Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Contractions with the bicommutant property


Author: Katsutoshi Takahashi
Journal: Proc. Amer. Math. Soc. 93 (1985), 91-95
MSC: Primary 47A45; Secondary 47A65, 47C05
DOI: https://doi.org/10.1090/S0002-9939-1985-0766534-7
MathSciNet review: 766534
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that if $ T$ is a contraction for which there is an operator $ W$ with dense range such that $ WT = SW$ for some unilateral shift $ S$, then $ T$ has the bicommutant property, that is, the double commutant of $ T$ is the weakly closed algebra generated by $ T$ and the identity. As an example of such a contraction we have a contraction $ T$ such that $ I - {T^ * }T$ is of trace class and the spectrum of $ T$ fills the unit disc.


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

  • [1] L. Kerchy, On invariant subspace lattices of $ {C_{11}}$-contractions, Acta Sci. Math. (Szeged) 43 (1981), 281-293. MR 640305 (83j:47010)
  • [2] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970. MR 0275190 (43:947)
  • [3] -, On the structure of intertwining operators, Acta Sci. Math. (Szeged) 35 (1973), 225-254. MR 0399896 (53:3737)
  • [4] K. Takahashi, $ {C_1}$-contractions with Hilbert-Schmidt defect operators, J. Operator Theory (to appear). MR 757438 (86a:47006)
  • [5] K. Takahashi and M. Uchiyama, Every $ {C_{00}}$-contraction with Hilbert-Schmidt defect operator is of class $ {C_0}$, J. Operator Theory 10 (1983), 331-335. MR 728912 (85g:47014)
  • [6] T. R. Turner, Double commutants of isometries, Tôhoku Math. J. 24 (1972), 547-549. MR 0322560 (48:922)
  • [7] M. Uchiyama, Double commutants of $ {C_0}$ contractions, Proc. Amer. Math. Soc. 69 (1978), 283-288. MR 0482301 (58:2374)
  • [8] -, Double commutants of C.q contractions. II, Proc. Amer. Math. Soc. 74 (1979), 271-277. MR 524299 (80b:47011)
  • [9] -, Contractions and unilateral shifts, Acta Sci. Math. (Szeged) 46 (1983), 345-356. MR 739054 (85e:47011)
  • [10] J. Wermer, On invariant subspaces of normal operators, Proc. Amer. Math. Soc. 3 (1952), 270-277. MR 0048700 (14:55g)
  • [11] P. Y. Wu, $ {C_{11}}$ contractions are reflexive. II, Proc. Amer. Math. Soc. 82 (1981), 226-230. MR 609656 (82j:47008)
  • [12] -, Approximate decompositions of certain contractions, Acta Sci. Math. (Szeged) 44 (1982), 137-149. MR 660520 (83j:47011)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A45, 47A65, 47C05

Retrieve articles in all journals with MSC: 47A45, 47A65, 47C05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0766534-7
Keywords: Contraction, the bicommutant property, double commutant, unilateral shift, Hilbert-Schmidt defect operator
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society