﻿ Bloch空间上加权复合算子的紧性

# Bloch空间上加权复合算子的紧性The Compactness of Weighted Composition Operator on Bloch Sapce

Abstract: If f is an analytic function in the unit disk D , the weighted composition operator is defined as following: . In this paper, using the function theoretic of Ψ and φ characterize the compactness of weighted composition operator on Bloch space.

1. 引言

$D=\left\{z:|z|<1\right\}$ 是复平面 $C$ 上的单位圆盘。 $H\left(D\right)$ 表示单位圆盘 $D$ 上的解析函数。设 $a\in D$${\sigma }_{a}\left(z\right)=\frac{a-z}{1-\stackrel{¯}{a}z}$ 表示单位圆盘 $D$ 上的莫比乌斯变换。

${\int }_{D}{|f\left(z\right)|}^{p}\text{d}\sigma \left(z\right)<\infty ,$

${‖f‖}_{\beta }={\mathrm{sup}}_{z\in D}|{f}^{\prime }\left(z\right)|\left(1-{|z|}^{2}\right)<\infty ,$

${‖f‖}_{\beta }\approx {\mathrm{sup}}_{a\in D}{‖f\circ {\sigma }_{a}-f\left(a\right)‖}_{{A}^{2}}.$

${\mathrm{sup}}_{z\in \partial D}|{f}^{\prime }\left(z\right)|\left(1-{|z|}^{2}\right)=0,$

${\mathrm{sup}}_{0

${‖f‖}_{*}^{2}={\mathrm{sup}}_{I\subset \partial D}\frac{1}{2\text{π}}{\int }_{I}{|f\left(\varsigma \right)-{f}_{I}|}^{2}\frac{\text{d}\varsigma }{2\text{π}}<\infty ,$

${‖f‖}_{*}\approx {\mathrm{sup}}_{a\in D}{‖f\circ {\sigma }_{a}-f\left(a\right)‖}_{{H}^{2}}.$

$\left({C}_{\phi }f\right)\left(z\right)=f\left(\phi \left(z\right)\right),\text{\hspace{0.17em}}\left({M}_{\psi }f\right)\left(z\right)=\psi \left(z\right)f\left(z\right).$

$\psi$$\phi$ 诱导的加权复合算子 ${W}_{\psi ,\phi }$ 定义如下：

$\left({W}_{\psi ,\phi }\right)f\left(z\right)=\psi \left(z\right)f\left(\phi \left(z\right)\right).$

$\underset{n\to \infty }{\mathrm{lim}}{‖{\phi }^{n}‖}_{*}=0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\underset{|\phi \left(a\right)|\to 1}{\mathrm{lim}}{‖{\sigma }_{a}\circ \phi ‖}_{*}=0.$

$\underset{n\to \infty }{\mathrm{lim}}{‖{\phi }^{n}‖}_{*}=0.$

$\underset{n\to \infty }{\mathrm{lim}}{‖\psi {\phi }^{n}‖}_{*}=0,\text{\hspace{0.17em}}\underset{|\phi \left(a\right)|\to 1}{\mathrm{lim}}\left(\mathrm{log}\frac{2}{1-{|\phi \left(a\right)|}^{2}}\right){‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{H}^{2}}=0.$

2. 主要结论

${\mathrm{sup}}_{a\in D}{‖f\circ {\sigma }_{a}-f\left(a\right)‖}_{{A}^{2}}\approx {\mathrm{sup}}_{a\in D}{‖f\circ {\sigma }_{a}-f\left(a\right)‖}_{{A}^{p}}.$

$\begin{array}{l}{‖\left(\psi \circ {\sigma }_{a}-\psi \left(a\right)\right)\cdot \left(f\circ \phi \circ {\sigma }_{a}-f\left(\phi \left(a\right)\right)\right)‖}_{{A}^{2}}^{2}\\ \le C{‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}\cdot {\mathrm{sup}}_{a\in D}{‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}.\end{array}$

$|f\left(z\right)|\le \mathrm{log}\frac{2}{1-{|z|}^{2}}{‖f‖}_{B}.$

$\underset{r\to 1}{\mathrm{lim}}{\mathrm{sup}}_{\left\{a\in D:|\phi \left(a\right)|>r\right\}}\alpha \left(\psi ,\phi ,a\right)=0,$ (1)

