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

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 unitary 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 $e_1,e_2\in E$, $z\in \Omega$, $\lambda\in \Omega^\prime$. 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 $\mathscr{H}$ 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.