Reducing subspaces for a class of multiplication operators on the Dirichlet space
Author:
Liankuo Zhao
Journal:
Proc. Amer. Math. Soc. 137 (2009), 30913097
MSC (2000):
Primary 47A15, 46E22; Secondary 47S99.
Published electronically:
March 11, 2009
MathSciNet review:
2506467
Fulltext PDF Free Access
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Abstract: In this paper, we discuss reducing subspaces of multiplication operators on the Dirichlet space defined by a Blaschke product with two zeros , in the unit disk and show that when , has two proper ones; otherwise it has none. This is different from the cases of the Hardy space and the Bergman space.
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 A. Brown, On a class of operators, Proc. Amer. Math. Soc. 4(1953), 723728. MR 0059483 (15:538c)
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 L. Carleson, A representation formula for the Dirichlet integral, Math. Z. 73(1960), 190196. MR 0112958 (22:3803)
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 C. Cowen. The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239(1978), 131. MR 0482347 (58:2420)
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 K. Guo and H. Huang, On multiplication operators of the Bergman space: Similarity, unitary equivalence and reducing subspaces, to appear in J. Operator Theory.
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 K. Guo, S. Sun, D. Zheng and C. Zhong, Multiplication operators on the Bergman space via the Hardy space of the bidisk, to appear in J. Reine Angew. Math.
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 P. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208(1961), 102112. MR 0152896 (27:2868)
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 J. Hu, S. Shun, X. Xu and D. Yu, Reducing subspace of analytic operators on the Bergman space, Integr. Equ. Oper. Theory 49(2004), 387395. MR 2068435 (2005e:47073)
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 M. Stessin and K. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130(2002), 26312639. MR 1900871 (2003c:47058)
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 S. Sun and Y. Wang, Reducing subspaces of certain analytic Toeplitz operators on the Bergman space, Northeast Math. J. 14(2)(1998), 147158. MR 1641059 (99i:47047)
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 12.
 K. Zhu, Reducing subspaces for a class of multiplication operators, J. London Math. Soc. 62(2)(2000), 553568. MR 1783644 (2001h:47044)
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Additional Information
Liankuo Zhao
Affiliation:
School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041004, People’s Republic of China
Email:
lkzhao@sxnu.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002993909098591
PII:
S 00029939(09)098591
Keywords:
Reducing subspace,
multiplication operator,
Dirichlet space
Received by editor(s):
June 25, 2008
Received by editor(s) in revised form:
December 17, 2008
Published electronically:
March 11, 2009
Communicated by:
Nigel J. Kalton
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