$\underset{r\to 1}{\mathrm{lim}}{\mathrm{sup}}_{\left\{a\in D:|\phi \left(a\right)|>r\right\}}\beta \left(\psi ,\phi ,a\right)=0,$ (2)

$\underset{r\to 1}{\mathrm{lim}}{\mathrm{sup}}_{\left\{a\in D:|\phi \left(a\right)|>r\right\}}{\int }_{E\left(\phi ,a,t\right)}{|\psi \circ {\sigma }_{a}\left(z\right)|}^{2}\text{d}\sigma \left(z\right)=0,$ (3)

$|\phi \left({a}_{n}\right)|\to 1\left(n\to \infty \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha \left(\psi ,\phi ,{a}_{n}\right)\ge \delta >0.$

$\alpha \left(\psi ,\phi ,{a}_{n}\right)\le {\mathrm{sup}}_{{a}_{n}\in D}{‖\psi \circ {\sigma }_{{a}_{n}}-\psi \left({a}_{n}\right)‖}_{{A}^{2}}+{‖{W}_{\psi ,\phi }{f}_{n}‖}_{B}.$

${\mathrm{sup}}_{{a}_{n}\in D}{‖\psi \circ {\sigma }_{{a}_{n}}-\psi \left({a}_{n}\right)‖}_{{A}^{2}}\le {\left(\mathrm{log}\frac{2}{1-{|\phi \left({a}_{n}\right)|}^{2}}\right)}^{-1}{\mathrm{sup}}_{a\in D}\beta \left(\psi ,\phi ,a\right)\to 0.$

$\underset{n\to \infty }{\mathrm{lim}}\alpha \left(\psi ,\phi ,{a}_{n}\right)=0,$

$|\phi \left({a}_{n}\right)|\to 1\left(n\to \infty \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta \left(\psi ,\phi ,{a}_{n}\right)\ge \delta >0.$

$\begin{array}{c}\beta \left(\psi ,\phi ,{a}_{n}\right)={‖\psi \circ {\sigma }_{{a}_{n}}-\psi \left({a}_{n}\right)\cdot {g}_{n}\left(\phi \left({a}_{n}\right)\right)‖}_{{A}^{2}}\\ \le C{‖{g}_{n}‖}_{B}{\left({‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}\cdot {\mathrm{sup}}_{{a}_{n}\in D}{‖\psi \circ {\sigma }_{{a}_{n}}-\psi \left({a}_{n}\right)‖}_{{A}^{2}}\right)}^{1/2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{‖\left({W}_{\psi ,\phi }{g}_{n}\right)\circ {\sigma }_{a}-\left({W}_{\psi ,\phi }{g}_{n}\right)\left(a\right)‖}_{{A}^{2}}+\alpha \left(\psi ,\phi ,{a}_{n}\right){‖{g}_{n}\circ {\sigma }_{a}-{g}_{n}\left(a\right)‖}_{{A}^{2}}.\end{array}$

${‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}\cdot {\mathrm{sup}}_{{a}_{n}\in D}{‖\psi \circ {\sigma }_{{a}_{n}}-\psi \left({a}_{n}\right)‖}_{{A}^{2}}\to 0$$\alpha \left(\psi ,\phi ,{a}_{n}\right)\to 0.$

$\underset{n\to \infty }{\mathrm{lim}}\beta \left(\psi ,\phi ,{a}_{n}\right)=0,$

$\underset{r\to 1}{\mathrm{lim}}{\mathrm{sup}}_{\left\{a\in D:|\phi \left(a\right)|>r\right\}}{\int }_{E\left(\phi ,a,t\right)}{|\psi \circ {\sigma }_{a}\left(z\right)|}^{2}\text{d}\sigma \left(z\right)\ge \delta >0.$

$\begin{array}{c}{‖{W}_{\psi ,\phi }{f}_{n}‖}_{B}^{2}\ge {‖\left(\psi \circ {\sigma }_{{a}_{n}}\right)\cdot {f}_{n}\circ \phi \circ {\sigma }_{{a}_{n}}‖}_{{A}^{2}}^{2}\\ \ge {\int }_{E\left(\phi ,{a}_{n},{t}_{n}\right)}{|\psi \circ {\sigma }_{a}\left(z\right)|}^{2}{t}_{n}^{2n}\text{d}\sigma \left(z\right)\ge {t}_{n}^{2n}\delta .\end{array}$

