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On the joint spectrum and $ H\sp \infty$-functional calculus for pairs of commuting contractions

Author: Alfredo Octavio
Journal: Proc. Amer. Math. Soc. 114 (1992), 497-503
MSC: Primary 47A13; Secondary 47A10, 47A60
MathSciNet review: 1069294
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Abstract: In this paper we show the existence of a pair of commuting completely nonunitary contractions $ (S,T)$ on a Hilbert space, whose joint Taylor spectrum contains the torus, such that there is a bounded analytic function $ h$ on the bidisk with $ h(S,T) = 0$.

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  • [1] C. Apostol, Ultraweakly closed operator algebras, J. Operator Theory 2 (1979), 49-61. MR 553863 (80k:47005)
  • [2] H. Bercovici, Operator theory and arithmetic in $ {H^\infty }$, Math. Surveys Monographs, Amer. Math. Soc., Providence, RI, 1988. MR 954383 (90e:47001)
  • [3] H. Bercovici, C. Foias, and C. Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conf. Series in Math., vol. 56, Amer. Math. Soc., Providence, RI, 1985. MR 787041 (87g:47091)
  • [4] E. Briem, A. M. Davie, and B. K. Øksendal, Functional calculus for commuting contractions, J. London Math. Soc. (2) 7 (1973), 709-718.
  • [5] S. Brown, B. Chevreau, and C. Pearcy, On the structure of contraction operators II, J. Funct. Anal. 76 (1988), 30-55. MR 923043 (90b:47030b)
  • [6] E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Univ. Press, London, 1966. MR 0231999 (38:325)
  • [7] R. E. Curto, Applications of several complex variables to multiparameter spectral theory, Surveys of Recent Results in Operator Theory, vol. II, Longman, London, 1988, pp. 25-90. MR 976843 (90d:47007)
  • [8] A. J. Lohwater and G. Piranian, Bounded analytic functions with large cluster sets, Ann. Acad. Sci. Fenn. Ser. A I 499 (1971), 3-6. MR 0293099 (45:2178)
  • [9] W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987. MR 924157 (88k:00002)
  • [10] -, Function theory on polydisks, Benjamin, New York, 1969.
  • [11] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970. MR 0275190 (43:947)

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Keywords: Contractions, joint spectrum, functional calculus
Article copyright: © Copyright 1992 American Mathematical Society

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