Similarities of $\omega$-accretive operators

Abstract

Given a number $0 \lt \omega \leq \frac{\pi}{2}$, an $\omega$-accretive operator is a sectorial operator $A$ on Hilbert space whose numerical range lies in the closed sector of all $z \in \mathbbC$ such that $\vert Arg(z)\vert \leq \omega$. It is easy to check that any such operator admits bounded imaginary powers, with $\Vert A^{it} \Vert \leq e^{\omega\vert t \vert$ for any $t \in \mathbbR$. We show that conversely, $A$ is similar to an $\omega$-accretive operator if $\Vert A^{it} \Vert \leq e^{\omega\vert t \vert$ for any $t \in \mathbbR$