Totally Accretive Operators

Abstract

Let $A$ be a (possibly unbounded) linear operator on a Banach space. We show that, when $A$ generates a uniformly bounded strongly continuous semigroup ${\left \{ {{e^{ - tA}}} \right \}_{t \geq 0}}$, then ${A^2}$ generates a bounded holomorphic semigroup (BHS) of angle $\theta$ if and only if $A$ generates a BHS of angle $\theta / 2 + \pi / 4$. We show that each power of $A$ generates a uniformly bounded strongly continuous semigroup if and only if $A$ generates a BHS of angle $\pi / 2$ if and only if each power of $A$ generates a BHS of angle $\pi / 2$. If $A$ is a linear operator on a Hilbert space, then each power of $A$ generates a strongly continuous contraction semigroup if and only if $A$ is positive selfadjoint.