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


Abstract in English

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.

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