${\mathrm{sup}}_{\left\{|\phi \left(a\right)|>r\right\}}\mathrm{max}\left\{\alpha \left(\psi ,\phi ,a\right)+\beta \left(\psi ,\phi ,a\right),{\left(\mathrm{log}\frac{1}{1-{|\phi \left(a\right)|}^{2}}\right)}^{1/2}\right\}<\epsilon ,$ (4)

${\mathrm{sup}}_{\left\{a\in D:|\phi \left(a\right)|\le r\right\}}{{\int }_{E\left(\phi ,a,t\right)}|\psi \circ {\sigma }_{a}\left(z\right)|}^{2}\text{d}\sigma \left(z\right)<{\epsilon }^{4},$ (5)

${\mathrm{sup}}_{\left\{n\ge {n}_{0}\right\}}{\mathrm{max}}_{w\in Q}|{f}_{n}\left(w\right)|<\epsilon <1,$ (6)

$\begin{array}{c}{‖{W}_{\psi ,\phi }{f}_{n}‖}_{B}\le |\left({W}_{\psi ,\phi }{f}_{n}\right)\left(0\right)|+{\mathrm{sup}}_{|\phi \left(a\right)|>r}{‖\left({W}_{\psi ,\phi }{f}_{n}\right)\circ {\sigma }_{a}-\left({W}_{\psi ,\phi }{f}_{n}\right)\left(a\right)‖}_{{A}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\mathrm{sup}}_{|\phi \left(a\right)|\le r}{‖\left({W}_{\psi ,\phi }{f}_{n}\right)\circ {\sigma }_{a}-\left({W}_{\psi ,\phi }{f}_{n}\right)\left(a\right)‖}_{{A}^{2}}\\ ={A}_{1}+{A}_{2}+{A}_{3},\end{array}$

${A}_{1}=|\left({W}_{\psi ,\phi }{f}_{n}\right)\left(0\right)|,$

${A}_{2}={\mathrm{sup}}_{|\phi \left(a\right)|>r}{‖\left({W}_{\psi ,\phi }{f}_{n}\right)\circ {\sigma }_{a}-\left({W}_{\psi ,\phi }{f}_{n}\right)\left(a\right)‖}_{{A}^{2}},$

${A}_{3}={\mathrm{sup}}_{|\phi \left(a\right)|\le r}{‖\left({W}_{\psi ,\phi }{f}_{n}\right)\circ {\sigma }_{a}-\left({W}_{\psi ,\phi }{f}_{n}\right)\left(a\right)‖}_{{A}^{2}}.$

$\begin{array}{c}{A}_{2}={\mathrm{sup}}_{|\phi \left(a\right)|>r}{‖\left({W}_{\psi ,\phi }{f}_{n}\right)\circ {\sigma }_{a}-\left({W}_{\psi ,\phi }{f}_{n}\right)\left(a\right)‖}_{{A}^{2}}\\ \le C{‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}\cdot {\mathrm{sup}}_{a\in D}{‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\mathrm{sup}}_{|\phi \left(a\right)|>r}\left(\alpha \left(\psi ,\phi ,a\right)+\beta \left(\psi ,\phi ,a\right)\right)\\ \le C{\mathrm{sup}}_{|\phi \left(a\right)|>r}{\left({‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}{\left(\mathrm{log}\frac{2}{1-{|\phi \left({a}_{n}\right)|}^{2}}\right)}^{-1}\beta \left(\psi ,\phi ,a\right)\right)}^{1/2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\mathrm{sup}}_{|\phi \left(a\right)|>r}\left(\alpha \left(\psi ,\phi ,a\right)+\beta \left(\psi ,\phi ,a\right)\right).\end{array}$

$\begin{array}{c}{A}_{3}\le {\mathrm{sup}}_{a\in D}{‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}\cdot {\mathrm{sup}}_{|\phi \left(a\right)|\le r}{‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\mathrm{sup}}_{|\phi \left(a\right)|\le r}{‖\left(\psi \circ {\sigma }_{a}\right){f}_{n}\circ \phi \circ {\sigma }_{a}-{f}_{n}\left(\phi \left(a\right)\right)‖}_{{A}^{2}}\\ \le {‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}·{\mathrm{max}}_{|w|\le r}|{f}_{n}|+{\left({A}_{4}+{A}_{5}\right)}^{1/2}\\ \le \epsilon +{\left({A}_{4}+{A}_{5}\right)}^{1/2},\end{array}$

