Spectral Radius Inequalities for Functions of Operators Dened by Power Series

Abstract

By the help of power series f (z) =
P
1n=0 anzn we can naturally
construct another power series that has as coe¢ cients the absolute values of
the coe¢ cients of f, namely fa (z) :=
P
1n=0 janj 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 kTk is its norm. When we have A and B two
commuting operators, then
r2 [f (AB)] fa