Closed sectorial forms and one-parameter contraction semigroups

Yu. M. Arlinskii

Matematičeskie zametki, Tome 61 (1997) no. 5, pp. 643-654

Résumé

Suppose that $s[u,v]$ is a closed sesquilinear sectorial form with vertex at zero, half-angle$\alpha\in[0,\pi/2)$, and dense domain $\mathscr D(s)$ in a Hilbert space $H$, $S$ is them-sectorial operator associated with $s$, $S_R$ is the real part of $S$, and $T(t)=\exp(-tS)$ is the contraction semigroup with generator $-S$, holomorphic in the sector $|\arg t|\pi/2-\alpha$. We characterizes in terms of $T(t)$. In particular, we prove that the following conditions: 1) $u\in\mathscr D(s)$; 2) the function $\|T(t)u\|$ is differentiable at zero; 3) the function $\bigl(T(t)u,u\bigr)$ is differentiable at zero.