论文标题
广义希尔伯特操作员在加权伯格曼的空间和迪里奇的空间上作用
Generalized Hilbert Operator Acting on Weighted Bergman Spaces and on Dirichlet 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$β> 0 $，概括的hankel矩阵$ \ mathcal {h} _ {μ，β} =（μ_{n，k，β}）_ {n，k \ geq0} $，带有条目$μ__{n，k，k，β} = \ int _ {[0.1）} \ frac {γ（n+β）} {n！γ（β）} t^{n+k}dμ（t）$，正式诱导操作员$ $ \ MATHCAL {MATHCAL {h} _ \ left（\ sum_ {k = 0}^\inftyμ_{n，k，k，β} a_k \ right）z^n $$在所有分析函数的空间上$ f（z）= \ sum_ {k = 0}^\ sum_ {k = 0}^\ inpty a_k z^n $在单位文件disc $ \ mathbbbb {d d} $中。在本文中，我们表征了$ [0,1）$的那些积极的borel措施，以使$ \ nathcal {h} _ {μ，β}（f）（z）= \ int _ {[0,1）} \ frac {f（f（f frac {f（t）}} $a_α^p（0 <p <\ infty，\;α> -1）$，其中我们描述了$ \ Mathcal {h} _ {μ，β}（β> 0）$的$ \ Mathcal {h} _ {h} _ {h} _ {useve（β> 0）$是一个有限的（spect。bergman space and dirhichletspace and dirice of the Bergman and compact）操作员。
Let $μ$ be a positive Borel measure on the interval [0,1). For $β> 0$, The generalized Hankel matrix $\mathcal{H}_{μ,β}= (μ_{n,k,β})_{n,k\geq0}$ with entries $μ_{n,k,β}= \int_{[0.1)}\frac{Γ(n+β)}{n!Γ(β)} t^{n+k}dμ(t)$, induces formally the operator $$\mathcal{H}_{μ,β}(f)(z)=\sum_{n=0}^\infty \left(\sum_{k=0}^\infty μ_{n,k,β}a_k\right)z^n$$ on the space of all analytic function $f(z)=\sum_{k=0}^ \infty a_k z^n$ in the unit disc $\mathbb{D}$. In this paper, we characterize those positive Borel measures on $[0,1)$ such that $\mathcal{H}_{μ,β}(f)(z)= \int_{[0,1)} \frac{f(t)}{(1-tz)^β} dμ(t)$ for all in weighted Bergman Spaces $A_α^p(0<p<\infty,\; α>-1)$, and among them we describe those for which $\mathcal{H}_{μ,β}(β>0)$ is a bounded(resp.,compact) operator on weighted Bergman spaces and Dirichlet spaces.
