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 $$f(A)=f(0)I+bA+{\int }_0^{\infty }(\frac{1}{\lambda }I-(\lambda I+A{)}^{-1})d\mu (\lambda ).$$ This is a generalization of fractional powers $$A^r=\frac{\sin r\pi }{\pi }{\int }_0^{\infty }(\frac{1}{\lambda }I-(\lambda I+A{)}^{-1}){\lambda }^{r}d\lambda (0<r<1).$$ 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 $$0\leqq \frac{1}{2}(A+A^{\ast })\leqq A\#A^{\ast }\leqq 2(A^{-1}+(A^{\ast }{)}^{-1}{)}^{-1},$$ and that $A\#A^{\ast }=|A|$ if and only if $A$ is normal. For a normal and strictly accretive operator $A$ we get \begin{align*} &|A|= \frac{1}π\int_0^{\infty}A (λA + A^*)^{-1} A^* λ^{-1/2} d λ, \\ &A + A^* \leqq A^{1-r} A^{*r} + A^r A^{*(1-r)} \quad (0\leqq r \leqq 1). \end{align*}