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 unitarily equivalent to the space $mathcal{B}^*$ and the following intertwining relations hold begin{equation*} mathscr{L} mathcal{U} = mathcal{U}mathscr{M}_z^* quadtext{and}quad mathscr{M}_zmathcal{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 relationbegin{equation*} langle(Psi^prime( bar{z}) e_1) (lambda),e_2rangle= langle e_1,(Psi( bar{ lambda}) e_2)(z)rangle end{equation*} for every $e_1,e_2in E$, $zin varOmega$, $lambdain varOmega^prime$. 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.