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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]

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]