Reducing subspaces for a class of multiplication operators on the Dirichlet space
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Abstract:
In this paper, we discuss reducing subspaces of multiplication operators $M_\phi$ on the Dirichlet space $\mathcal {D}$ defined by a Blaschke product $\phi$ with two zeros $a$, $b$ in the unit disk $\mathbb {D}$ and show that when $a+b=0$, $M_\phi$ has two proper ones; otherwise it has none. This is different from the cases of the Hardy space and the Bergman space.References
- Arlen Brown, On a class of operators, Proc. Amer. Math. Soc. 4 (1953), 723–728. MR 59483, DOI 10.1090/S0002-9939-1953-0059483-2
- Lennart Carleson, A representation formula for the Dirichlet integral, Math. Z. 73 (1960), 190–196. MR 112958, DOI 10.1007/BF01162477
- Carl C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978), 1–31. MR 482347, DOI 10.1090/S0002-9947-1978-0482347-9
- K. Guo and H. Huang, On multiplication operators of the Bergman space: Similarity, unitary equivalence and reducing subspaces, to appear in J. Operator Theory.
- 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.
- Paul R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102–112. MR 152896, DOI 10.1515/crll.1961.208.102
- Junyun Hu, Shunhua Sun, Xianmin Xu, and Dahai Yu, Reducing subspace of analytic Toeplitz operators on the Bergman space, Integral Equations Operator Theory 49 (2004), no. 3, 387–395. MR 2068435, DOI 10.1007/s00020-002-1207-7
- William T. Ross, The classical Dirichlet space, Recent advances in operator-related function theory, Contemp. Math., vol. 393, Amer. Math. Soc., Providence, RI, 2006, pp. 171–197. MR 2198379, DOI 10.1090/conm/393/07378
- Michael Stessin and Kehe Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2631–2639. MR 1900871, DOI 10.1090/S0002-9939-02-06382-7
- Shanli Sun and Yuejian Wang, Reducing subspaces of certain analytic Toeplitz operators on the Bergman space, Northeast. Math. J. 14 (1998), no. 2, 147–158. MR 1641059
- Shan Li Sun and Yue Jian Wang, The commutants of a class of analytic Toeplitz operators on Bergman spaces, Acta Sci. Natur. Univ. Jilin. 2 (1997), 4–8 (Chinese, with English and Chinese summaries). MR 1604545
- Kehe Zhu, Reducing subspaces for a class of multiplication operators, J. London Math. Soc. (2) 62 (2000), no. 2, 553–568. MR 1783644, DOI 10.1112/S0024610700001198
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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 3091-3097
- MSC (2000): Primary 47A15, 46E22; Secondary 47S99
- DOI: https://doi.org/10.1090/S0002-9939-09-09859-1
- MathSciNet review: 2506467