Spectral radius inequalities for functions of operators defined by power series

Abstract

By the help of power series f(z)=??,n=0 anzn we can naturally construct another power series that has as coefficients the absolute values of the coefficients of f , namely fa(z):= ??,n=0 |an|zn. Utilising these functions we show among others that r[f(T)] ? fa [r(T)] where r (T) denotes the spectral radius of the bounded linear operator T on a complex Hilbert space while ||T|| is its norm. When we have A and B two commuting operators, then r2[f(AB)]? fa(r2(A)) fa(r2(B)) and r[f(AB)]?1/2[fa(||AB||)+fa(||A2||1/2||B2||1/2)].