论文标题
广义希尔伯特操作员作用于Bloch型空间
Generalized Hilbert Operator Acting on Bloch Type Spaces
论文作者
论文摘要
在间隔[0,1）上设置$μ$为阳性的borel量度。对于$α> 0 $，hankel矩阵$ \ mathcal {h} _ {μ，α} =（μ__{n，k，α}）_ {n，k \ geq 0} $带有条目美元 $ \ MATHCAL {h} _ {μ，α}（f）（z）= \ sum_ {n = 0}^{\ infty} \ left（\ sum_ {\ sum_ {k = 0}^{\ infty} {\ infty}μ__{ $ f（z）= \ sum_ {k = 0}^{\ infty} a_ {k} z^{k} $ in nime disc $ \ mathbb {d} $。在本文中，我们表征了$ \ Mathcal {h} _ {μ，α} $（$α\ geq 2 $）的度量$μ$是一个有限的（spect。，紧凑）的操作员，来自bloch type space $ \ mathscr {b} $ \ mathscr {b} _ {α-1} $。我们还提供了必要的条件，$ \ MATHCAL {H} _ {μ，α} $是一个有界操作员，是通过在Bloch类型的一般情况下作用的一般情况。
Let $μ$ be a positive Borel measure on the interval [0,1). For $α>0$, the Hankel matrix $\mathcal{H}_{μ,α}=(μ_{n,k,α})_{n,k\geq 0}$ with entries $μ_{n,k,α}=\int_{[0,1)}\frac{Γ(n+α)}{n!Γ(α)}t^{n+k}dμ(t)$ formally induces the operator $$\mathcal{H}_{μ,α}(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n, k,α} a_{k}\right)z^{n} $$ on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ in the unit disc $\mathbb{D}$. In this paper, we characterize the measures $μ$ for which $\mathcal{H}_{μ,α}$ ($α\geq 2$) is a bounded (resp., compact) operator from the Bloch type space $\mathscr{B}_β$ ($0<β<\infty$) into $\mathscr{B}_{α-1}$. We also give a necessary condition for which $\mathcal{H}_{μ,α}$ is a bounded operator by acting on Bloch type spaces for general cases.
