Functions and Means of Accretive Operators

Abstract

Let $A$ be a bounded accretive operator on a Hilbert space and $f(t)$ an operator monotone function on $(0, \infty)$ with $f(0)>-\infty$. Then, for $\epsilon >0$, analytic function $f (A+\epsilon I) $ is defined by Riesz-Dunford integral. We define $f(A)$ as the norm limit of it and show $$ f(A) = f(0)I + b A + \int_0^{\infty} (\frac{1}{\lambda} I - (\lambda I + A)^{-1}) d\mu(\lambda).$$ This is a generalization of fractional powers $$A^r = \frac{\sin r \pi}{\pi} \int_0^{\infty} (\frac{1}{\lambda} I - (\lambda I + A)^{-1}) \lambda ^{r} d\lambda \quad (0<r<1).$$ Let $A$ and $B$ be strictly accretive matrices, namely those real parts are positive definite. The geometric mean $A\# B$ has been introduced in Drury[6] and subsequently general matrix mean $A\sigma_f B$ in Bedrani-Kittaneh-Sababheh [3]. We extend these means to accretive, not necessarily strictly accretive, operators $A$ and $B$, and verify that $$A\# B= A^{1/2} B^{1/2}$$ if $A$ and $B$ are normal and commutative. Let $A$ be a strictly accretive operator. Then we show that $$0 \leqq \frac{1}{2} (A + A^*) \leqq A \# A^* \leqq 2(A^{-1} + (A^*)^{-1})^{-1},$$ and that $A \# A^* = | A |$ if and only if $A$ is normal. For a normal and strictly accretive operator $A$ we get \begin{align*} &|A|= \frac{1}{\pi}\int_0^{\infty}A (\lambda A + A^*)^{-1} A^* \lambda^{-1/2} d \lambda, \\ &A + A^* \leqq A^{1-r} A^{*r} + A^r A^{*(1-r)} \quad (0\leqq r \leqq 1). \end{align*}