单位球Hardy空间上加权复合算子的交换子

1 江苏海洋大学理学院 江苏连云港 222005

2 天津大学数学学院 天津 300072

Commutators of Weighted Composition Operators on Hardy Space of the Unit Ball

1 School of Science, Jiangsu Ocean University, Jiangsu Lianyungang 222005

2 Department of Mathematics, Tianjin University, Tianjin 300072

 基金资助: 国家自然科学基金.  11771323江苏海洋大学博士科研启动金.  KQ17006

 Fund supported: the NSFC.  11771323the Doctoral Research Launch Fund of Jiangsu Ocean University.  KQ17006 Abstract

In this paper, we study commutators of weighted composition operators with linear fractional non-automorphisms on Hardy space of the unit ball. First, we obtain the formula of commutators of weighted composition operators. Then, we characterize compactness of commutators according to two special situations of linear fractional maps. Finally, we obtain that commutators are compact when linear fractional maps are parabolic and commutators are not compact when linear fractional maps are hyperbolic.

Keywords： Hardy spaces ; Weighted composition operators ; Commutators

Xu Ning, Zhou Zehua, Ding Ying. Commutators of Weighted Composition Operators on Hardy Space of the Unit Ball. Acta Mathematica Scientia[J], 2021, 41(6): 1606-1615 doi:

${\Bbb B}_n $${\Bbb C}^n 上单位开球, \varphi$$ {\Bbb B}_n$上解析自映射. 记$H({\Bbb B}_n) $${\Bbb B}_n 上所有解析函数全体, H^\infty$$ {\Bbb B}_n$上所有有界解析函数全体. 设$u \in H^\infty$, $f\in H({\Bbb B}_n)$, 定义加权复合算子$W_{u, \varphi}f = u\cdot(f \circ \varphi)$. 特别地, 当$u = 1$时, 即为复合算子$C_\varphi$. 注意到, $W_{u, \varphi} = M_uC_\varphi$, 其中, $M_u$为乘积算子. 记${\cal B}(X)$为可分复Hilbert空间上所有有界线性算子全体. 当$S, T\in {\cal B}(X)$, $ST $$TS 非零且都不是紧时, 若 [S, T] = ST-TS 紧, 则称交换子 [S, T] 是非平凡紧. 若 [T^*, T] = 0 , 则称 T\in {\cal B}(X) 正规. 若 [T^*, T] 紧, 则称 T\in {\cal B}(X) 本性正规. 许多学者都对一维以及高维线性分式复合算子的交换子感兴趣. Clifford等研究了当 \varphi$$ \psi$是单位圆盘${\Bbb D}$的线性分式映射时, Hardy空间$H^2({\Bbb D})$上复合算子交换子$[C^*_\psi, C_\varphi]$非平凡紧性. Jung等研究了当$\varphi $$\psi 是单位圆盘非自同构线性分式映射时, Hardy空间 H^2({\Bbb D}) 上加权复合算子交换子 [W^*_{v, \psi}, W_{u, \varphi}] 紧性. Lacruz等研究了单位圆盘上线性分式映射复合算子二次交换子性质. 对于多变量情形, 江良英等研究了线性分式复合算子的本性正规性, 然后第一作者在文献 中研究了单位球Hardy空间 H({\Bbb B}_n) 上自同构复合算子交换子 [C^*_\psi, C_\varphi] 紧性. 本文将研究 \varphi$$ \psi$是单位球${\Bbb B}_n$的非自同构线性分式映射时, Hardy空间$H({\Bbb B}_n)$上加权复合算子交换子$[W^*_{v, \psi}, W_{u, \varphi}]$紧性. 第2节给出主要定理的一些准备引理. 第3节给出线性分式加权复合算子交换子$[W^*_{v, \psi}, W_{u, \varphi}]$的计算公式, 然后讨论当$\|\varphi\|_\infty = 1 $$[W^*_{v, \psi}, W_{u, \varphi}] 的紧性. 本文重点讨论当 \varphi 在单位球边界 \partial{\Bbb B}_n 上有一个不动点 e_1 情形. 显然, 通过酉变换, 不妨设 e_1 = (1, 0, \cdots, 0) . 2 预备知识 \sigma$$ \partial{\Bbb B}_n$上正规Lebesgue测度. 当$n\geq1$, 定义Hardy空间为

