The Spectrum of a Composition Operator and Calderón’s Complex Interpolation
Using the method of complex interpolation due to A.P. Calderón, we give a general theorem for identifying the spectrum of an operator acting on a family of interpolation spaces. We then use this to determine the spectrum of certain composition operators acting on the weighted Dirichlet and analytic Besov spaces of the unit disk.
KeywordsComposition operators complex interpolation spectra
Mathematics Subject Classification (2000)Primary 46B70 47B33
Unable to display preview. Download preview PDF.
- B. Aupetit, A Primer on Spectral Theory, Springer-Verlag, Berlin-Heidelberg, 1991.Google Scholar
- B. Barnes, Interpolation of spectrum of bounded operators on Lebesgue spaces, Rocky Mountain J. Math. 20 (1987), 359–378.Google Scholar
- J. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1990.Google Scholar
- R. Donaway, Norm and essential norm estimates of composition operators on Besov type spaces, Ph.D. thesis, The University of Virginia, 1999.Google Scholar
- E. Gallardo-Gutiérrez and A. Montes-Rodríguez, The role of the spectrum in the cyclic behavior of composition operators, Mem. Amer. Math. Soc. 167 (2004), no. 791, x + 81 pp.Google Scholar
- E. Hille and R. Phillips, Functional Analysis and Semi-groups, revised ed., American Math. Society, Providence, 1957.Google Scholar
- M. Pons, Composition operators on Besov and Dirichlet type spaces, Ph.D. thesis, The University of Virginia, 2007.Google Scholar
- K. Saxe, On complex interpolation and spectral continuity, Stud. Math. 130 (1998), no. 3, 223–229.Google Scholar
- M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Ph.D. thesis, Michigan State University, 1996.Google Scholar
- K. Zhu, Operator Theory in Function Spaces, 2nd ed., American Mathematical Society, Providence, 2007.Google Scholar