${A}_{4}={\mathrm{sup}}_{\left\{|\phi \left(a\right)|\le r\right\}}{\int }_{\partial D\E\left(\phi ,a,t\right)}{|\psi \circ {\sigma }_{a}\left(z\right){F}_{n,a}\left(z\right)|}^{2}\text{d}\sigma \left(z\right),$

${A}_{5}={\mathrm{sup}}_{\left\{|\phi \left(a\right)|\le r\right\}}{\int }_{E\left(\phi ,a,t\right)}{|\psi \circ {\sigma }_{a}\left(z\right){F}_{n,a}\left(z\right)|}^{2}\text{d}\sigma \left(z\right).$

$a\in D$$n\ge {n}_{0}$ ，令 ${G}_{n,a}=f\circ {\sigma }_{\phi \left(a\right)}-{f}_{n}\left(\phi \left(a\right)\right)$${\lambda }_{a}={\sigma }_{\phi \left(a\right)}\circ \phi \circ {\sigma }_{a}-{f}_{n}\left(\phi \left(a\right)\right)$

${G}_{n,a}\left(0\right)=0$${F}_{n,a}={G}_{n,a}\circ {\sigma }_{\phi \left(a\right)}$ 。根据文献 [9] 中的(3-19)式，对任意的 $z\in \partial D\E\left(\phi ,a,t\right)$ 可得

${F}_{n,a}\left(z\right)={G}_{n,a}\left({\lambda }_{a}\left(z\right)\right)\le 2|{\lambda }_{a}\left(z\right)|{\mathrm{max}}_{|w|\le t}|{G}_{n,a}\left(w\right)|.$

$\begin{array}{l}{\mathrm{sup}}_{\left\{|\phi \left(a\right)|\le r\right\}}{\int }_{\partial D\E\left(\phi ,a,t\right)}{|\psi \circ {\sigma }_{a}\left(z\right){F}_{n,a}\left(z\right)|}^{2}\text{d}\sigma \left(z\right)\\ \le 4{\left({\mathrm{max}}_{|w|\le t}|{G}_{n,a}\left(w\right)|\right)}^{2}{‖\left(\psi \circ {\sigma }_{a}\right)\cdot {\lambda }_{a}‖}_{{A}^{2}},\end{array}$

${\mathrm{max}}_{|w|\le t}|{G}_{n,a}\left(w\right)|\le {\mathrm{max}}_{|w|\le t}|{f}_{n}\left({\sigma }_{\phi \left(a\right)}\left(w\right)\right)|+|{f}_{n}\left(\phi \left(a\right)\right)|\le 2\epsilon$

$\begin{array}{c}{‖\left(\psi \circ {\sigma }_{a}\right)\cdot {\lambda }_{a}‖}_{{A}^{2}}\le {‖\left(\psi \circ {\sigma }_{a}-\psi \left(a\right)\right)\cdot {\lambda }_{a}‖}_{{A}^{2}}+{‖\psi \left(a\right)\cdot {\lambda }_{a}‖}_{{A}^{2}}\\ \le {‖\left(\psi \circ {\sigma }_{a}-\psi \left(a\right)\right)‖}_{{A}^{2}}{‖{\lambda }_{a}‖}_{\infty }+{\mathrm{sup}}_{a\in D}\alpha \left(\psi ,\phi ,a\right)<\infty .\end{array}$

${A}_{5}$ ，根据Holder不等式和(5)，有

$\begin{array}{c}{A}_{5}\le {\mathrm{sup}}_{\left\{|\phi \left(a\right)|\le r\right\}}{\left({\int }_{E\left(\phi ,a,t\right)}{|\psi \circ {\sigma }_{a}\left(z\right)|}^{2}\text{d}\sigma \left(z\right)\right)}^{1/2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot {\mathrm{sup}}_{\left\{|\phi \left(a\right)|\le r\right\}}{\left({\int }_{E\left(\phi ,a,t\right)}{|\psi \circ {\sigma }_{a}\left(z\right)|}^{2}{|{F}_{n,a}\left(z\right)|}^{4}\text{d}\sigma \left(z\right)\right)}^{1/2}\\ \le {\epsilon }^{2}{\mathrm{sup}}_{\left\{|\phi \left(a\right)|\le r\right\}}{‖\left(\psi \circ {\sigma }_{a}\right)\cdot {F}_{n,a}‖}_{{A}^{4}}{‖{F}_{n,a}‖}_{{A}^{4}}.\end{array}$

