Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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^* \quad\text{and}\quad
  \mathscr{M}_z\mathcal{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 relation\begin{equation*}
  \langle(\Psi^\prime( \bar{z}) e_1) (\lambda),e_2\rangle= \langle e_1,(\Psi(
\bar{ \lambda}) e_2)(z)\rangle \end{equation*} for every $e_1,e_2\in E$, $z\in
\varOmega$, $\lambda\in \varOmega^\prime$.
  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.

11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg