Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Contraction operators quasisimilar to a unilateral shift


Author: V. T. Alexander
Journal: Trans. Amer. Math. Soc. 283 (1984), 697-703
MSC: Primary 47A45
DOI: https://doi.org/10.1090/S0002-9947-1984-0737893-X
MathSciNet review: 737893
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {U_n}$ denote the unilaterial shift of finite multiplicity $ n$. It is shown that a contraction operator $ T$ is quasisimilar to $ {U_n}$ if and only if $ T$ is of Class $ {C_1}$., the canonical isometry $ V$ associated with $ T$ is pure and $ T$ is $ n$-cyclic with analytically independent vectors. For this, the notions of operators of analytic type and analytic independence of vectors are introduced. A characterization of the cyclic vectors of the Backward Shift is also presented.


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

  • [1] W. S. Clary, Quasisimilarity and subnormal operators, Ph.D. thesis, The University of Michigan, 1973.
  • [2] R. G. Douglas, On the operator equation $ {S^{\ast}}XT = X$, Acta. Sci. Math. (Szeged) 30 (1969), 19-32. MR 0250106 (40:3347)
  • [3] R. G. Douglas, H. S. Shapiro and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970), 37-76. MR 0270196 (42:5088)
  • [4] P. L. Duren, Theory of $ {H^p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [5] W. W. Hastings, Subnormal operators quasisimilar to an isometry, Trans. Amer. Math. Soc. 256 (1979), 145-161. MR 546912 (80k:47028)
  • [6] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR 0133008 (24:A2844)
  • [7] B. Sz.-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space, American Elsevier, New York, 1970. MR 0275190 (43:947)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47A45

Retrieve articles in all journals with MSC: 47A45


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0737893-X
Keywords: Contraction, unilateral shift, operator of analytic type, isometry, quasisimilarity
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society