论文标题
组成运算符的beurling类型不变子空间
Beurling type invariant subspaces of composition operators
论文作者
论文摘要
令$ \ mathbb {d} $为$ \ mathbb {c} $中的打开单位磁盘，让$ h^2 $表示在$ \ mathbb {d} $上表示hardy空间，让$φ：\ mathbb {d} \ rightArrow \ rightArrow \ rightArrow \ mathbb {d} $ holomorphic $ holomorphic $ self $ ho holomorphic $ a $ $ h^2 $上的组成操作员$c_φ$由\ [（c_φf）（z）（z）= f（φ（z））\ quad \ quad（f \ in H^2，h^2，\，z \，z \ in \ in \ mathbb {d}）。 \]用$ \ Mathcal {s}（\ Mathbb {d}）$表示所有具有holomorphic且由Modulus在$ \ Mathbb {d} $上的函数的集合，即\ [\ nathcal {\ Mathcal {s}（s}（s}（s}）） h^\ infty（\ mathbb {d}）：\ |ψ\ | _ {\ infty}：= \ sup_ {z \ in \ mathbb {d}} |ψ（z）| \ leq 1 \}。 \] $ \ Mathcal {s}（\ Mathbb {d}）$的元素称为Schur函数。本文的目的是回答以下有关综合操作员不变子空间的问题：表征$φ$，$ \ mathbb {d} $的holomorphic自态自我地图和内在函数$θ\ in H^\ in h^\ infty in h^\ infty in h h^\ infty（\ mathbb {d}） $C_φ$。我们证明了以下结果：$c_φ（θh^2）\subseteqθh^2 $，并且仅当\ [\ frac {\ frac {frac {θ\circcφ}θ\ in \ mathcal {s}（s}}（\ mathbb {d}））。 \]此分类还允许我们在组成运算符的Beurling类型不变子空间上恢复或改善一些已知结果。
Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, let $H^2$ denote the Hardy space on $\mathbb{D}$ and let $φ: \mathbb{D} \rightarrow \mathbb{D}$ be a holomorphic self map of $\mathbb{D}$. The composition operator $C_φ$ on $H^2$ is defined by \[ (C_φ f)(z)=f(φ(z)) \quad \quad (f \in H^2,\, z \in \mathbb{D}). \] Denote by $\mathcal{S}(\mathbb{D})$ the set of all functions that are holomorphic and bounded by one in modulus on $\mathbb{D}$, that is \[ \mathcal{S}(\mathbb{D}) = \{ψ\in H^\infty(\mathbb{D}): \|ψ\|_{\infty} := \sup_{z \in \mathbb{D}} |ψ(z)| \leq 1\}. \] The elements of $\mathcal{S}(\mathbb{D})$ are called Schur functions. The aim of this paper is to answer the following question concerning invariant subspaces of composition operators: Characterize $φ$, holomorphic self maps of $\mathbb{D}$, and inner functions $θ\in H^\infty(\mathbb{D})$ such that the Beurling type invariant subspace $θH^2$ is an invariant subspace for $C_φ$. We prove the following result: $C_φ (θH^2) \subseteq θH^2$ if and only if \[ \frac{θ\circ φ}θ \in \mathcal{S}(\mathbb{D}). \] This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of composition operators.
