On Cauchy dual operator and duality for Banach spaces of analytic functions
Pawe{\l} Pietrzycki

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 ( consisting of a reflexive Banach spaces of vector-valued analytic functions on which a left-invertible multiplication operator acts and an operator-valued holomorphic function . We prove that there exist a dual pair ( such that the space is unitarily equivalent to the space 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 is the unitary operator between…
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Taxonomy
TopicsAdvanced Banach Space Theory · Holomorphic and Operator Theory · Advanced Harmonic Analysis Research
