Sectorial operator
= Sectorial operator
{title2=$A$}
A densely defined <closed operator> is sectorial of angle $\theta$ if its <spectrum> lies in $\{z:|\arg z|\leq\theta\}\cup\{0\}$ and, for every larger angle $\phi<\pi$, its <resolvent> outside that larger sector satisfies $\|(z-A)^{-1}\|\leq C_\phi/|z|$. If $\theta<\pi/2$, then $-A$ generates an <analytic semigroup>. A real shift of the generator allows exponential growth without changing the short-time smoothing estimates.