Spectral Radius Inequalities for Functions of Operators Defined by Power Series

S. S. Dragomir

By the help of power series $f(z)=\sum_{n=0}^{\infty}a_nz^n$ we can naturally construct another power series that has as coefficients the absolute values of the coefficients of $f$, namely $f_a(z):=\sum_{n=0}^{\infty}|a_n|z^n$. Utilising these functions we show among others that \[ r[f(T)]eq f_a[r(T)] \] 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 \[ r^2[f(AB)]eq f_a(r^2(A))f_a(r^2(B)) \] and \[ r[f(ab)]eqfrac12[f_a(\|AB\|)+f_a(\|A^2\|^{1/2}\|B\|^{1/2})]. \]