$u $$\partial{\Bbb B}_n 上有界可测复值函数, H^2({\Bbb B}_n) 上Toeplitz算子定义为 T_u(f) = P(uf) , 其中 P$$ L^2(\partial{\Bbb B}_n) $$H^2({\Bbb B}_n) 的正交投影. 如果 u 是解析的, 则 T_u$$ M_u$. Cowen等定义单位球${\Bbb B}_n$上线性分式映射如下:

$\begin{equation} \varphi(z) = \frac{Az+B}{\langle z, C\rangle+D} \end{equation}$

${\Bbb B}_n$的线性分式映射, 其中$\langle\cdot, \cdot\rangle$表示${\Bbb C}^n$中欧式内积, $A$表示$n\times n$矩阵, $B $$C 表示 n\times 1 列向量, D 是一个复数. C_\varphi$$ H^2({\Bbb B}_n)$上由$\varphi$诱导的复合算子. 则$C_\varphi$的伴随算子$C^*_\varphi$

$\begin{equation} C^*_\varphi = T_gC_{\sigma_\varphi}T^*_h, \end{equation}$

$\begin{equation} \sigma_\varphi(z) = \frac{A^*z-C}{\langle z, -B\rangle+\overline{D}} \end{equation}$

$\varphi$的Kreǐn伴随, $g(z) = (\langle z, -B\rangle+\overline{D})^{-n}$, $h(z) = (\langle z, C\rangle+D)^n$, $T_g, T_h$分别是解析Toeplitz算子.

$m^*_{\sigma_\varphi}$是Hilbert空间上$m_\varphi$伴随. 要验证两个复合算子Kreǐn伴随, 只要证明$m_\varphi $$m_{\sigma_\varphi} 是Kreǐn伴随矩阵即可. 下面回顾不动点及其分类. 定理2.2 设 \varphi\in LFT({\Bbb B}_n)$$ {\Bbb B}_n$内部无不动点, 则存在唯一一个不动点$\zeta\in \partial{\Bbb B}_n$使得$\varphi(\zeta) = \zeta$以及$\langle {\rm d}\varphi_\zeta(\zeta), \zeta\rangle = \lambda$其中$0<\lambda\leq1$.

设$\sigma_\varphi$如定理2.1中是$\varphi$伴随, 则$\sigma_\varphi(e_1) = e_1$. 由文献, 对于$\overline{\langle z, C\rangle}\in C(\overline{{\Bbb B}_n})$, 任意非负整数$j$以及紧算子$K$, 有$T_{(\overline{\langle z, C\rangle})^j} = (T_{\overline{\langle z, C\rangle}})^j+K$. 从而有

$\begin{eqnarray} |a_2|<1, \ \mbox{Re}c_2>\frac{|b_2|^2}{4}, \ \mbox{Re}(\overline{c_1}-2\overline{d_1}d_2+c_2)>|\overline{a_1}d_2-\frac{\overline{b_1}}{2}|^2. \end{eqnarray}$

$\sigma_\psi $$\sigma_\varphi\circ\psi 满足式(2.4). 直接计算可得 其中 于是 \varphi, \psi 的伴随映射 \sigma_\varphi, \sigma_\psi 分别为 我们可以得到分别与 \sigma_\varphi, \sigma_\psi 共轭的 H_2 的解析自映射 \Delta, \Gamma , 形式如下 由文献中定理4.3, 若 z = (z_1, z_2) = \sigma_C^{-1}(w)\in\partial{\Bbb B}_2 , 则 w = (w_1, w_2) = \sigma_C(z)\in\partial H_2 且满足 \mbox{Re} w_1 = |w_2|^2 , 于是可得 与文献中命题3.1和定理4.3的证明类似, 选取 \begin{eqnarray} k<\min\Big\{\frac{1-|a_2|^2}{|a_2|^2}, \frac{\mbox{Re}\overline{c_2}-\frac{|b_2|^2}{4}}{1+\frac{|b_2|^2}{4}}\Big\}, \end{eqnarray} 1-|\sigma_\psi(z)|^2\geq k|e_1-\sigma_\psi(z)|^2$$ e_1\in\partial{\Bbb B}_2$的某个邻域内成立.

