On Cauchy dual operator and duality for Banach spaces of analytic functions
Paweł Pietrzycki
TL;DR
The paper investigates dualities in two intertwined settings: left-invertible operators and Banach spaces of vector-valued analytic functions, linking them via a Cauchy-dual framework and an operator-valued holomorphic map $\Psi$. It develops an analytic-model approach whereby a left-invertible operator $T$ and its Cauchy dual $T'$ become multiplication and backward-shift operators on reproducing kernel Hilbert spaces, with intertwining relations $\mathscr{L} \mathcal{U} = \mathcal{U} \mathscr{M}_z^*$ and $\mathscr{M}_z \mathcal{U} = \mathcal{U} \mathscr{L}^*$ realized by a unitary $\mathcal{U}$. The authors construct a dual Banach-space pair $(\mathcal{B}^\u00u-upright prime,\Psi^\u00u-upright prime)$ from any admissible $(\mathcal{B},\Psi)$, proving the required intertwining and establishing a bridge between the analytic-model duality and the Cauchy pairing via a precise inner-product relation between $\Psi$ and $\Psi^\u00u-upright prime$. The framework encompasses classical spaces (Hardy, Bergman, Dirichlet) and annulus-model spaces, and extends to weighted composition operators, clarifying when the dualities coincide with the Cauchy pairing on dense subspaces, thereby unifying operator-theoretic and function-theoretic dualities with concrete, computable structures.
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 ($\mathcal{B},Ψ)$ 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 $Ψ$. We prove that there exist a dual pair ($\mathcal{B}^\prime,Ψ^\prime)$ 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 $Ψ$ and $Ψ^\prime$ are connected through the relation\begin{equation*} \langle(Ψ^\prime( \bar{z}) e_1) (λ),e_2\rangle= \langle e_1,(Ψ( \bar{ λ}) e_2)(z)\rangle \end{equation*} for every $e_1,e_2\in E$, $z\in \varOmega$, $λ\in \varOmega^\prime$. 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 $(\mathscr{H},Ψ)$ coincide with $\mathscr{H}^\prime$, where $Ψ$ 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.