### 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

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