## 调和Hardy空间上的Toeplitz算子的酉等价性

1重庆工商大学数学与统计学院 重庆 400067

2经济社会应用统计重庆市重点实验室 重庆 400067

## The Unitary Equivalence of the Toeplitz Operators on the Harmonic Hardy Space

Ding Xuanhao,1,2, Huang Yuhao,1, Li Yongning,1,2,*

1School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067

2Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing 400067

 基金资助: 国家自然科学基金.  11871122国家自然科学基金.  12101092重庆市自然科学基金.  cstc2020jcyj-msxmX0318重庆工商大学基金.  2056008重庆工商大学校级科研项目.  yjscxx2022-112-73

 Fund supported: The NSFC.  11871122The NSFC.  12101092Natural Science Foundation of Chongqing.  cstc2020jcyj-msxmX0318Chongqing Technology and Business University.  2056008Scientific research project in Chongqing Technology and Business University-level.  yjscxx2022-112-73

$H^{2}$是单位圆盘${\Bbb D}=\{\xi\in{\Bbb C}:|\xi|<1\}$上的经典Hardy空间. 设$u$$v是内函数且至少其中一个是非常值的, 调和Hardy空间H_{u,v}^{2}定义为H_{u,v}^{2}=uH^{2}\oplus\overline{v}(H^{2})^{\perp}=uH^{2}\oplus\overline{vzH^{2}}. 对任意的x\in H_{u,v}^{2}, 定义H_{u,v}^{2}上的调和Toeplitz算子 \widehat{T}_{\varphi}x=Q_{u,v}(\varphi x), 其中, Q_{u,v}:L^{2}\rightarrow H_{u,v}^{2}为正交投影. 该文刻画了调和Toeplitz算子和对偶截断Toeplitz算子的酉等价性, 并给出了两个调和Toeplitz算子可交换的充要条件, 调和Toeplitz代数的性质以及\widehat{T}_{z}的换位子的刻画. 最后, 该文还得到了有限多个连续符号的调和 Toeplitz算子乘积的本质谱. 关键词： 调和Hardy空间 ; 调和Toeplitz算子 ; 酉等价 ; 对偶截断Toeplitz算子 ; 本质谱 Abstract Let H^{2} be the Hardy space on the unit disk {\Bbb D}=\{\xi\in{\Bbb C}:|\xi|<1\}. Suppose u and v are inner functions and at least one of them is nonconstant, the harmonic Hardy space H_{u,v}^{2} is defined by H_{u,v}^{2}=uH^{2}\oplus\overline{v}(H^{2})^{\bot}=uH^{2}\oplus\overline{vzH^{2}}. For any x\in H_{u,v}^{2}, define the Toeplitz operator on the H_{u,v}^{2} by \widehat{T}_{\varphi}x=Q_{u,v}(\varphi x), where Q_{u,v} is the orthogonal projection from L^{2}\rightarrow H_{u,v}^{2}. In this paper, the unitary equivalence of the harmonic Toeplitz operator and the dual truncated Toeplitz operator is obtained, moreover, the sufficient and necessary conditions for when two Toeplitz operators commute is given, and the properties of the harmonic Toeplitz algebra and the commutant of \widehat{T}_{z} are described. Finally, the essential spectrum for the product of finitely many harmonic Toeplitz operators with continuous symbols is obtained in this paper. Keywords： Harmonic Hardy space ; Harmonic Toeplitz operator ; Unitary equivalence ; Dual truncated Toeplitz operator ; Essential spectrum 导出 EndNote| Ris| Bibtex 本文引用格式 丁宣浩, 黄雨浩, 李永宁. 调和Hardy空间上的Toeplitz算子的酉等价性. 数学物理学报[J], 2023, 43(1): 27-34 doi: Ding Xuanhao, Huang Yuhao, Li Yongning. The Unitary Equivalence of the Toeplitz Operators on the Harmonic Hardy Space. Acta Mathematica Scientia[J], 2023, 43(1): 27-34 doi: ## 1 引言 {\Bbb D}=\{\xi\in{\Bbb C}:|\xi|<1\}是复平面{\Bbb C}上的单位圆盘, \partial{\Bbb D}=\{\xi\in{\Bbb C}:|\xi|=1\} 是单位圆周. 