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

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 ${\mathcal{B}}^{\ast }$ 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 ${\mathcal{B}}^{\ast }$. In addition we show that $\Psi$ and ${\Psi }^{\prime }$ 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 $e_1,e_2\in E$, $z\in \Omega$, $\lambda \in {\Omega }^{\prime }$. 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 $\mathcal{H}$ and ${\mathcal{H}}^{\prime }$, respectively. We prove that Hilbert space of the dual pair of $(\mathcal{H},\Psi )$ coincide with ${\mathcal{H}}^{\prime }$, where $\Psi$ is a certain operator-valued holomorphic function. Moreover, we characterize when the duality between spaces $\mathcal{H}$ and ${\mathcal{H}}^{\prime }$ obtained by identifying them with $\mathcal{H}$ is the same as the duality obtained from the Cauchy pairing.

References in corpus (5)

• The Cauchy dual and 2-isometric liftings of concave operators
• An Analytic Model for Left-Invertible Weighted Shifts on Directed Trees
• Hilbert spaces of analytic functions with a contractive backward shift
• A Shimorin-type analytic model on an annulus for left-invertible operators and applications
• Generalized multipliers for left-invertible operators and applications