|
A characterization of compact perturbations of Toeplitz operators
Author(s):
Jingbo
Xia
Journal:
Trans. Amer. Math. Soc.
361
(2009),
5163-5175.
MSC (2000):
Primary 47A55, 47B35
Posted:
April 7, 2009
MathSciNet review:
2515807
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
Let  be a bounded operator on the Hardy space  of the unit circle. It has been a longstanding problem to determine whether the condition that  is compact for every inner function  implies that  is a compact perturbation of a Toeplitz operator. We show that the answer is affirmative.
References:
-
- 1.
- A. Brown and P. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89-102. MR 0160136 (28:3350)
- 2.
- K. Davidson, On operators commuting with Toeplitz operators modulo the compact operators, J. Funct. Anal. 24 (1977), 291-302. MR 0454715 (56:12963)
- 3.
- R. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, 49, Academic Press, New York-London, 1972. MR 0361893 (50:14335)
- 4.
- R. Douglas, Banach algebra techniques in operator theory, 2nd ed., Graduate Texts in Mathematics, 179, Springer-Verlag, New York, 1998. MR 1634900 (99c:47001)
- 5.
- R. Douglas, H. Shapiro and A. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970), 37-76. MR 0270196 (42:5088)
- 6.
- J. Garnett, Bounded analytic functions, Academic Press, New York-London, 1981. MR 628971 (83g:30037)
- 7.
- B. Johnson and S. Parrott, Operators commuting with a von Neumann algebra modulo the set of compact operators, J. Funct. Anal. 11 (1972), 39-61. MR 0341119 (49:5869)
- 8.
- D. Marshall, Blaschke products generate
 , Bull. Amer. Math. Soc. 82 (1976), 494-496. MR 0402054 (53:5877) - 9.
- D. Sarason, On products of Toeplitz operators, Acta Sci. Math. (Szeged) 35 (1973), 7-12. MR 0331109 (48:9443)
- 10.
- J. Xia, On the essential commutant of
 QC , Trans. Amer. Math. Soc. 360 (2008), 1089-1102. MR 2346484
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2000):
47A55, 47B35
Retrieve articles in all Journals with
MSC (2000):
47A55, 47B35
Additional Information:
Jingbo
Xia
Affiliation:
Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
Email:
jxia@acsu.buffalo.edu
DOI:
10.1090/S0002-9947-09-04736-9
PII:
S 0002-9947(09)04736-9
Received by editor(s):
August 20, 2007
Posted:
April 7, 2009
Additional Notes:
This work was supported by National Science Foundation grant DMS-0456448.
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