记H^{2}是经典的Hardy空间, (H^{2})^{\perp}=\overline{zH^{2}}是其正交补空间. 记L^{2}=L^{2}(\partial{\Bbb D}) 是单位圆周 \partial{\Bbb D}上的Lebesgue平方可积函数构成的空间, L^{\infty}是单位圆周 \partial{\Bbb D}上的本性有界的可测函数全体构成的空间, H^{\infty}为单位开圆盘{\Bbb D}上的有界解析函数全体构成的空间[1]. 如果u \in H^{\infty}$$|u({\rm e}^{{\rm i}t})|=1$$\partial{\Bbb D}上几乎处处成立, 则称u 是内函数. P:L^{2}\rightarrow H^{2} 为正交投影, 则对于 \varphi\in L^{\infty}, 符号为\varphi的Toeplitz算子T_{\varphi}定义为 T_{\varphi}f=P(\varphi f), \forall f\in H^{2}. Hankel算子H_{\varphi}定义为 H_{\varphi}g=(I-P)(\varphi g), \forall g\in H^{2}. 对偶Toeplitz算子S_{\varphi}定义为 S_{\varphi}h=(I-P)(\varphi h), \forall \in (H^{2})^{\perp}. 容易得到 H_{\varphi}^{\ast} h=P(\overline{\varphi} h), \forall h\in (H^{2})^{\perp}. M_{\varphi}$$L^{2}$上的乘法算子, 且$M_{\varphi} x=\varphi x.$$M_{\varphi}在空间分解L^{2}=H^{2}\oplus (H^{2})^{\bot} 下可表示为如下2\times 2算子矩阵形式 M_{\varphi}=\left[\begin{array}{cccccc}T_\varphi& & H_{\overline{\varphi}}^{*} \\ H_\varphi && S_\varphi \\ \end{array}\right]. 由于M_{\varphi}M_{\psi}=M_{\varphi\psi}, 则有 T_{\varphi\psi}=T_{\varphi}T_{\psi}+H_{\overline{\varphi}}^{*}H_{\psi}; H_{\varphi\psi}=H_{\varphi}T_{\psi}+S_{\varphi}H_{\psi}; S_{\varphi\psi}=S_{\varphi}S_{\psi}+H_{\varphi}H_{\overline{\psi}}^{*}. \psi\in H^{\infty}, 则H_{\psi}=0, 故有 H_{\varphi\psi}=H_{\varphi}T_{\psi}. Toeplitz算子理论发展至今, 已形成一个庞大的知识体系[2-4], 在物理学, 概率论, 通信理论及控制论等领域中均有重要的应用, 吸引着相关领域的学者们的关注和兴趣. 更多关于Toeplitz算子的相关问题研究可参考文献[5-9]. u是一个非常值内函数, 称K_{u}^{2}=H^{2}\ominus uH^{2}为模空间. 显然地, H^{2}=K_{u}^{2}\oplus uH^{2}, L^{2}=K_{u}^{2}\oplus (K_{u}^{2})^{\bot}=K_{u}^{2}\oplus(uH^{2}\oplus\overline{zH^{2}}). 2007 年, Sarason在文献[10] 中引入了模空间K_{u}^{2}上的Toeplitz算子, 即截断Toeplitz算子, 引起了众多学者的关注和研究[11-14]. P_{u}:L^{2}\rightarrow K_{u}^{2}为正交投影, 则对于\varphi\in L^{2}, 符号为\varphi的截断Toeplitz算子A_{\varphi}定义为 A_{\varphi}h=P_{u}(\varphi h), \forall h\in K_{u}^{2}\cap L^{\infty}. 截断Hankel算子B_{\varphi}定义为 B_{\varphi}h=(I-P_{u})(\varphi h), \forall h\in K_{u}^{2}\cap L^{\infty}. 2018年, Sang等[15]引入了(K_{u}^{2})^{\bot} 上的Toeplitz算子, 即对偶截断 Toeplitz算子, 给出了两个对偶截断Toeplitz算子的乘积为一个有限秩算子的充要条件. 对偶截断Toeplitz算子D_{\varphi}定义为 D_{\varphi}x=(I-P_{u})(\varphi x), \forall x\in (K_{u}^{2})^{\bot}\cap L^{\infty}. 容易得到 B_{\varphi}^{\ast} x=P_{u}(\overline{\varphi} x), \forall x\in (K_{u}^{2})^{\bot}\cap L^{\infty}. 与经典的Hardy空间类似, L^{2}上的乘法算子M_{\varphi} 在空间分解L^{2}=K_{u}^{2}\oplus (K_{u}^{2})^{\bot}下可表示为如下2\times 2算子矩阵形式 M_{\varphi}=\left[\begin{array}{cccccc} A_\varphi && B_{\overline{\varphi}}^{*} \\ B_\varphi && D_\varphi \\ \end{array}\right]. 由于M_{\varphi}M_{\psi}=M_{\varphi\psi}, 则有 A_{\varphi\psi}=A_{\varphi}A_{\psi}+B_{\overline{\varphi}}^{*}B_{\psi}; B_{\varphi\psi}=B_{\varphi}A_{\psi}+D_{\varphi}B_{\psi}; D_{\varphi\psi}=D_{\varphi}D_{\psi}+B_{\varphi}B_{\overline{\psi}}^{*}. 在2020年, Qin等[16]引入了调和函数Hardy空间及其上的Toeplitz 算子. 设u$$v$是内函数且至少有一个是非常值的, 定义调和Hardy空间$H_{u,v}^{2}$

