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Linear relations in the Calkin algebra for composition operators


Authors: Thomas Kriete and Jennifer Moorhouse
Journal: Trans. Amer. Math. Soc. 359 (2007), 2915-2944
MSC (2000): Primary 47B33; Secondary 47B32
DOI: https://doi.org/10.1090/S0002-9947-07-04166-9
Published electronically: January 4, 2007
MathSciNet review: 2286063
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Abstract: We consider this and related questions: When is a finite linear combination of composition operators, acting on the Hardy space or the standard weighted Bergman spaces on the unit disk, a compact operator?


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  • 1. A. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (1987), no. 5, 490-503, 515. MR 0931885 (89e:30058)
  • 2. E. Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981), 230-232. MR 0593463 (82f:47039)
  • 3. P. S. Bourdon, Components of linear-fractional composition operators, J. Math. Anal. Appl. 279 (2003), 228-245. MR 1970503 (2004c:47047)
  • 4. P. S. Bourdon, D. Levi, S. K. Narayan, and J. H. Shapiro, Which linear fractional composition operators are essentially normal? J. Math. Anal. Appl. 280 (2003), 30-53. MR 1972190 (2003m:47042)
  • 5. J. A. Cima and A. L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math. 179 (1997), 59-64. MR 1452525 (98e:47047)
  • 6. D. N. Clark, One-dimensional perturbations of restricted shifts, J. D'Analyse Math. 25 (1972), 169-191. MR 0301534 (46:692)
  • 7. C. C. Cowen, Composition operators on $ H^2$, J. Operator Theory 9 (1983), 77-106. MR 0695941 (84d:47038)
  • 8. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026 (97i:47056)
  • 9. P. L. Duren, Theory of $ H^p$ Spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • 10. P. Gorkin and R. Mortini, Norms and essential norms of linear combinations of endomorphisms, Trans. Amer. Math. Soc. 358 (2006), no. 2, 553-571. MR 2177030 (2006h:47055)
  • 11. P. R. Halmos, Measure Theory, Van Nostrand, New York, 1950. MR 0033869 (11:504d)
  • 12. K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, NJ, 1962. MR 0133008 (24:A2844)
  • 13. B. D. MacCluer, Components in the space of composition operators, Integral Equations and Operator Theory 12 (1989), 725-738. MR 1009027 (91b:47070)
  • 14. B. D. MacCluer, S. Ohno, and R. Zhao, Topological structure of the space of composition operators on $ H^\infty$, Integral Equations and Operator Theory 40 (2001), 481-494. MR 1839472 (2002d:47039)
  • 15. B. D. MacCluer and J. H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canadian J. Math. 38 (1986), 878-906. MR 0854144 (87h:47048)
  • 16. J. Moorhouse, Compact differences of composition operators, J. Functional Analysis 219 (2005), 70-92. MR 2108359 (2005i:47037)
  • 17. J. Moorhouse and C. Toews, Differences of composition operators, Trends in Banach Spaces and Operator Theory (Memphis, TN, 2001), 207-213, Contemp. Math.  321, Amer. Math. Soc., Providence, RI, 2003. MR 1978818 (2004b:47046)
  • 18. N. K. Nikolski, Operators, Functions, and Systems: An Easy Reading, Volume 1: Hardy, Hankel and Toeplitz, Mathematical Surveys and Monographs, 92, Amer. Math. Soc., Providence, RI, 2002. MR 1864396 (2003i:47001a)
  • 19. P. J. Nieminen and E. Saksman, On the compactness of the difference of composition operators, J. Math. Anal. Appl. 298 (2004), 501-522. MR 2086972 (2005e:30065)
  • 20. A. G. Poltoratski, Boundary behavior of pseudocontinuable functions (Russian), Algebra i Analiz 5 (1993), 189-210; translation in St. Petersburg Math. J. 5 (1994), 389-406. MR 1223178 (94k:30090)
  • 21. S. Saitoh, Integral Transforms, Reproducing Kernels and their Applications, Pitman Research Notes in Mathematics, 369, Longman, Harlow, 1997. MR 1478165 (98k:46041)
  • 22. D. E. Sarason, Composition operators as integral operators, Analysis and Partial Differential Equations, C. Sadosky (ed.), 545-565, Marcel-Dekker, New York, 1990. MR 1044808 (92a:47040)
  • 23. D. E. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, University of Arkansas Lecture Notes in the Mathematical Sciences, Volume 10, John Wiley and Sons, New York, 1994. MR 1289670 (96k:46039)
  • 24. J. E. Shapiro, Aleksandrov measures used in essential norm inequalities for composition operators, J. Operator Theory 40 (1998), 133-146. MR 1642538 (99i:47062)
  • 25. J. H. Shapiro, The essential norm of a composition operator, Annals Math. 125 (1987), 375-404. MR 0881273 (88c:47058)
  • 26. J. H. Shapiro and C. Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990), 117-152. MR 1066401 (92g:47041)
  • 27. J. H. Shapiro and C. Sundberg, Compact composition operators on $ L^1$, Proc. Amer. Math. Soc. 108 (1990), 443-449. MR 0994787 (90d:47035)
  • 28. J. H. Shapiro and P. D. Taylor, Compact, nuclear and Hilbert-Schmidt composition operators on $ H^2$, Indiana Univ. Math. J. 23 (1973), 471-496. MR 0326472 (48:4816)

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Additional Information

Thomas Kriete
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: tlk8q@virginia.edu

Jennifer Moorhouse
Affiliation: Department of Mathematics, Colgate University, Hamilton, New York 11346
Email: jmoorhouse@mail-colgate.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04166-9
Keywords: Composition operator, Hardy space, Bergman spaces, linear relations modulo the compact operators.
Received by editor(s): July 14, 2004
Received by editor(s) in revised form: July 7, 2005
Published electronically: January 4, 2007
Additional Notes: Work of the first author was supported in part by a sesquicentennial associateship at the University of Virginia.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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