Berger, Marc A.; Wang, Yang Bounded semigroups of matrices. (English) Zbl 0818.15006 Linear Algebra Appl. 166, 21-27 (1992). 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. Cited in 9 ReviewsCited in 164 Documents MSC: 15A18 Eigenvalues, singular values, and eigenvectors 15A30 Algebraic systems of matrices 20M20 Semigroups of transformations, relations, partitions, etc. Keywords:bounded semigroup; joint spectral radii; bounded set of matrices; spectral radius; operator norm Citations:Zbl 0746.15015 PDFBibTeX XMLCite \textit{M. A. Berger} and \textit{Y. Wang}, Linear Algebra Appl. 166, 21--27 (1992; Zbl 0818.15006) Full Text: DOI 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 [3] Jacobson, N., Structure of Rings, (Amer. Math. Soc. Colloq. Publ. XXXVII (1968)), Providence · JFM 65.1131.01 [4] Kaplansky, I., Fields and Rings (1972), Univ. of Chicago Press: Univ. of Chicago Press Chicago · Zbl 1001.16500 [5] Micchelli, C. A.; Prautzsch, H., Uniform refinement of curves, Linear Algebra Appl., 114/115, 841-870 (1989) · Zbl 0668.65011 [6] Random Matrices and Their Applications, (Cohen, J. E.; etal., Contemp. Math., 50 (1986), Amer. Math. Soc: Amer. Math. Soc Providence) [7] Rota, G. C.; Strang, W. G., A note on the joint spectral radius, Indag. Math., 22, 379-381 (1960) · Zbl 0095.09701 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.