Markus Seidel

Fredholm theory for band-dominated and related operators: A survey

Abstract This paper presents the Fredholm theory on l p -spaces for band-dominated operators and important subclasses, such as operators in the Wiener algebra. It particularly closes several gaps in the previously known results for the case p = ∞ and addresses the open questions raised by Chandler-Wilde and Lindner in [S.N. Chandler-Wilde, M. Lindner, Limit operators, collective compactness and the spectral theory of infinite matrices, Mem. Amer. Math. Soc. 210 (989) (2011)]. The main tools are provided by the limit operator method and an algebraic framework for the description and adaption of Fredholmness and convergence. A comprehensive overview of this approach is given.

An affirmative answer to a core issue on limit operators

Abstract An operator on an l p -space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. It is known that a rich band-dominated operator is P -Fredholm (which is a generalization of the classical Fredholm property) if and only if all of its so-called limit operators are invertible and their inverses are uniformly bounded. We show that the condition on uniform boundedness is redundant in this statement.

Finite Sections of Band-dominated Operators – Norms, Condition Numbers and Pseudospectra

This paper is devoted to the asymptotic behavior of the norms, condition numbers and pseudospectra of the finite sections of band-dominated operators acting on the spaces \( \iota^{p} ({\mathbb{Z}}, X) \ {\rm{with}} \ 1 \leq p \leq \infty\)and X being a Banach space.

On Semi-Fredholm Band-Dominated Operators

In this paper we study the semi-Fredholm property of band-dominated operators A and prove that it already implies the Fredholmness of A in all cases where this is not disqualified by obvious reasons. Moreover, this observation is applied to show that the Fredholmness of a band-dominated operator already follows from the surjectivity of all its limit operators.

Norms, Condition Numbers and Pseudospectra of Convolution Type Operators on Intervals

The results in this paper describe the asymptotic behavior of convolution type operators on finite intervals as the length of these intervals tends to infinity. The family of operators under consideration here is generated (among others) by Fourier convolutions with slowly oscillating, almost periodic, bounded and uniformly continuous, and quasi-continuous multipliers, as well as operators of multiplication by slowly oscillating, almost periodic, and piecewise continuous functions. The focus is on the convergence of norms, condition numbers and pseudospectra.

Approximation sequences to operators on Banach spaces: a rich approach: APPROXIMATION SEQUENCES TO OPERATORS

Criteria for the stability of finite sections of a large class of convolution type operators on

Banach algebras of structured matrix sequences

L^p(\mathbb{R})

Banach algebras of operator sequences

are obtained. In this class almost all classical symbols are permitted, namely operators of multiplication with functions in

Essential pseudospectra and essential norms of band-dominated operators

[\textrm{PC} ,\textrm{SO}, L^\infty_0]

Finite Sections of Band-Dominated Operators: lp-Theory

and convolution operators (as well as Wiener-Hopf and Hankel operators) with symbols in