A field theoretic operator model and Cowen-Douglas class

TL;DR

This paper develops a functional model for specific bounded operators with rank-one self-commutator, utilizing a geometric framework and field theory interpretation to analyze their resolvent behavior outside the spectrum.

Contribution

It introduces a novel operator model based on a geometric and field theory perspective, extending the understanding of hyponormal operators with rank-one self-commutator.

Findings

Model transfers operator action outside the spectrum

Uses analytic multipliers and Cauchy transforms

Provides a field theory interpretation of the resolvent

Abstract

In resonance to a recent geometric framework proposed by Douglas and Yang, a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space is developed. By taking advantage of the refined existing theory of the principal function of a hyponormal operator we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. A natural field theory interpretation of the resulting resolvent functional model is proposed.