Vector norm inequalities for power series of operators in Hilbert spaces

Abstract

In this paper, vector norm inequalities that provides upper bounds for the Lipschitz quantity $\left\Vert f\left( T\right) x-f\left( V\right)x\right\Vert $ for power series $f\left( z\right) =\sum_{n=0}^{\infty}a_{n}z^{n},$ bounded linear operators $T,V$ on the Hilbert space $H$ and vectors $x\in H$ are established. Applications in relation to Hermite-Hadamard type inequalities and examples for elementary functions of interest are given as well.