Showing papers by "Albrecht Böttcher published in 1996"
01 Jan 1996
TL;DR: In this article, all irreducible representations of Banach algebras generated by N idempotents which satisfy some additional relations are described and constructed a symbol theory with applications to singular integral operators.
Abstract: It is well known that for Banach algebras generated by two idempotents and the identity all irreducible representations are of order not greater than two. These representations have been described completely and have found important applications to symbol theory. It is also well known that without additional restrictions on the idempotents these results do not admit a natural generalization to algebras generated by more than two idempotents and the identity. In this paper we describe all irreducible representations of Banach algebras generated by N idempotents which satisfy some additional relations. These representations are of order not greater than N and allow us to construct a symbol theory with applications to singular integral operators.
27 citations
01 Jan 1996
TL;DR: In this article, a systematic introduction to the theory of Toeplitz operators with oscillating symbols on the Hardy space H 2 of the unit circle is presented. But the focus of this paper is on the question of whether the argument of the symbol implies that the operator is normally solvable, semi-Fredholm, Fredholm, or even invertible.
Abstract: This paper is a systematic introduction (with a number of new results) to some aspects of the theory of Toeplitz operators with oscillating symbols on the Hardy space H 2 of the unit circle. What we are interested in is obtaining answers to the question which geometric and/or algebraic properties of the argument of the symbol imply that the operator is normally solvable, semi-Fredholm, Fredholm, or even invertible. Our discussion includes well known results on symbols in C + H ∞ , SAP, or PQC and also less known and new insights into operators whose streched symbols have arguments behaving like x λ, exp(x λ), (log x)λ, or sin(x λ) with λ > 0 at infinity. We also present some new results on the finite section method for Toeplitz operators.
22 citations
18 citations
01 Jan 1996
TL;DR: In this article, it was shown that a Carleson Jordan curve can transform circular arcs in the essential spectra of Toeplitz operators into logarithmic double-spirals.
Abstract: This paper is concerned with the spectra of Toeplitz operators with piecewise continuous symbols and with the symbol calculus for singular integral operators with piecewise continuous coefficients on L P (Γ) where 1 < p < ∞ and Γ is a Carleson Jordan curve. It is well known that piecewise smooth curves lead to the appearance of circular arcs in the essential spectra of Toeplitz operators, and only recently the authors discovered that certain Carleson curves metamorphose these circular arcs into logarithmic double-spirals. In the present paper we dispose of the matter by determining the local spectra produced by a general Carleson curve. These spectra are of a qualitatively new type and may, in particular, be heavy sets — until now such a phenomenon has only be observed for spaces with general Muckenhoupt weights.
16 citations
TL;DR: In this article, the authors considered the problem of asymptotically inverting infinite Toeplitz matrices by having recourse to their finite principal sections, and established criteria for the applicability of the finite section method to ToePlitz operators with symbols in $C + H + H^\infty, in $PC$ or with locally sectorial symbols.
Abstract: Infinite Toeplitz matrices with operator-valued entries arise, for example, when interpreting Wiener–Hopf integral operators on $L^2 (0,\infty )$ as matrices acting on the direct sum of countably many copies of $L^2 (0,1)$ . This paper concerns the question of asymptotically inverting such infinite Toeplitz matrices by having recourse to their finite principal sections. As expected from the corresponding theories for the scalar and matrix-valued cases, this problem leads to the investigation of compactness properties of infinite Hankel matrices. By introducing the concept of $Q_n $-compact operators on spaces of square-summable sequences with values in a separable Hilbert space, criteria for the applicability of the finite section method to Toeplitz operators with symbols in $C + H^\infty $, in $PC$, or with locally sectorial symbols are established.
9 citations
TL;DR: In this article, the authors apply the theory of submultiplicative functions to the problem of deciding whether a weight satisfies the Muckenhoupt condition and to the spectral theory of Toeplitz operators with piecewise cintinuous symnols.
Abstract: This paper is concerned with the application of the theory of submultiplicative fuctions to the problem of deciding whether a weight satisfies the Muckenhoupt condition and secondly,to the spectral theory of Toeplitz operators with piecewise cintinuous symnols.We establish criteria for the membership of integrable functions in the Muckenhoupt class in terms of the indices of suitably constructed submultiplicative functions.These criteribe together with licalization techniques and Winener-Hopf factorization methods enablle us to describe the spectra of Toeplitz operators with picewise continuous symbols on Lp space with arbitary Muckenhoupt weights over arbitary Carleson Jordan curves.
5 citations
TL;DR: In this article, the convergence of polynomial collocation for solving Toeplitz integral equations on the Bergman space of the complex unit disk was studied and it was shown that the convergence depends on the choice of the system of collocation points.