Powers of generators of holomorphic semigroups

Abstract

We show that when the (possibly unbounded) linear operator − A - A generates a bounded holomorphic semigroup of angle θ \theta , and n ( π / 2 − θ ) > π / 2 n\left ( {\pi /2 - \theta } \right ) > \pi /2 , then − A n - {A^n} generates a bounded holomorphic semigroup of angle π / 2 − n ( π / 2 − θ ) \pi /2 - n\left ( {\pi /2 - \theta } \right ) . When − A - A generates a bounded holomorphic semigroup of angle π / 2 \pi /2 , then, for all n n , − A n - {A^n} generates a bounded holomorphic semigroup of angle π / 2 \pi /2 .