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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]

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]