A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Volume 271 / 2023

Paweł Pietrzycki Studia Mathematica 271 (2023), 121-150 MSC: Primary 47B38; Secondary 47B32, 47B33. DOI: 10.4064/sm210907-19-9 Published online: 15 June 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.

Authors

  • Paweł PietrzyckiWydział Matematyki i Informatyki
    Uniwersytet Jagielloński
    30-348 Kraków, Poland
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image