Functional calculus for generators of symmetric contraction semigroups

TL;DR

This paper establishes an optimal H"ormander-type functional calculus for generators of symmetric contraction semigroups on $L^p$ spaces, extending the understanding of their spectral properties.

Contribution

It proves that such generators admit an optimal holomorphic functional calculus in a specific sector of the complex plane for all $1<p< abla$, a significant advancement in operator theory.

Findings

Proves the existence of a H"ormander-type calculus for generators of symmetric contraction semigroups.

Shows the angle of the sector is optimal for the functional calculus.

Extends the calculus to all $L^p$ spaces with $1<p< abla$.

Abstract

We prove that every generator of a symmetric contraction semigroup on a $\sigma$-finite measure space admits, for $1<p<\infty$, a H\"ormander-type holomorphic functional calculus on $L^p$ in the sector of angle $\phi^*_p=\arcsin|1-2/p|$. The obtained angle is optimal.