher, Grudsky and Silbermann for the case of continuous generating functions.
Numerical ranges of large Toeplitz matrices
Linear Algebra Applications 282(1998), 185 - 198.
Abstract:
A Banach algebraic approach is proposed to study the asymptotic behaviour of the numerical ranges of certain (finite) approximation matrices of (infinite) operators. The approach works for large classes of approximation methods; it is examined in detail here for the finite sections of Toeplitz operators and of operators which are generated by Toeplitz operators. The basic ingredient is a precise knowledge of the finite section method for Toeplitz and related operators.
Banach algebras generated by two idempotents and
one flip
with Tilo Finck and Bernd Silbermann; Mathematische Nachrichten
216(2000), 73 - 94.
Abstract:
The paper is concerned with the invertibility of elements in a Banach algebra generated by two idempotents and one flip element. A symbol in form of a 2 times 2 matrix function which is defined on some Hausdorff compact will be constructed and a complete description of this compact will be given.
Index calculus for approximation methods, and singular value decomposition
with Bernd Silbermann; Journ. Math. Anal. Appl. 225(1998), 401 -
426.
Abstract:
We introduce two classes of sequences (A_n) of approximation operators, Moore-Penrose and Fredholm sequences, which correspond in a sense to normally solvable and Fredholm operators, respectively. For Moore-Penrose sequences, we derive a certain qualitative behaviour of the singular values of the approximation operators A_n, and we describe this behaviour quantitatively for Fredholm sequences by defining the kernel dimension and the index of an approximation method.
A note on singular values of Cauchy-Toeplitz matrices
with Bernd Silbermann; Linear Algebra Applications 275-276(1998),
531 - 536.
Abstract:
We consider the asymptotic behaviour of the smallest singular values of the n times n sections of a general infinite Cauchy-Toeplitz matrix.
Fredholm theory and the finite section method for band-dominated operators
with Volodya Rabinovich and Bernd Silbermann, Integral Equations
Operator Theory 30(1998), 452 - 495.
Abstract:
The topics of this paper are Fredholm properties and the applicability of the finite section method for band operators on l^p-spaces as well as for their norm limits which we call band-dominated operators. The derived criteria will be established in terms of the limit operators of the given band-dominated operator. After presenting the general theory, we present its specifications to concrete classes of band-dominated operators.
Finite Section Method in some algebras of Multiplication and Convolution
Operators and a Flip
with Pedro Santos and Bernd Silbermann; Zeitschrift für Analysis
Anwend. 16(1997), 3, 575 - 606.
Abstract:
This paper is concerned with the applicability of the finite section method to operators belonging to the closed subalgebra of L(L^2(R)) generated by operators of multiplication by piecewise continuous functions, convolution operators with piecewise continuous generating functions and the flip operator (Ju)(x) = u(-x). For this, an algebra of sequences is introduced, which contains the special sequences we are interested in. There is a direct relationship between the applicability of the finite section method for a given operator and invertibility of the corresponding sequence in this algebra. Employing local techniques we derive necessary and sufficient conditions for the stability of the approximation methods under consideration. Finally examples are presented, including explicit conditions for the applicability of the finite section method to a Wiener-Hopf plus Hankel operator with piecewise continuous symbols, and some relations between the approximation operators and the limit operator are discussed.
Spline approximation methods for Wiener-Hopf operators
In: Gohberg, Lancaster, Shivakumar (Eds.): Recent developments
in Operator Theory and its Applications, OT 87, Birkhäuser Verlag
1996, 282 - 308.
Abstract:
In the present paper, we introduce an algebra of approximation sequences both for singular integral operators with piecewise continuous coefficients and for Wiener-Hopf operators with piecewise continuous generating function. By means of localization techniques and of the two-projections-theorem, necessary and sufficient conditions for the stability of sequences in this algebra are derived.
Asymptotic Moore-Penrose invertibility of singular integral operators
with Bernd Silbermann; Integral Equations Operator Theory 26(1996),
1, 81 - 101.
Abstract:
The topic of the paper is the asymptotic Moore-Penrose inversion of singular integral operators by means of the trigonometric collocation and the finite section method. Two versions (a "weak" and a "strong" one) of these methods are considered. Necessary and sufficient conditions for the applicability of the weak method, and sufficient conditions for the strong method, are derived. These conditions involve a certain asymptotic resp. exact kernel structure of the approximation matrices.
C^*-algebra techniques in numerical analysis
with Bernd Silbermann; Journ. Operator Theory 35(1996), 2, 241 - 280.
Abstract:
The topic of the present paper is a general approach of studying invertibility problems in certain Banach algebras which, in particular, can possess a trivial center. For these algebras, a symbol map is introduced which is responsible for invertibility. Special attention is paid to the invertibility in algebras the elements of which are sequences of operators. These sequences can be viewed as approximation sequences for a given operator, and the proposed approach allows to relate properties of the approximation sequence (stable convergence, limiting sets of spectra, Moore-Penrose invertibility, asymptotic behaviour of the condition numbers) with corresponding properties of the symbol of the sequence. This method applies to practically relevant approximation methods such as the finite section method for Toeplitz operators and spline projection methods (Galerkin, collocation, qualocation) and quadrature methods for singular integral equations with piecewise continuous coefficients as well as for Mellin operators.
N projections theorems
with Albrecht Böttcher, Israel Gohberg, Yurii Karlovich, Naum
Krupnik, Bernd Silbermann, Ilya Spitkovsky; In: Operator Theory:
Advances and Applications 90, Birkhäuser Verlag, Basel 1996, 19 - 54.
Abstract:
We consider Banach algebras which are generated by N idempotents (and the identity). For N=2, such algebras have been thoroughly studied and spectrum-preserving homomorphisms (which are isometric in the C^*-algebra case) have been constructed by many mathematicians. It is also well known that for N > 2 these results have no useful analogues unless additional axioms for the generating idempotents are required. In this paper, we propose a system of axioms for algebras generated by N idempotents which is strong enough to ensure that all irreducible representations of the algebra are finite-dimensional and is flexible enough to have nontrivial realizations by algebras of singular integral operators.