On the Cauchy dual operator and duality for Banach spaces of analytic functions
Volume 271 / 2023
Abstract
Two related types of duality are investigated. The first is the duality for left-invertible operators and the second is the duality for Banach spaces of vector-valued analytic functions. We will examine a pair ($\mathcal B,\varPsi)$ consisting of a reflexive Banach space $\mathcal B$ of vector-valued analytic functions on an open set $\varOmega \subset \mathbb C$ on which a left-invertible multiplication operator acts and an operator-valued holomorphic function $\varPsi $ on an open set $\varOmega ^\prime \subset \mathbb C$. We prove that there exists a dual pair ($\mathcal B^\prime ,\varPsi ^\prime )$ such that the spaces $\mathcal B^\prime $ and $\mathcal B^*$ are isometrically isomorphic while the multiplication operator on $\mathcal B^\prime $ is isometrically equivalent to the adjoint of the left inverse of the multiplication operator on $\mathcal B$. In addition we show that $\varPsi $ and $\varPsi ^\prime $ are connected through the relation \[\langle (\varPsi^\prime ( \bar z e_1) (\lambda ),e_2\rangle = \langle e_1,(\varPsi ( \bar \lambda ) e_2)(z)\rangle\] for all $e_1,e_2\in E$, $z\in \varOmega $, $\lambda \in \varOmega ^\prime $, where $E$ is a Hilbert space.
If a left-invertible operator $T\in \boldsymbol B(\mathcal H)$ satisfies certain conditions, then both $T$ and the Cauchy dual operator $T^\prime $ can be modelled as the multiplication operator on reproducing kernel Hilbert spaces of vector-valued analytic functions $\mathscr H$ and $\mathscr H^\prime $, respectively. We prove that the Hilbert space of the dual pair of $(\mathscr H,\varPsi )$ coincides with $\mathscr H^\prime $, where $\varPsi $ is a certain operator-valued holomorphic function. Moreover, we characterize when the duality between $\mathscr H$ and $\mathscr H^\prime $ obtained by identifying them with $\mathcal H$ is the same as the duality obtained from the Cauchy pairing.