$\begin{eqnarray} k<\min\bigg\{\frac{1-|a_1a_2|^2}{|a_1a_2|^2}, \frac{\mbox{Re}(\overline{c_1}-2\overline{d_1}d_2+c_2)-|\overline{a_1}d_2-\frac{\overline{b_1}}{2}|^2} {1+|\overline{a_1}d_2-\frac{\overline{b_1}}{2}|^2}\bigg\}, \end{eqnarray}$

$1-|\sigma_\varphi\circ\psi(z)|^2\geq k|e_1-\sigma_\varphi\circ\psi(z)|^2 $$e_1\in\partial{\Bbb B}_2 的某个邻域内成立. 结合式(2.5), (2.7)与(2.8), 可得 引理2.6证毕. 引理2.7 设 \varphi, \psi\in LFM({\Bbb B}_2)\backslash Aut({\Bbb B}_2) 是双曲型, 具有相同的边界固定点 e_1 . 假设 \varphi, \psi 分别与 H_2 的解析自映射 \Phi, \Psi 共轭, \Phi, \Psi 具有如下形式 其中 a_j, b_j, c_j, d_j\in {\Bbb C} 满足 \begin{eqnarray} |a_2|^2<\lambda_2, \ \lambda_2\mbox{Re}c_2>\frac{|b_2|^2}{4}, \ \frac{\lambda_1}{\lambda_2}\mbox{Re}(\overline{c_1}-2\overline{d_1}d_2+c_2)>|\frac{\overline{a_1}d_2}{\lambda_2}-\frac{\overline{b_1}}{2}|^2, \end{eqnarray} \sigma_\psi$$ \sigma_\varphi\circ\psi$满足式(2.4).

与引理2.6同理, 选取

$\begin{equation} k<\min\bigg\{\frac{\lambda_2-|a_2|^2}{|a_2|^2}, \frac{\lambda_2\mbox{Re}\overline{c_2}-\frac{|b_2|^2}{4}}{1+\frac{|b_2|^2}{4}}\bigg\}, \end{equation}$

$1-|\sigma_\psi(z)|^2\geq k|e_1-\sigma_\psi(z)|^2 $$e_1\in\partial{\Bbb B}_2 的某个邻域内成立. 选取 \begin{equation} k<\min\bigg\{\frac{\lambda_1\lambda_2-\lambda_2|a_1a_2|^2}{|a_1a_2|^2}, \frac{\frac{\lambda_1}{\lambda_2}\mbox{Re}(\overline{c_1}-2\overline{d_1}d_2+c_2)-|\frac{\overline{a_1}d_2}{\lambda_2}-\frac{\overline{b_1}}{2}|^2} {1+|\frac{\overline{a_1}d_2}{\lambda_2}-\frac{\overline{b_1}}{2}|^2}\bigg\}, \end{equation} 1-|\sigma_\varphi\circ\psi(z)|^2\geq k|e_1-\sigma_\varphi\circ\psi(z)|^2$$ e_1\in\partial{\Bbb B}_2$的某个邻域内成立. 结合式(2.5), (2.10) 以及(2.11), 可得

由引理2.5知$C^*_\psi = ({\cal D}_1\psi_1(e_1))^{-n} C_{\sigma_\psi}+K.$因此

(ⅰ) 在定理3.1中取$u = v, \varphi = \psi$即可.

(ⅱ) 在定理3.1中取$u = v = 1$即可.

(ⅰ) $u(e_1)v(e_1) = 0$,

(ⅱ) $\varphi\circ\sigma_\psi = \sigma_\psi\circ\varphi$.

由定理3.1知, $[W^*_{v, \psi}, W_{u, \varphi}]$紧当且仅当$({\cal D}_1\psi_1(e_1))^{-n}u(e_1)\overline{v(e_1)}[C_{\sigma_\psi}, C_\varphi]$是紧的.

假设$\varphi $$\psi 分别是抛物型与双曲型. 由定理3.1, 可得 因为 \psi 是双曲型, 所以 {\cal D}_1({\sigma_\psi})_1(e_1) = \frac{1}{\lambda}>1 , 即 \sigma_\psi 不是双曲型. 由文献 中定理3.2可知, \sigma_\psi 有一个孤立不动点 \alpha\in{\Bbb B}_n . \sigma_\psi\circ\varphi = \varphi\circ\sigma_\psi , 则 \sigma_\psi\circ\varphi(\alpha) = \varphi\circ\sigma_\psi(\alpha) = \varphi(\alpha), 上式说明 \varphi(\alpha) 也是 \sigma_\psi 的一个不动点. 由于 \varphi$$ {\Bbb B}_n$内部没有不动点, 即$\varphi(\alpha)\neq \alpha$. 由引理2.3可知若$\psi $$\partial {\Bbb B}_n 上无不动点, 所以 \sigma_\psi$$ \partial {\Bbb B}_n$上也没有不动点. 综上讨论说明$\sigma_\psi$存在$\alpha\in\partial{\Bbb B}_n $$\varphi(\alpha)\in{\Bbb B}_n 两个不动点, 与 \sigma_\psi 有一个不动点矛盾. 因此, \sigma_\psi\circ\varphi\neq\varphi\circ\sigma_\psi . 通过定理3.2知 [W^*_{v, \psi}, W_{u, \varphi}] 非紧. 当 \varphi 为双曲型, \psi 为抛物型时, 同理可证. 注3.1 由推论3.2, 我们判断 [W^*_{v, \psi}, W_{u, \varphi}] 的紧性, 只要考虑 \varphi$$ \psi$都是抛物型或都是双曲型即可.

