Hyponormal operators quasisimilar to an isometry
HTML articles powered by AMS MathViewer
- by Pei Yuan Wu PDF
- Trans. Amer. Math. Soc. 291 (1985), 229-239 Request permission
Abstract:
An expression for the multiplicity of an arbitrary contraction is presented. It is in terms of the isometries which can be densely intertwined to the given contraction. This is then used to obtain a generalization of a result of Sz.-Nagy and Foiaş concerning the existence of a $C{._0}$ contraction which is a quasiaffine transform of a contraction. We then consider the problem when a hyponormal operator is quasisimilar to an isometry or, more generally, when two hyponormal contractions are quasisimilar to each other. Our main results in this respect generalize previous ones obtained by Hastings and the author. For quasinormal and certain subnormal operators, quasisimilarity or similarity to an isometry may even imply unitary equivalence.References
- V. T. Alexander, Contraction operators quasisimilar to a unilateral shift, Trans. Amer. Math. Soc. 283 (1984), no. 2, 697–703. MR 737893, DOI 10.1090/S0002-9947-1984-0737893-X
- Stuart Clary, Equality of spectra of quasi-similar hyponormal operators, Proc. Amer. Math. Soc. 53 (1975), no. 1, 88–90. MR 390824, DOI 10.1090/S0002-9939-1975-0390824-7
- John B. Conway, On quasisimilarity for subnormal operators, Illinois J. Math. 24 (1980), no. 4, 689–702. MR 586807 —, Subnormal operators, Pitman, London, 1981.
- John B. Conway, On quasisimilarity for subnormal operators, Illinois J. Math. 24 (1980), no. 4, 689–702. MR 586807
- John B. Conway and Pei Yuan Wu, The structure of quasinormal operators and the double commutant property, Trans. Amer. Math. Soc. 270 (1982), no. 2, 641–657. MR 645335, DOI 10.1090/S0002-9947-1982-0645335-6
- R. G. Douglas, On the operator equation $S^{\ast } XT=X$ and related topics, Acta Sci. Math. (Szeged) 30 (1969), 19–32. MR 250106
- Robert Goor, On Toeplitz operators which are contractions, Proc. Amer. Math. Soc. 34 (1972), 191–192. MR 293443, DOI 10.1090/S0002-9939-1972-0293443-3
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952 W. W. Hastings, Subnormal operators quasisimilar to an isometry, Notices Amer. Math. Soc. 22 (1975), A-189.
- William W. Hastings, Subnormal operators quasisimilar to an isometry, Trans. Amer. Math. Soc. 256 (1979), 145–161. MR 546912, DOI 10.1090/S0002-9947-1979-0546912-3
- T. B. Hoover, Quasi-similarity of operators, Illinois J. Math. 16 (1972), 678–686. MR 312304
- Richard V. Kadison and I. M. Singer, Three test problems in operator theory, Pacific J. Math. 7 (1957), 1101–1106. MR 92123
- Bernard B. Morrel, A decomposition for some operators, Indiana Univ. Math. J. 23 (1973/74), 497–511. MR 343079, DOI 10.1512/iumj.1973.23.23042
- C. R. Putnam, Hyponormal contractions and strong power convergence, Pacific J. Math. 57 (1975), no. 2, 531–538. MR 380493
- Béla Sz.-Nagy and Ciprian Foiaş, Vecteurs cycliques et quasi-affinités, Studia Math. 31 (1968), 35–42 (French). MR 236756, DOI 10.4064/sm-31-1-35-42 —, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970. K. Takahashi, On the reflexivity of some contractions (preprint).
- L. R. Williams, Equality of essential spectra of quasisimilar quasinormal operators, J. Operator Theory 3 (1980), no. 1, 57–69. MR 565751 P. Y. Wu, On the quasi-similarity of hyponormal contractions, Illinois J. Math. 25 (1981), 498-503.
- Pei Yuan Wu, Approximate decompositions of certain contractions, Acta Sci. Math. (Szeged) 44 (1982), no. 1-2, 137–149. MR 660520
- Pei Yuan Wu, On the reflexivity of $C_{1{\bf \cdot }}$ contractions and weak contractions, J. Operator Theory 8 (1982), no. 2, 209–217. MR 677412
- Pei Yuan Wu, Multiplicities of isometries, Integral Equations Operator Theory 7 (1984), no. 3, 436–439. MR 756766, DOI 10.1007/BF01208384
- Pei Yuan Wu, Contractions with a unilateral shift summand are reflexive, Integral Equations Operator Theory 7 (1984), no. 6, 899–904. MR 774731, DOI 10.1007/BF01195874
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 291 (1985), 229-239
- MSC: Primary 47B20; Secondary 47A45
- DOI: https://doi.org/10.1090/S0002-9947-1985-0797056-X
- MathSciNet review: 797056