Cowen-Douglas Operator and Shift on Basis

TL;DR

This paper explores the relationship between Cowen-Douglas operators and shift operators on various bases, establishing conditions under which such operators can be represented as adjoints of backward shifts and highlighting limitations on their basis representations.

Contribution

It demonstrates that Cowen-Douglas operators can be realized as adjoints of backward shifts on general bases and shows they cannot be shifts on Markushevicz bases for rank n ≥ 2.

Findings

Cowen-Douglas operators are adjoints of backward shifts on certain bases.

They cannot be shifts on Markushevicz bases for n ≥ 2.

Provides a basis-theoretic model for these operators.

Abstract

In this paper we show a Cowen-Douglas operator $T \in \mathcal{B}_{n}(\Omega)$ is the adjoint operator of some backward shift on a general basis by choosing nice cross-sections of its complex bundle $E_{T}$. Using the basis theory model, we show that a Cowen-Douglas operator never be a shift on some Markushevicz basis for $n \ge 2$.