论文标题
Hausdorff操作员在Fock空间上
Hausdorff operators on Fock Spaces
论文作者
论文摘要
令$μ$为正真实轴上的正骨量度。我们研究积分操作员 $$ \ Mathcal {h}_μ（f）（z）= \ int_ {0}^{\ infty} \ frac {1} {t} {t} f \ left（\ frac {z} {z} {t} {t} {t} \ right） $$ 作用在[1，\ infty]，\，α> 0 $的Fock空间上$ f^{p}_α$，$ p \。它的作用很容易被视为瞬间序列的系数乘法 $$ μ_n= \ int_ {1}^{\ infty} \ frac {1} {t^{n+1}}} \，dμ（t）。 $$ 我们证明了这一点 \ begin {equation*} || \ Mathcal {h}_μ|| _ {f^{p}_α\ to f^{p}_α} = \ sup_ {n \ in \ MathBb {n}}μ_n，\ ,, \ ,, \ ,, \ ,, \ ,, \，\，\，\，\，1 \，\， \ end {equation*}一个小o，条件描述了每个$ f^{p}_α，\，p \ in（1，\ infty）$上$ \ mathcal {h}_μ$的紧凑性。此外，我们完全表征了$ \ Mathcal {H}_μ$的Schatten类成员资格。
Let $μ$ be a positive Borel measure on the positive real axis. We study the integral operator $$ \mathcal{H}_μ(f)(z)=\int_{0}^{\infty}\frac{1}{t}f\left(\frac{z}{t}\right)\,dμ(t),\quad z\in \mathbb{C}\,, $$ acting on the Fock spaces $F^{p}_α$, $p\in [1,\infty],\,α>0$. Its action is easily seen to be a coefficient multiplication by the moment sequence $$ μ_n= \int_{1}^{\infty}\frac{1}{t^{n+1}}\,dμ(t) . $$ We prove that \begin{equation*} ||\mathcal{H}_μ||_{F^{p}_α\to F^{p}_α}=\sup_{n\in\mathbb{N}}μ_n,\,\,\,\,\,1\leq p\leq \infty\,\,. \end{equation*} A little-o,condition describes the compactness of $\mathcal{H}_μ$ on every $F^{p}_α,\,p\in (1,\infty )$. In addition, we completely characterize the Schatten class membership of $\mathcal{H}_μ$.
