On the numerical range of sectorial forms

TL;DR

This paper establishes a precise bound for the sectorial angle of elliptic differential operators, improving understanding of their functional calculus properties across various boundary conditions and coefficient types.

Contribution

It provides a sharp, optimal bound for the sectorial form angle of non-symmetric elliptic operators, enhancing the analysis of their $ ext{H}^ ext{o}$-angle and calculus.

Findings

Sharper $ ext{H}^ ext{o}$-angle bounds for elliptic operators.

Optimal bounds for sectorial forms with various boundary conditions.

Improved $ ext{H}^ ext{o}$-calculus results for real-valued coefficients.

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.