Polynomials of generators of integrated semigroups

Abstract:

We give general sufficient conditions on $p$ and $A$, for $p(A)$ to generate an exponentially bounded holomorphic $k$-times integrated semigroup, where $p$ is a polynomial and $A$ is a linear operator on a Banach space. Corollaries include the following. (1) If $iA$ generates a strongly continuous group and $p$ is a polynomial of even degree with positive leading coefficient, then $- p(A)$ generates a strongly continuous holomorphic semigroup of angle $\frac {\pi } {2}$. (2) If $- A$ generates a strongly continuous holomorphic semigroup of angle $\theta$ and $p$ is an $n$th degree polynomial with positive leading coefficient, with $n\left ( {\tfrac {\pi } {2} - \theta } \right ) < \tfrac {\pi } {2}$, then $- p(A)$ generates a strongly continuous holomorphic semigroup of angle $\tfrac {\pi } {2} - n(\tfrac {\pi } {2} - \theta )$. (3) If $( - A)$ generates an exponentially bounded holomorphic $k$-times integrated semigroup of angle $\theta$, and $p$ and $\theta$ are as in (2), then $- p(A)$ generates an exponentially bounded holomorphic $(k + 1)$-times integrated semigroup of angle $\tfrac {\pi } {2} - n(\tfrac {\pi } {2} - \theta )$.