Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (6): 2225-2248.doi: 10.1007/s10473-024-0610-4

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MULTIPLICATION OPERATORS ON WEIGHTED DIRICHLET SPACES

Kaikai HAN1,†, Yucheng LI2, Maofa WANG3   

  1. 1. School of Statistics and Mathematics, Hebei University of Economics and Business, Shijiazhuang 050061, China;
    2. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China;
    3. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • Received:2023-08-15 Published:2024-12-06
  • Contact: † Kaikai HAN, E-mail: kkhan.math@whu.edu.cn
  • About author:Yucheng LI, E-mail: liyucheng@hebtu.edu.cn; Maofa WANG, E-mail: mfwang.math@whu.edu.cn
  • Supported by:
    National Natural Science Foundation of China (12101179, 12171138, 12171373) and the Natural Science Foundation of Hebei Province of China (A2022207001).

Abstract: In this paper, we study multiplication operators on weighted Dirichlet spaces Dβ (βR). Let n be a positive integer and βR, we show that the multiplication operator Mzn on Dβ is similar to the operator 1nMz on the space 1nDβ. Moreover, we prove that Mzn (n2) on Dβ is unitarily equivalent to 1nMz on 1nDβ if and only if β=0. In addition, we completely characterize the unitary equivalence of the restrictions of Mzn to different invariant subspaces zkDβ (k1), and the unitary equivalence of the restrictions of Mzn to different invariant subspaces Sj (0j<n).
Abkar, Cao and Zhu [Complex Anal Oper Theory, 2020, 14: Art 58] pointed out that it is an important, natural, and difficult question in operator theory to identify the commutant of a bounded linear operator. They characterized the commutant A(Mzn) of Mzn on a family of analytic function spaces Aα2 (αR) on D (in fact, the family of spaces Aα2 (αR) is the same with the family of spaces Dβ (βR)) in terms of the multiplier algebra of the underlying function spaces. In this paper, we give a new characterization of the commutant A(Mzn) of Mzn on Dβ, and characterize the self-adjoint operators and unitary operators in A(Mzn). We find that the class of self-adjoint operators (unitary operators) in A(Mzn) when β0 is different from the class of self-adjoint operators (unitary operators) in A(Mzn) when β=0.

Key words: multiplication operator, weighted Dirichlet space, similarity, unitary equivalence, commutant

CLC Number: 

  • 47B35
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