Self-commutators of Toeplitz operators and isoperimetric inequalities

TL;DR

This paper explores the geometric bounds of self-commutators of Toeplitz operators, linking operator theory with classical geometric inequalities like isoperimetric, Faber-Krahn, and Saint-Venant, and proposes conjectures for improved bounds.

Contribution

It extends the analysis of self-commutators of Toeplitz operators to Bergman spaces, deriving new geometric bounds and connecting them to fundamental inequalities in shape optimization.

Findings

Derived lower bounds reflecting domain geometry for Toeplitz operators on Bergman space.

Connected operator inequalities to classical geometric inequalities like Faber-Krahn and Saint-Venant.

Proposed a conjecture for an improved version of Putnam's inequality in this setting.

Abstract

For a hyponormal operator, C. R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality. For a nontrivial domain, we compare these estimates to exact results. Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain. When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants). We conjecture an improved version of Putnam's inequality within this restricted setting.