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Bounded semigroups of matrices. (English) Zbl 0818.15006

Summary: In this note we prove two conjectures of I. Daubechies and J. C. Lagarias [ibid. 161, 227-263 (1992; Zbl 0746.15015)]. The first asserts that if \(\Sigma\) is a bounded set of matrices such that all left infinite products converge, then \(\Sigma\) generates a bounded semigroup. The second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices. One definition involves the conventional spectral radius, and one involves the operator norm.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A30 Algebraic systems of matrices
20M20 Semigroups of transformations, relations, partitions, etc.

Citations:

Zbl 0746.15015
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References:

[1] M.A. Berger, Random affine iterated function systems: Smooth curve generation, to appear.; M.A. Berger, Random affine iterated function systems: Smooth curve generation, to appear. · Zbl 0759.58021
[2] I. Daubechies and J.C. Lagarias, Set of matrices all infinite products of which converge, to appear.; I. Daubechies and J.C. Lagarias, Set of matrices all infinite products of which converge, to appear. · Zbl 0746.15015
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[7] Rota, G. C.; Strang, W. G., A note on the joint spectral radius, Indag. Math., 22, 379-381 (1960) · Zbl 0095.09701
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