Bernd Silbermann

Spectral Theory of Approximation Methods for Convolution Equations

Given a linear bounded operator A on a Banach space X, it is in general rather difficult to decide whether the equation Ax = y with x, y ∈ X is uniquely solvable for each right side y, and to solve this equation practically, or, in other words, to decide whether the operator A is invertible, and to determine its inverse A −1. For that reason, besides this “classical” invertibility of A, other invertibility concepts are in discussion.

Toeplitz matrices and determinants with Fisher-Hartwig symbols

Abstract This paper is concerned with Toeplitz matrices generated by symbols of the form a(t) = Φ r=1 R | t−t r | 2α r (− t ) t r β r b(t) (| t |=1) where t 1 ,…, t R are pairwise distinct points on the unit circle, b is sufficiently smooth, b(t) ≠ 0 (¦t¦ = 1) , and ind b = 0, (− t ) t r β r is defined as exp {iβ r arg ( − t t r )} with ¦arg ( − t t r )¦ , and α r , β r are complex numbers satisfying ¦ Re α r ¦ 1 2 , ¦ Re β r ¦ 1 2 . Weighted Hardy spaces L + 2 ( ϱ 1 ) and L + 2 ( ϱ 2 ) are defined such that the Toeplitz operator T ( a ): L + 2 ( ϱ 1 ) → L + 2 ( ϱ 2 ) is bounded and invertible and that the finite section method is applicable to T ( a ) considered as acting from L + 2 ( ϱ 1 ) onto L + 2 ( ϱ 2 ). These results are then applied to prove a conjecture of Fisher and Hartwig ( Adv. in Chem. Phys. 15 (1968), 333–353) which asserts that the determinants of the n × n sections of T ( a ) are asymptotically equal to G(b) n n q E (a) (n → ∞) , where q = ∑ ( α r 2 − β r 2 ) and G ( b ), E (a) are certain nonzero constants. Under quite general conditions on the smoothness of b, these constants are completely identified.

A general look at local principles with special emphasis on the norm computation aspect

Local principles associate with every element of a Banach algebra a family of local objects in terms of which the properties of the original element can be studied. In this paper some general relations between three such principles, usually affiliated with the names of Simonenko, Allan/Douglas, and Gohberg/Krupnik, are discussed. Special attention is paid to the question on how the norm of an element can be expressed in terms of the norms of the local objects associated with it. The general theory is illustrated by some concrete results on singular integral operators and Toeplitz operators.

The C*-algebra generated by Toeplitz and Hankel operators with piecewise quasicontinuous symbols

The present paper concerns the description of the image in the Calkin algebra of the C*-algebra generated by Toeplitz and Hankel operators with piecewise quasicontinuous symools.

Local objects in the theory of Toeplitz operators

This paper is aimed at the local theory of Toeplitz operators including harmonic extension, harmonic approximation and finite section method.

Local theory of the collocation method for the approximate solution of singular integral equations, I

A new approach for investigating the collocation method for singular integral equations is given. Using Banach algebra techniques and a local principle established by I. Gohberg and N. Krupnik under suitable conditions the convergence of the Lagrange- and Mul thopp-collocation method is proved for one-dimensional singular integral equations and systems of such equations with piecewise continuous coefficients having a countable infinite number of discontinuities.

Finite sections of band-dominated operators with almost periodic coefficients

We consider the sequence of the finite sections R n AR n of a band-dominated operator A on l 2(ℤ) with almost periodic coefficients. Our main result says that if the compressions of A onto ℤ+ and ℤ− are invertible, then there is a distinguished subsequence of (R n AR n) which is stable. Moreover, this subsequence proves to be fractal, which allows us to establish the convergence in the Hausdorff metric of the singular values and pseudoeigenvalues of the finite section matrices.

A symbol calculus for finite sections of singular integral operators with shift and piecewise continuous coefficients

Abstract This paper is concerned with a symbol calculus for the finite section method for the approximate solution of the equation Ax = y when A is specified to be the operator A = PaP + PbQ + QcP + Q dQ + ( Pa ′ P + Pb ′ Q + Qc ′ P + Q d ′ Q ) J . Herein P denotes the Riesz projection of L 2 onto H 2 , Q = I − P , and J is the shift operator ( Jf)(t) = ( 1 t ) f( 1 t ) The functions a ,…, d ′ are assumed to be m × m matrix-valued and piecewise continuous. The finite section method for A proves to converge if and only if the operator A itself is invertible, if the operator P a P + Q d Q + (P a ′ P + Q d ′Q)J with a (t)

Harmonic approximation of Toeplitz operators and index formulas

= a( 1 t ) is invertible, and if a certain operator-valued function A ( τ ) (the symbol of A relative to the finite section method) is invertible at each point τ of the upper half of the unit circle.

In Memoriam Georg Heinig (1947–2005)

On demontre des formules de lindice pour des operateurs de Toeplitz par extension harmonique