Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hermitian weighted composition operators on $ H^2$


Authors: Carl C. Cowen and Eungil Ko
Journal: Trans. Amer. Math. Soc. 362 (2010), 5771-5801
MSC (2010): Primary 47B38; Secondary 47B15, 47B33, 47D03
DOI: https://doi.org/10.1090/S0002-9947-2010-05043-3
Published electronically: June 9, 2010
MathSciNet review: 2661496
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, we identify the Hermitian weighted composition operators on $ H^{2}$ and compute their spectral measures. Some relevant semigroups are studied. The resulting ideas can be used to find the polar decomposition, the absolute value, and the Aluthge transform of some composition operators on $ H^{2}$.


References [Enhancements On Off] (What's this?)

  • 1. J. B. Conway, ``A Course in Functional Analysis'', Springer-Verlag, New York, 1990. MR 1070713 (91e:46001)
  • 2. C. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978), 1-31. MR 0482347 (58:2420)
  • 3. C. C. Cowen, An analytic Toeplitz operator that commutes with a compact operator, J. Functional Analysis 36(2) (1980), 169-184. MR 569252 (81d:47020)
  • 4. C. C. Cowen, Composition operators on $ H^2$, J. Operator Theory 9 (1983), 77-106. MR 695941 (84d:47038)
  • 5. C. C. Cowen, Linear fractional composition operators on $ H^2$, Integral Equations Operator Theory 11 (1988), 151-160. MR 928479 (89b:47044)
  • 6. C. C. Cowen and E. A. Gallardo Gutierrez, The adjoint of a composition operator, preprint, 1/31/2005.
  • 7. C. C. Cowen and E. A. Gallardo Gutierrez, Projected and multiple valued weighted composition operators, J. Functional Analysis 238 (2006), 447-462. MR 2253727 (2007e:47033)
  • 8. C. C. Cowen and B. D. MacCluer, ``Composition Operators on Spaces of Analytic Functions'', CRC Press, Boca Raton, 1995. MR 1397026 (97i:47056)
  • 9. C. C. Cowen and B. D. MacCluer, Linear fractional maps of the ball and their composition operators, Acta. Sci. Math. (Szeged) 66 (2000), 351-376. MR 1768872 (2001g:47041)
  • 10. F. Forelli, The isometries of $ H^p$, Canadian J. Math. 16 (1964), 721-728. MR 0169081 (29:6336)
  • 11. G. Gunatillake, ``Weighted Composition Operators'', Thesis, Purdue University, 2005.
  • 12. E. Hille and R. S. Phillips ``Functional Analysis and Semigroups'', revised ed., American Math. Society, Providence, 1957. MR 0089373 (19:664d)
  • 13. I. B. Jung, E. Ko, and C. Pearcy, Aluthge transforms of operators, Integral Equations Operator Theory 37 (2000), 437-448. MR 1780122 (2001i:47035)
  • 14. W. Konig, Semicocycles and weighted composition semigroups on $ H^p$, Michigan Math. J. 37 (1990), 469-476. MR 1077330 (91m:47057)
  • 15. A. Pazy, ``Semigroups of Linear Operators and Applications to Partial Differential Equations'', Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
  • 16. H. Sadraoui, ``Hyponormality of Toeplitz and Composition Operators'', Thesis, Purdue University, 1992.
  • 17. A. G. Siskakis, Weighted composition semigroups on Hardy spaces, Linear Alg. Appl. 84 (1986), 359-371. MR 872296 (88b:47058)
  • 18. A. G. Siskakis, Semigroups of composition operators on spaces of analytic functions, a review, in ``Studies on Composition Operators'', Contemporary Math. 213 (1998), 229-252. MR 1601120 (98m:47049)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47B38, 47B15, 47B33, 47D03

Retrieve articles in all journals with MSC (2010): 47B38, 47B15, 47B33, 47D03


Additional Information

Carl C. Cowen
Affiliation: Department of Mathematical Sciences, Indiana University, Purdue University, Indianapolis, Indianapolis, Indiana 46202
Email: ccowen@iupui.edu

Eungil Ko
Affiliation: Department of Mathematics, Ewha Women’s University, Seoul 120-750, Korea
Email: eiko@ewha.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-2010-05043-3
Keywords: Weighted composition operator, composition operator, Hermitian operator, operator semigroup, spectral theory
Received by editor(s): June 8, 2007
Received by editor(s) in revised form: June 22, 2008
Published electronically: June 9, 2010
Additional Notes: The second author was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00461).
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society