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Simultaneous contractibility. (English) Zbl 0912.15033

Authors’ summary: Let C be a set of n×n complex matrices. For m=1,2,, C m is the set of all products of matrices in C of length m. Denote by r ^(C) the joint spectral radius of C, that is,

r ^(C):=lim sup m [sup AC m A] 1 m ·

We call C simultaneously contractible if there is an invertible matrix S such that

sup{S -1 AS;AC}<1,

where · is the spectral norm. This paper is primarily devoted to determining the optimal joint spectral radius range for simultaneous contractibility of bounded sets of n×n complex matrices, that is, the maximum subset J of [0,1) such that if C is a bounded set of n×n complex matrices and r ^(C)J, then C is simultaneously contractible. The central result proved in this paper is that this maximum subset is [0,1 n)· Our method of proof is based on a matrix-theoretic version of complex John’s ellipsoid theorem and the generalized Gelfand spectral radius formula.

MSC:
15A60Applications of functional analysis to matrix theory
15A18Eigenvalues, singular values, and eigenvectors
47A10Spectrum and resolvent of linear operators