Simultaneous Schur stability Original Research Article

Let Σ be a set of n × n complex matrices. Denote by script capital l(∑) the multiplicative semi-group generated by Σ. For an n × n complex matrix A, the spectrum of A and spectral radius of A are denoted by σ(A) and r(A), respectively. Motivated by a phenomenon in the closed unit disc of the complex plane, we give a rigorous definition of simultaneous Schur stability as follows. Σ is said to be simultaneously Schur stable if r(A)less-than-or-equals, slant1(Aset membership, variantscript capital l(∑)) and 1negated set membershipσ(A)(A)set membership, variantscript capital l(∑)). Σ is said to be asymptotically stable if there exists a norm · on Cn such that sup A; A ε Σ < 1. It is proved that for a bounded set Σ of n × n complex matrices, Σ is asymptotically stable if and only if it is simultaneously Schur stable. By way of “simultaneous Schur stability”, some applications are illustrated, especially an analytic-combinatorial proof of a recent result of considerable depth: Generalized Gelfand spectral radius formula.