|
Reducing subspaces for a class of multiplication operators on the Dirichlet space
Author(s):
Liankuo
Zhao
Journal:
Proc. Amer. Math. Soc.
137
(2009),
3091-3097.
MSC (2000):
Primary 47A15, 46E22;
Secondary 47S99.
Posted:
March 11, 2009
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 1.
- A. Brown, On a class of operators, Proc. Amer. Math. Soc. 4(1953), 723-728. MR 0059483 (15:538c)
- 2.
- L. Carleson, A representation formula for the Dirichlet integral, Math. Z. 73(1960), 190-196. MR 0112958 (22:3803)
- 3.
- C. Cowen. The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239(1978), 1-31. MR 0482347 (58:2420)
- 4.
- K. Guo and H. Huang, On multiplication operators of the Bergman space: Similarity, unitary equivalence and reducing subspaces, to appear in J. Operator Theory.
- 5.
- 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.
- 6.
- P. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208(1961), 102-112. MR 0152896 (27:2868)
- 7.
- J. Hu, S. Shun, X. Xu and D. Yu, Reducing subspace of analytic operators on the Bergman space, Integr. Equ. Oper. Theory 49(2004), 387-395. MR 2068435 (2005e:47073)
- 8.
- W. Ross, The classical Dirichlet space, Recent advances in operator-related function theory, Contemp. Math. 393, Amer. Math. Soc., Providence, RI, 2006, 171-197. MR 2198379 (2006k:31007)
- 9.
- M. Stessin and K. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130(2002), 2631-2639. MR 1900871 (2003c:47058)
- 10.
- S. Sun and Y. Wang, Reducing subspaces of certain analytic Toeplitz operators on the Bergman space, Northeast Math. J. 14(2)(1998), 147-158. MR 1641059 (99i:47047)
- 11.
- S. Sun and Y. Wang, The commutants of a class of analytic Toeplitz operators on Bergman spaces, Acta Sci. Natur. Univ. Jilinensis 2(1997), 4-8. MR 1604545 (98m:47034)
- 12.
- K. Zhu, Reducing subspaces for a class of multiplication operators, J. London Math. Soc. 62(2)(2000), 553-568. MR 1783644 (2001h:47044)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
47A15, 46E22,
47S99.
Retrieve articles in all Journals with MSC
(2000):
47A15, 46E22,
47S99.
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:
10.1090/S0002-9939-09-09859-1
PII:
S 0002-9939(09)09859-1
Keywords:
Reducing subspace,
multiplication operator,
Dirichlet space
Received by editor(s):
June 25, 2008,
Received by editor(s) in revised form:
December 17, 2008
Posted:
March 11, 2009
Communicated by:
Nigel J. Kalton
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org