论文标题
关于Cauchy双重操作员和双重性，用于分析功能的Banach空间
On Cauchy dual operator and duality for Banach spaces of analytic functions
论文作者
论文摘要
在本文中，研究了两种相关的二元性类型。第一个是剩余的可逆运算符和第二个之间的二元性是矢量值分析函数的Banach空间之间的二元性。我们将检查一对（$ \ MATHCAL {B}，ψ）$由反射Banach空间组成的$ \ Mathcal {B} $的矢量值分析函数，在该功能上，剩余的可连续性乘法操作员行为和操作员带有操作员的holomorphic函数$ψ$ψ$。我们证明存在一对（$ \ Mathcal {b}^\ prime，ψ^\ prime）$，使得空间$ \ nathcal {b}^\ prime $在单位上等同于space $ \ nathcal {b} \ begin {equation*} \ Mathscr {l} \ Mathcal {u} = \ Mathcal {u} \ Mathscr {m} _z^* \ Quad \ text \ text \ text {and} \ quad \ Mathscr {m} _z \ Mathcal {u} = \ Mathcal {u} \ Mathscr {l}^*， \ end {equation*}其中$ \ mathcal {u} $是$ \ Mathcal {b}^\ prime $和$ \ MATHCAL {B}^*$之间的单一操作员。此外，我们表明$ψ$和$ψ^\ prime $是通过关系\ begin {equation*}连接的 \ langle（ψ^\ prime（\ bar {Z}） $λ\ in \ varomega^\ prime $。 如果剩下的可逆操作员$ t $满足某些条件，则可以将$ t $和cauchy双运算符$ t^\ prime $建模为乘法运算符，用于重现矢量 - valuew分析函数$ \ mathscr {h} $和$ \ mathscr＆$ \ mathscr $ \ \ \ \ \ \ \ prime $ prime的矢量 - 值分析功能的内核hilbert空间。我们证明，双对的希尔伯特空间$（\ mathscr {h}，ψ）$与$ \ mathscr {h}^\ prime $重合，其中$ψ$是某个操作员值得称赞的全态函数。此外，我们表征何时空间之间的二元性$ \ mathscr {h} $和$ \ mathscr {h}^\ prime $通过用$ \ mathcal {h} $识别它们获得的二元性与从牛chy配对中获得的二元性相同。
In this paper, two related types of dualities are investigated. The first is the duality between left-invertible operators and the second is the duality between Banach spaces of vector-valued analytic functions. We will examine a pair ($\mathcal{B},Ψ)$ 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 $Ψ$. We prove that there exist a dual pair ($\mathcal{B}^\prime,Ψ^\prime)$ 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 $Ψ$ and $Ψ^\prime$ are connected through the relation\begin{equation*} \langle(Ψ^\prime( \bar{z}) e_1) (λ),e_2\rangle= \langle e_1,(Ψ( \bar{ λ}) e_2)(z)\rangle \end{equation*} for every $e_1,e_2\in E$, $z\in \varOmega$, $λ\in \varOmega^\prime$. 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 $(\mathscr{H},Ψ)$ coincide with $\mathscr{H}^\prime$, where $Ψ$ 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.
