On the joint spectrum and -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

DOI:
https://doi.org/10.1090/S0002-9939-1992-1069294-1

MathSciNet review:
1069294

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we show the existence of a pair of commuting completely nonunitary contractions on a Hilbert space, whose joint Taylor spectrum contains the torus, such that there is a bounded analytic function on the bidisk with .

**[1]**Constantin Apostol,*Ultraweakly closed operator algebras*, J. Operator Theory**2**(1979), no. 1, 49–61. MR**553863****[2]**Hari Bercovici,*Operator theory and arithmetic in 𝐻^{∞}*, Mathematical Surveys and Monographs, vol. 26, American Mathematical Society, Providence, RI, 1988. MR**954383****[3]**Hari Bercovici, Ciprian Foias, and Carl Pearcy,*Dual algebras with applications to invariant subspaces and dilation theory*, CBMS Regional Conference Series in Mathematics, vol. 56, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR**787041****[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]**Scott W. Brown, Bernard Chevreau, and Carl Pearcy,*On the structure of contraction operators. II*, J. Funct. Anal.**76**(1988), no. 1, 30–55. MR**923043**, https://doi.org/10.1016/0022-1236(88)90047-X**[6]**E. F. Collingwood and A. J. Lohwater,*The theory of cluster sets*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR**0231999****[7]**Raúl E. Curto,*Applications of several complex variables to multiparameter spectral theory*, Surveys of some recent results in operator theory, Vol. II, Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 25–90. MR**976843****[8]**A. J. Lohwater and George Piranian,*Bounded analytic functions with large cluster sets*, Ann. Acad. Sci. Fenn. Ser. A I**499**(1971), 7. MR**0293099****[9]**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157****[10]**-,*Function theory on polydisks*, Benjamin, New York, 1969.**[11]**Béla Sz.-Nagy and Ciprian Foiaş,*Harmonic analysis of operators on Hilbert space*, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. MR**0275190**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47A13,
47A10,
47A60

Retrieve articles in all journals with MSC: 47A13, 47A10, 47A60

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1069294-1

Keywords:
Contractions,
joint spectrum,
functional calculus

Article copyright:
© Copyright 1992
American Mathematical Society