We prove the reducibility of analytic multipliers $M_\phi$ with a class of finite Blaschke products symbol $\phi$ on the Sobolev disk algebra $R(\mathbb{D})$. We also describe their nontrivial minimal reducing subspaces.
Yong CHEN
,
Ya LIU
,
Chuntao QIN
. REDUCIBILITY FOR A CLASS OF ANALYTIC MULTIPLIERS ON SOBOLEV DISK ALGEBRA[J]. Acta mathematica scientia, Series B, 2021
, 41(2)
: 361
-370
.
DOI: 10.1007/s10473-021-0202-5
[1] Adams R A. Sobolev Spaces. New York:Academic Press, 1975
[2] Cowen C. The commutant of an analytic Toeplitz oprator. Trans Amer Math Soc, 1978, 239:1-31
[3] Chen Y, Xu H M. Reducibility and unitarily equivalence for a class of analytic multipliers on the Dirichlet space. Complex Anal Oper Theory, 2013, 7(6):1897-1908
[4] Chen Y, Lee Y J, Yu T. Reducibility and unitary equivalence for a class of analytic multiplication operator on the Dirichlet space. Studia Math, 2014, 220(2):141-156
[5] Chen Y, Qin C T, Wu Q. Reducibility and unitary equivalence of analytic multipliers on Sobolev disk algebra. J Math Anal Appl, 2017, 455:1249-1256
[6] Chen Y, Xu X M, Zhao Y L. Reducibility for a class of analytic multipliers on the Dirichlet space. Complex Anal Oper Theory, 2018, 12:1781-1790
[7] Douglas R, Putinar M, Wang K. Reducing subspaces for analytic multipliers of the Bergman space. J Funct Anal, 2012, 263(6):1744-1765
[8] Gu C X, Luo S B, Xiao J. Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surfaces. Complex Manifolds, 2017, 4:84-119
[9] Guo K Y, Sun S H, Zheng D C, Zhong C Y. Multiplication operators on the Bergman space via the Hardy space of the bidisk. J Reine Angew Math, 2009, 628:129-168
[10] Hu J Y, Sun S H, Xu X M, Yu D H. Reducing subspace of analytic Toeplitz operators on the Bergman space. Integr Equ Oper Theory, 2004, 49:387-395
[11] Jiang C L, Wang Z Y. Structure of Hilbert space operators. Singapore:World Scientic, 2006
[12] Li Y C, Liu Q J, Lan W H. On similarity and reducing subspaces of multiplication operator on Sobolev disk algebra. J Math Anal Appl, 2014, 419(2):1161-1167
[13] Luo S B. Reducing subspaces of multiplication operators on the Dirichlet space. Integr Equ Oper Theory, 2016, 85(4):539-554
[14] Stessin M, Zhu K H. Reducing subspaces of weighted shift operators. Proc Amer Math Soc, 2002, 130:2631-2639
[15] Stessin M, Zhu K H. Generalized factorization in Hardy spaces and the commutant of Toeplitz operators. Canad J Math, 2003, 55(2):379-400
[16] Sun S H, Zheng D C, Zhong C Y. Classification of reducing subspaces of a class of multiplication operators on the Bergman space via the Hardy space of the bidisk. Canad J Math, 2010, 62(2):415-438
[17] Thomson J. The commutant of a class of analytic Toeplitz operators Ⅱ. Indiana Univ Math J, 1976, 25:793-800
[18] Thomson J. The commutant of a class of analytic Toeplitz operators. Amer J Math, 1977, 99:522-529
[19] Zhao L K. Reducing subspaces for a class of multiplication operators on the Dirichlet space. Proc Amer Math Soc, 2019, 137:3091-3097