$H_{u,v}^{2}=uH^{2}\oplus\overline{v}(H^{2})^{\bot}=uH^{2}\oplus\overline{vzH^{2}}.$

$(K_{uv}^{2})^{\bot}=uvH^{2}+(H^{2})^{\bot}=uvH^{2}+\overline{zH^{2}}=H_{uv,1}^{2}.$

$H^2_{u,v}$上的调和 Toeplitz 算子$\widehat{T}_{f}$$H^2_{v,u}上的调和Toeplitz算子\widehat{\mathsf{T}}_{f}是酉等价的. 由于(K_{uv}^2)^{\bot}=(K_{vu}^2)^{\bot}, 根据定理2.1可知H^2_{u,v}上的调和Toeplitz算子\widehat{T}_{f}$$H^2_{v,u}$上的调和Toeplitz算子$\widehat{\mathsf{T}}_{f}$均与$(K_{uv}^2)^{\bot}$上的对偶截断Toeplitz算子$D_{f}$是酉等价的, 从而$\widehat{T}_{f}$$\widehat{\mathsf{T}}_{f}是酉等价的. 证毕. 特别地, 当内函数v=1时, 结合定理2.1和上述推论则有如下推论. 推论2.2f \in L^{\infty}, u是非常值的内函数, 则H_{1,u}^2(=H^2 \oplus \overline{uzH^2})上的调和 Toeplitz算子\widehat{\mathsf{T}}_{f}$$(K_u^2)^{\bot}(=uH^2 \oplus \overline{zH^2})$上的对偶截断Toeplitz算子$D_{f}$是酉等价的.

