Simultaneous Contractibility

Let C be a set of n x n complex matrices. For m = 1,2,...,\ Cm is the set of all products of matrices in C of length m. Denote by r̂( C) the joint spectral radius of C, that is, r̂( C) def=m→∞[_A∈ Cᵐ\| A\| ]1/m. We call Csimultaneously contractible if there is an invertible matrix S such that \\| S⁻¹AS\|;\ A∈ C\<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 x n complex matrices, that is, the maximum subset J of [0,1) such that if C is a bounded set of n x n complex matrices and r̂( 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.