An introduction of Berezin sectorial operators and its application to Berezin number inequalities

TL;DR

This paper introduces Berezin sectorial operators, a new class generalizing sectorial operators, and explores their properties, inequalities, and potential applications in operator theory on function spaces.

Contribution

It defines Berezin sectorial operators, compares them with classical sectorial operators, and derives new Berezin number inequalities with applications to specific operators.

Findings

Berezin sectorial operators can differ from classical sectorial operators.

Derived Berezin number inequalities including a weak power inequality.

Studied geometric properties of Berezin range for specific operators.

Abstract

We introduce a new class of operators, called Berezin sectorial operators, which generalizes classical sectorial operators. We provide examples on the Hardy-Hilbert space showing that there exist operators that are Berezin sectorial but not sectorial and that the Berezin sectorial index can be strictly smaller than the classical one. We derive Berezin number inequalities for this class, including a weak version of the power inequality, and study geometric properties of the Berezin range for finite-rank and weighted shift operators on the Dirichlet space. We also raise the question of whether similar constructions are possible for composition-differentiation operators on the Dirichlet space.