## 3 若干推论

(1) 存在$\lambda\in {\Bbb C}$, 使得$f, g, \bar{f}(uv-\lambda)$$\bar{g}(uv-\lambda)均在H^2中; (2) 存在\lambda\in {\Bbb C}, 使得\bar{f}, \bar{g}, f(uv-\lambda)$$g(uv-\lambda)$均在$H^2$中;

(3) 存在$a, b, c \in {\Bbb C}$$|a|+|b| \neq 0, 使得af+bg=c. u, v是内函数且至少其中一个是非常值的, 记B(H_{u,v}^2)$$H_{u,v}^2$上的所有有界线性算子的全体. 设$\mathfrak{X}$$L^{\infty}的闭自伴子代数, 记 {\cal T}_{\mathfrak{X}}=clos\bigg\{\sum_{i=1}^{n}\prod_{j=1}^{j=n}\widehat{T}_{\varphi_{ij}}: \varphi_{ij} \in \mathfrak{X}\bigg\} B(H_{u,v}^2)的包含\{\widehat{T}_{\varphi}: \varphi \in \mathfrak{X}\}且在范数意义下封闭的最小子代数, 从而{\cal T}_{\mathfrak{X}}是由\{\widehat{T}_{\varphi}: \varphi \in \mathfrak{X}\}所生成的C^{*}-代数. {\cal T}_{\mathfrak{X}}中由所有的半换位子 [\widehat{T}_{\varphi},\widehat{T}_{\psi}):= \widehat{T}_{\varphi}\widehat{T}_{\psi}-\widehat{T}_{\varphi\psi}, \varphi, \psi \in \mathfrak{X} 所生成的{\cal T}_{\mathfrak{X}}的一个闭理想称为半换位子理想, 并记为\mathfrak{S}{\cal T}_{\mathfrak{X}}. {\cal T}_{\mathfrak{X}}中由所有的换位子 [\widehat{T}_{\varphi},\widehat{T}_{\psi}]:= \widehat{T}_{\varphi}\widehat{T}_{\psi}-\widehat{T}_{\psi}\widehat{T}_{\varphi}, \varphi, \psi \in \mathfrak{X} 所生成的{\cal T}_{\mathfrak{X}}的一个闭理想称为换位子理想, 并记为\mathfrak{C}{\cal T}_{\mathfrak{X}}. 根据Sang等[9]所给出的对偶截断Toeplitz C^{*} -代数的刻画以及调和Toeplitz算子和对偶截断Toeplitz算子的酉等价性, 我们得到了如下关于调和Toeplitz代数{\cal T}_{L^{\infty}}$${\cal T}_{C({\Bbb T})}$的结构定理.

$0 \rightarrow \mathfrak{S}{\cal T}_{L^{\infty}} \rightarrow {\cal T}_{L^{\infty}} \rightarrow L^{\infty} \rightarrow 0$

$0 \rightarrow {\cal K} \rightarrow {\cal T}_{C({\Bbb T})} \rightarrow C({\Bbb T}) \rightarrow 0$

