On Cauchy dual operator and duality for Banach spaces of analytic functions

TL;DR

This paper explores duality relationships between left-invertible operators and Banach spaces of vector-valued analytic functions, establishing a framework for their dualities and model representations via multiplication operators.

Contribution

It introduces a duality framework connecting Banach spaces of analytic functions and operators, and characterizes when these dualities align with Cauchy pairings.

Findings

Existence of dual pairs with intertwining relations

Modeling of operators as multiplication operators on reproducing kernel Hilbert spaces

Characterization of duality consistency with Cauchy pairing

Abstract

In this paper, two related types of dualities are investigated. The first is the duality between left-invertible operators and the second is the duality between Banach spaces of vector-valued analytic functions. We will examine a pair ($\mathcal{B},\Psi)$ consisting of a reflexive Banach spaces $\mathcal{B}$ of vector-valued analytic functions on which a left-invertible multiplication operator acts and an operator-valued holomorphic function $\Psi$. We prove that there exist a dual pair ($\mathcal{B}^\prime,\Psi^\prime)$ such that the space $\mathcal{B}^\prime$ is unitarily equivalent to the space $\mathcal{B}^*$ and the following intertwining relations hold \begin{equation*} \mathscr{L} \mathcal{U} = \mathcal{U}\mathscr{M}_z^* \quad\text{and}\quad \mathscr{M}_z\mathcal{U} = \mathcal{U} \mathscr{L}^*, \end{equation*} where $\mathcal{U}$ is the unitary operator between $\mathcal{B}^\prime$ and $\mathcal{B}^*$. In addition we show that $\Psi$ and $\Psi^\prime$ are connected through the relation\begin{equation*} \langle(\Psi^\prime( \bar{z}) e_1) (\lambda),e_2\rangle= \langle e_1,(\Psi( \bar{ \lambda}) e_2)(z)\rangle \end{equation*} for every $e_1,e_2\in E$, $z\in \varOmega$, $\lambda\in \varOmega^\prime$. If a left-invertible operator $T$ satisfies certain conditions, then both $T$ and the Cauchy dual operator $T^\prime$ can be modelled as a multiplication operator on reproducing kernel Hilbert spaces of vector-valued analytic functions $\mathscr{H}$ and $\mathscr{H}^\prime$, respectively. We prove that Hilbert space of the dual pair of $(\mathscr{H},\Psi)$ coincide with $\mathscr{H}^\prime$, where $\Psi$ is a certain operator-valued holomorphic function. Moreover, we characterize when the duality between spaces $\mathscr{H}$ and $\mathscr{H}^\prime$ obtained by identifying them with $\mathcal{H}$ is the same as the duality obtained from the Cauchy pairing.