Some numerical radius inequalities for power series of operators in Hilbert spaces

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|z n. Utilizing these functions, we show among others that w[f(T)] ≤ fa [w(T)] and w[f(T)] ≤ 1/2[fa(||T||) + f a(||T2||1/2)], where w(T) denotes the numerical radius of the bounded linear operator T on a complex Hilbert space, while ||T|| is its norm. © 2013 Dragomir.