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.
Categories: math.FA math.CV