Relationship between Cauchy dual operator and Cauchy dual for Banach spaces of analytic functions.

In this paper, two related types of dualities are investigated. The first is duality between left-invertible operators and the second is duality beetwen Banach spaces of vector-valued analytic functions. A left-invertible operator $T\in \boldsymbol{B}(\mathcal{H})$, which satisfies certain conditions can be modelled as a multiplication operator on a reproducing kernel Hilbert space $\mathscr{H}$ of vector-valued analytic functions on an annulus. If the Cauchy dual operator $T^\prime$ also satisfy these conditions, then for both operators $T$ and $T^\prime$ one can construct Hilbert spaces $\mathscr{H}$ and $\mathscr{H}^\prime$. In this paper, a characterization is given of a left-invertible operator $T\in \boldsymbol{B}(\mathcal{H})$ such that the duality between $\mathscr{H}$ and $\mathscr{H}^\prime$ spaces obtained by identifying them with $\mathcal{H}$ is the same as the duality obtained from the Cauchy pairing. Moreover, certain Banach spaces of vector-valued analytic functions on which a left-invertible multiplication operator acts are investigated. A construction of Cauchy dual space for these spaces is provided.