Generalized multipliers for left-invertible analytic operators and its applications to commutant and reflexivity

TL;DR

This paper introduces generalized multipliers for left-invertible analytic operators, characterizes their commutants, and provides criteria for reflexivity of weighted shifts on directed trees, advancing operator theory understanding.

Contribution

It develops a new framework of generalized multipliers, characterizes commutants in this context, and establishes reflexivity criteria for weighted shifts on directed trees.

Findings

Generalized multipliers form a Banach algebra.

The commutant of such operators is characterized via these multipliers.

Two criteria for reflexivity of weighted shifts on directed trees are proven.

Abstract

We introduce generalized multipliers for left-invertible analytic operators. We show that they form a Banach algebra and characterize the commutant of such operators in its terms. In the special case, we describe the commutant of balanced weighted shift only in terms of its weights. In addition, we prove two independent criteria for reflexivity of weighted shifts on directed trees.