由引理2.6可知, $\sigma_\psi \ \mbox{与}\ \sigma_\varphi\circ\psi$都满足式(2.4). 通过具体计算

由引理2.7可知, $\sigma_\psi $$\sigma_\varphi\circ\psi 满足式(2.4). 由定理3.2, 只要证明 \varphi\circ\sigma_\psi\neq\sigma_\psi\circ\varphi 即可. 假设 \varphi\circ\sigma_\psi = \sigma_\psi\circ\varphi . 由定理条件知, \psi 只有一个不动点 e_1\in\partial{\Bbb B}_2 , 所以 \psi 的伴随映射 \sigma_\psi 也只有一个不动点 e_1 . 因为 \psi 是双曲型, 所以它的边界扩张系数 \lambda_2<1 . \sigma_\psi 的边界扩张系数 \frac{1}{\lambda_2}>1 . 由文献 定理3.2知, \sigma_\psi 有一个孤立不动点 \alpha\in{\Bbb B}_2 . 如果 \sigma_\psi\circ\varphi = \varphi\circ\sigma_\psi , 则 \sigma_\psi\circ\varphi(\alpha) = \varphi\circ\sigma_\psi(\alpha) = \varphi(\alpha). 上式说明 \varphi(\alpha)$$ \sigma_\psi$一个不动点. 然而, $\varphi$${\Bbb B}_2$内无不动点, 因此$\varphi(\alpha)\neq \alpha$. 由引理2.3, 若$\psi$没有其它不动点, 则$\sigma_\psi$也无其它不动点. 通过上述讨论$\sigma_\psi$有三个不动点$e_1\in\partial{\Bbb B}_2$, $\alpha\in{\Bbb B}_2$以及$\varphi(\alpha)\in{\Bbb B}_2$, 这与$\sigma_\psi$只有一个不动点$\alpha\in{\Bbb B}_2$矛盾. 因此$\varphi\circ\sigma_\psi\neq\sigma_\psi\circ\varphi$, 由定理3.2可知$[W^*_{v, \psi}, W_{u, \varphi}]$非紧.

参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Clifford J , Levi D , Narayan S .

Commutators of composition operators with adjoints of composition operators

Complex Var Elliptic Equ, 2012, 57, 677- 686

Jung S , Kim Y , Ko E .

Commutators of weighted composition operators

Internat J Math, 2014, 25 (6): 1450053

Lacruz M , León-Saavedra F , Petrovic S , et al.

The double commutant property for composition operators

Collect Math, 2019, 70, 501- 532

Jiang L Y , Ouyang C H .

Essential normality of linear fractional composition operators in the unit ball of CN

Science in China: Mathematics, 2009, 52 (12): 2668- 2678

Jiang L Y .

Commutators of composition operators with adjoints of composition operators on the ball

Complex Var Elliptic Equ, 2016, 61 (3): 405- 421

Cowen C C , MacCluer B D .

Linear fractional maps of the ball and their composition operators

Acta Sci Math(Szeged), 2000, 66 (1/2): 351- 376

Bisi C , Bracci F .

Linear fractional maps of the unit ball: a geometric study

Adv Math, 2002, 167 (2): 265- 287

MacCluer B D , Weir R J .

Linear-fractional composition operators in several variables

Integral Equations and Operator Theory, 2005, 53 (3): 373- 402

McDonald G .

Fredholm properties of a class of Toeplitz operators on the ball

Indiana Univ Math J, 1977, 26, 567- 576

Bracci F , Contreras M , Diaz-Madrigal S .

Classification of semigroups of linear fractional maps in the unit ball

Adv Math, 2007, 208, 318- 350

Heller K , MacCluer B D , Weir R J .

Compact differences of composition operators in several variables

Integral Equations and Operator Theory, 2011, 69 (2): 247- 268

Cowen C C , MacCluer B D . Composition Operators on Spaces of Analytic Functions. Boca Raton, Fla: CRC Press, 1995

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