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This note contains reflections on a type of factorization that came up in the work of analytic operator valued functions by I.C. Gohberg and E.I. Sigal (cf. [14] , [10] , [11]). The determining characteristic of the factorizations in question is that they are obtained by the extraction of elementary factors of the form I − P + λ P where P is an idempotent in the underlying Banach algebra. It will be demonstrated here that there is a fundamental difference between the finite dimensional (matrix) case and the situation where infinite dimensional Banach algebras are taken into consideration. [ABSTRACT FROM AUTHOR]

The classical Rouché Theorem from complex function theory has been generalized to a result involving Fredholm operator valued functions (see [9] and [6]). The setting in question is (generally) a non-commutative one, so actually there are two versions of the generalization. In this paper, a unifying enhanced variant is presented involving the notion of the spectral radius. Its proof is based on a maximum principle for the spectral radius. An example is given showing that the enhanced Rouché Theorem presented here is a genuine improvement over the earlier version of the result. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe asymptotic spectral properties of sequences of variable-coefficient Toeplitz matrices. These sequences, A N ( a ) , with a being in a Wiener type algebra and defined on an annular cylinder ( [ 0 , 1 ] 2 × T ) , widely generalize the sequences of finite sections of a Toeplitz operator. We prove that if a ( x , x , t ) does not vanish for every ( x , t ) ∈ [ 0 , 1 ] × T then the singular values of A N ( a ) have the k -splitting property, which means that, there exists an integer k such that, for N large enough, the first k -singular values of A N ( a ) converge to zero as N → ∞ , while the others are bounded away from zero, with k = dim ⁡ ker ⁡ T ( a ( 0 , 0 , t ) ) + dim ⁡ ker ⁡ T ( a ( 1 , 1 , t − 1 ) ) , the sum of the kernel dimensions of two Toeplitz operators. In the end of the paper we discuss Fredholm properties of the mentioned sequences and describe them completely. [ABSTRACT FROM AUTHOR]

An explicit index formula for Toeplitz plus Hankel operators with piecewise quasicontinuous generating functions is obtained. Moreover, for some classes of the operators mentioned conditions of one-sided invertibility are established. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator , , where a is a piecewise continuous function on the unit circle. We prove that the behavior of the approximation numbers of the finite sections depends heavily on the Fredholm properties of the operators and . In particular, if the operators and are Fredholm on , then the approximation numbers of have the so-called k-splitting property. But, in contrast with the case of continuous symbols, the splitting number k is in general larger than . [Copyright &y& Elsevier]

We study approximation methods for the Muskhelishvili integral equations on curves with corner points and establish necessary and sufficient conditions for their stability. [Copyright &y& Elsevier]

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see [12]. The proof involves elements from Integer Programming (totally unimodular systems of equations playing a role in particular) and employs Farkas' Lemma. The linear space of block upper triangular matrices can be viewed as being determined by a special pattern of zeros. The present paper is concerned with the question whether the decomposition result can be extended to situations where other, less restrictive, zero patterns play a role. It is shown that such generalizations do indeed hold for certain directed graphs determining the pattern of zeros. The graphs in question are what will be called L -free. This notion is akin to other graph theoretical concepts available in the literature, among them the one of being N -free in the sense of [16]. [ABSTRACT FROM AUTHOR]

Abstract: A new class of Banach algebra valued functions is identified for which the logarithmic residue with respect to a Cauchy domain vanishes (if and) only if the functions take invertible values in . Trace conditions and the extraction of elementary factors of the type play an important role. The class contains the Fredholm operator valued functions and the Banach algebra valued functions possessing a simply meromorphic resolvent as special instances. An example is given to show that new ground is covered and a long standing open problem is discussed from a fresh angle. [Copyright &y& Elsevier]

Let T(bjk) be a finite collection of infinite Toeplitz band matrices, let Tn(bjk) denote their (n + 1) × (n + 1) truncations, and put [This symbol cannot be presented in ASCII format] where . ·. p stands for the operator norm associated with the lp-norm 1 ≤ p < ∝ ) on Cn+1. We establish tight two-sided estimates for the difference Mp - . An. p. It is well known that if T(b) is a Hermitian Toeplitz band matrix and An = Tn(b), then M2 - . An. 2 goes to zero with polynomial speed. We show that such a slow convergence rate is, in a sense, an exceptional case, and we prove that in the generic case Mp - . An. p approaches zero with exponential speed. Our results yield good error estimates when computing the norms of certain infinite matrices via their large truncations and, conversely, when determining the norms of certain large matrices by having recourse to known norms of infinite matrices. [ABSTRACT FROM AUTHOR]