${‖{F}_{n,a}‖}_{{A}^{4}}\le C{‖{f}_{n}\circ \phi \circ {\sigma }_{a}-{f}_{n}\left(\phi \left(a\right)\right)‖}_{{A}^{2}}\le C.$

$\begin{array}{c}{\mathrm{sup}}_{\left\{|\phi \left(a\right)|\le r\right\}}{‖\left(\psi \circ {\sigma }_{a}\right)\cdot {F}_{n,a}‖}_{{A}^{4}}\le {\mathrm{sup}}_{\left\{|\phi \left(a\right)|\le r\right\}}{‖\left({W}_{\psi ,\phi }{f}_{n}\right)\circ {\sigma }_{a}-\left({W}_{\psi ,\phi }{f}_{n}\right)\left(a\right)‖}_{{A}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\mathrm{sup}}_{\left\{|\phi \left(a\right)|\le r\right\}}{‖\left(\psi \circ {\sigma }_{a}-\psi \left(a\right)\right)\cdot {f}_{n}\left(\phi \left(a\right)\right)‖}_{{A}^{4}}\\ \le C\left({‖{W}_{\psi ,\phi }{f}_{n}‖}_{B}{‖\psi \circ {\sigma }_{a}-\psi \left(a\right)‖}_{{A}^{2}}\right).\end{array}$

${\mathrm{sup}}_{n\ge {n}_{0}}{‖{W}_{\psi ,\phi }{f}_{n}‖}_{B}\le {A}_{1}+{A}_{2}+{A}_{3}\le C\epsilon ,$

[1] Axler, S. (1986) The Bergman space, the Bloch Space and Commutators of Multiplication Operators. Duke Mathematical Journal, 53, 315-332.
https://doi.org/10.1215/S0012-7094-86-05320-2

[2] Garnett, J. (1981) Bounded Analytic Functions. Academic Press, New York.

[3] Colonna, F. (2013) New Criteria for Boundedness and Compactness of Weighted Composition Operators Mapping into the Bloch Space. Central European Journal of Mathematics, 11, 55-73.

[4] Colonna, F. (2013) Weighted Composition Operators between and BMOA. Bulletin of the Korean Mathematical Society, 50.

[5] Galindo, P., Laitila, J., Lindsrom, M. (2013) Essential Norm Estimates for Composition Operators on BMOA. Journal of Functional Analysis, 265, 629-643.
https://doi.org/10.1016/j.jfa.2013.05.002

[6] Bourdon, P., Cima, J. and Matheson, A. (1999) Compact Composition Operators on BMOA. Transactions of the American Mathematical Society, 351, 2183-2196.
https://doi.org/10.1090/S0002-9947-99-02387-9

[7] Hyyarinen, O. and Lindstrom, M. (2012) Estimates of Essential Norms of Weighted Composition Operators between Bloch-Type Spaces. Journal of Mathematical Analysis & Applications, 393, 38-44.
https://doi.org/10.1016/j.jmaa.2012.03.059

[8] Laitiala, J. (2009) Weighted Composition Operators on BMOA. Computational Methods & Function Theory, 9, 27-46.

[9] Laitiala, J. and Lindstrom, M. (2015) The Essential Norm of a Weighted Composition Operator on BMOA. Mathematische Zeitschrift, 279, 423-434.
https://doi.org/10.1007/s00209-014-1375-6

[10] Liu, X. and Li, S. (2017) Norm and Essential Norm of a Weighted Composition Operator on the Bloch Space. Integral Equations & Operator Theory, 1-17.

[11] Smith, W. (1999) Compactness of Composition Operators on BMOA. Proceedings of the American Mathematical Society, 127, 2715-2725.
https://doi.org/10.1090/S0002-9939-99-04856-X

[12] Wulan, H. (2007) Compactness of Composition Operators on BMOA and VMOA. Science in China (Series A: Mathematics), 50, 997.
https://doi.org/10.1007/s11425-007-0061-0

[13] Wulan, H., Zheng, D. and Zhu, K. (2009) Compact Composition Operators on BMOA and the Bloch Space. Proceedings of the American Mathematical Society, 137, 3861-3868.
https://doi.org/10.1090/S0002-9939-09-09961-4

Top