数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (6): 2225-2248.doi: 10.1007/s10473-024-0610-4
Kaikai HAN1,†, Yucheng LI2, Maofa WANG3
Kaikai HAN1,†, Yucheng LI2, Maofa WANG3
摘要: In this paper, we study multiplication operators on weighted Dirichlet spaces . Let be a positive integer and , we show that the multiplication operator on is similar to the operator on the space . Moreover, we prove that on is unitarily equivalent to on if and only if . In addition, we completely characterize the unitary equivalence of the restrictions of to different invariant subspaces , and the unitary equivalence of the restrictions of to different invariant subspaces .
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 of on a family of analytic function spaces on (in fact, the family of spaces is the same with the family of spaces ) in terms of the multiplier algebra of the underlying function spaces. In this paper, we give a new characterization of the commutant of on , and characterize the self-adjoint operators and unitary operators in . We find that the class of self-adjoint operators (unitary operators) in when is different from the class of self-adjoint operators (unitary operators) in when .
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