$\theta=uv$, 显然$\theta$为内函数. 因为$\varphi$$\partial{\Bbb D}上连续, 由文献[3] 可知H_{\varphi}$$H_{\overline{\varphi}}^{*}$ 是紧的, 则有$H_{\theta\varphi}(=H_{\varphi}T_{\theta})$$H_{\theta\bar{\varphi}}^{\ast} (=T_{\bar{\theta}}H_{\bar{\varphi}}^{\ast}) 也是紧的. 由于 M_{\varphi}=\left[\begin{array}{cccccc} T_\varphi& & H_{\overline{\varphi}}^{*} \\ H_\varphi && S_\varphi \\ \end{array}\right], 则对于(K_{\theta}^{2})^{\bot}上的对偶截断Toeplitz算子D_{\varphi}, 结合引理4.1可得 U_{\theta}^{\ast}D_{\varphi}U_{\theta}-M_{\varphi}=\left[\begin{array}{cccccc} 0 && H_{\theta\bar{\varphi}-\bar{\varphi}}^{\ast} \\ H_{\theta\varphi-\varphi}& & 0 \\ \end{array}\right] 是一个紧算子, 故D_{\varphi}模掉紧算子后与M_{\varphi}是酉等价的, 即U_{\theta}^{*}D_{\varphi}U_{\theta}=M_{\varphi}{\rm mod} ({\cal K}), 这里{\cal K}表示L^2上的紧算子全体. 由定理2.1知D_{\varphi}={\cal A}\widehat{T}_{\varphi}{\cal A}^{*}, 因此 ({\cal A}^{*}U_{\theta})^{*}\widehat{T}_{\varphi}({\cal A}^{*}U_{\theta})=M_{\varphi}{\rm mod} ({\cal K}), 显然{\cal A}^{*}U_{\theta}是酉算子, 故\widehat{T}_{\varphi}模掉紧算子后与M_{\varphi}是酉等价的.证毕. 下述推论可直接由定理3.3得到. 为了文章的完整性, 这里我们给出其另一种证明. 推论4.2u, v为内函数且至少其中一个是非常值的, \varphi_{i} (i=1, 2,\cdots, n)$$\partial{\Bbb D}$上的连续函数, 对于$H_{u, v}^2$上的调和 Toeplitz算子$\widehat{T}_{\varphi_{i}} (i=1, 2,\cdots, n)$, 则有

$\sigma_{e}(\widehat{T}_{\varphi_{1}}\widehat{T}_{\varphi_{2}}\cdots \widehat{T}_{\varphi_{n}})={\rm R}(\varphi_{1}\varphi_{2}\cdots \varphi_{n}).$

$\theta=uv$$U={\cal A}^{*}U_{\theta}, 记\widetilde{T}_{\varphi}=U^{*}\widehat{T}_{\varphi}U. 由推论4.1知: 若\varphi$$\partial{\Bbb D}$上连续, 则 $\widetilde{T}_{\varphi}=M_{\varphi}+K,$ 这里$K$$L^2$上的紧算子. 故可设$\widetilde{T}_{\varphi_{i}}=M_{\varphi_{i}}+K_{i}$, 其中$K_{i} (i=1, 2,\cdots, n)$是紧算子, 则

$\begin{eqnarray*} \widetilde{T}_{\varphi_{1}}\widetilde{T}_{\varphi_{2}}\cdots \widetilde{T}_{\varphi_{n}} =(M_{\varphi_{1}}+K_{1})(M_{\varphi_{2}}+K_{2})\cdots (M_{\varphi_{n}}+K_{n}) =M_{\varphi_{1}\varphi_{2}\cdots \varphi_{n}}+\widetilde{K}, \end{eqnarray*}$ 这里$\widetilde{K}$是紧算子, 所以 $\sigma_{e}(\widetilde{T}_{\varphi_{1}}\widetilde{T}_{\varphi_{2}}\cdots \widetilde{T}_{\varphi_{n}}) =\sigma_{e}( M_{\varphi_{1}\varphi_{2}\cdots \varphi_{n}}) ={\rm R}(\varphi_{1}\varphi_{2}\cdots \varphi_{n}).$

$\begin{eqnarray*} \widetilde{T}_{\varphi_{1}}\widetilde{T}_{\varphi_{2}}\cdots \widetilde{T}_{\varphi_{n}} = (U^{*}\widehat{T}_{\varphi_{1}}U)(U^{*}\widehat{T}_{\varphi_{2}}U)\cdots (U^{*}\widehat{T}_{\varphi_{n}}U) =U^{*}(\widehat{T}_{\varphi_{1}}\widehat{T}_{\varphi_{2}}\cdots \widehat{T}_{\varphi_{n}})U. \end{eqnarray*}$ 于是 $\sigma_{e}(\widehat{T}_{\varphi_{1}}\widehat{T}_{\varphi_{2}}\cdots \widehat{T}_{\varphi_{n}})={\rm R}(\varphi_{1}\varphi_{2}\cdots \varphi_{n}).$ 证毕.

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