On the numerical range of sectorial forms

Abstract

We provide a sharp and optimal generic bound for the angle of the sectorial form associated to a non-symmetric second-order elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper ${\mathcal{H}}^{\infty }$-angle for the ${\mathcal{H}}^{\infty }$-calculus on $L_p$ for all $p\in (1,\infty )$ if the coefficients are real valued.