Acta mathematica scientia,Series A ›› 2023, Vol. 43 ›› Issue (1): 27-34.

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

Ding Xuanhao1,2(),Huang Yuhao1(),Li Yongning1,2,*()   

  1. 1School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067
    2Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing 400067
  • Received:2021-12-22 Revised:2022-08-17 Online:2023-02-26 Published:2023-03-07
  • Supported by:
    The NSFC(11871122);The NSFC(12101092);Natural Science Foundation of Chongqing(cstc2020jcyj-msxmX0318);Chongqing Technology and Business University(2056008);Scientific research project in Chongqing Technology and Business University-level(yjscxx2022-112-73)


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.

Key words: Harmonic Hardy space, Harmonic Toeplitz operator, Unitary equivalence, Dual truncated Toeplitz operator, Essential spectrum

CLC Number: 

  • O177.1