MULTIPLICATION OPERATORS ON WEIGHTED DIRICHLET SPACES  

作　　者：Kaikai HAN Yucheng LI Maofa WANG 韩凯凯;李玉成;王茂发(School of Statistics and Mathematics,Hebei University of Economics and Business,Shijiazhuang 050061,China;School of Mathematical Sciences,Hebei Normal University,Shijiazhuang 050024,China;School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China)

机构地区：[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

出　　处：《Acta Mathematica Scientia》2024年第6期2225-2248,共24页数学物理学报（B辑英文版）

基　　金：supported by the National Natural Science Foundation of China(12101179,12171138,12171373);the Natural Science Foundation of Hebei Province of China(A2022207001)。

摘　　要：In this paper,we study multiplication operators on weighted Dirichlet spaces D_(β)(β∈R).Let n be a positive integer and β∈R,we show that the multiplication operator M_(z)^(n) on D_(β) is similar to the operator ⊕_(1)^(n)M_(z)on the space⊕_(1)^(n)D_(β).Moreover,we prove that M_(z)^(n)(≥2)on Dβis unitarily equivalent to ⊕_(1)^(n)M_(z) on⊕_(1)^(n)D_(β) if and only if β=0.In addition,we completely characterize the unitary equivalence of the restrictions of M_(z)^(n) to different invariant subspaces z^(k)D_(β)(k≥1),and the unitary equivalence of the restrictions of M_(z)^(n) to different invariant subspaces S_(j)(0≤j<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′( M_(z)^(n))of M_(z)^(n)on a family of analytic function spaces A_(α)^(2)(α∈R)on D(in fact,the family of spaces A_(α)^(2)(α∈R)is the same with the family of spaces D_(β)(β∈R))in terms of the multiplier algebra of the underlying function spaces.In this paper,we give a new characterization of the commutant A′( M_(z)^(n))of M_(z)^(n)on D_(β),and characterize the self-adjoint operators and unitary operators in A'(M_(z)^(n)).We find that the class of self-adjoint operators(unitary operators)in A'(M_(z)^(n))when β≠0 is different from the class of self-adjoint operators(unitary operators)in A′( M_(z)^(n))when β=0.

关 键 词：multiplication operator weighted Dirichlet space SIMILARITY unitary equivalence COMMUTANT 

分 类 号：O177[理学—数学]