Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON k-QUASI-CLASS A CONTRACTIONS

  • Jeon, In Ho (Department of Mathematics Education Seoul National University of Education) ;
  • Kim, In Hyoun (Department of Mathematics Incheon National University)
  • Received : 2014.03.03
  • Accepted : 2014.03.13
  • Published : 2014.03.30
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Abstract

A bounded linear Hilbert space operator T is said to be k-quasi-class A operator if it satisfy the operator inequality $T^{*k}{\mid}T^2{\mid}T^k{\geq}T^{*k}{\mid}T{\mid}^2T^k$ for a non-negative integer k. It is proved that if T is a k-quasi-class A contraction, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator $D=T^{*k}({\mid}T^2{\mid}-{\mid}T{\mid}^2)T^k$ is strongly stable.

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References

  1. B. P. Duggal, On charcterising contractions with Hilbert-Schmidt defect operator and $C_{.0}$ completely non-unitary part, Integr. Equat. Op. Th. 27 (1997), 314-323. https://doi.org/10.1007/BF01324731
  2. B. P. Duggal, I. H. Jeon and C. S. Kubrusly, Contractions satisfying the absolute value property $|A|^2\;{\leq}\;|A^2|$, Integr. Equat. Op. Th. 49 (2004), 141-148. https://doi.org/10.1007/s00020-002-1202-z
  3. B. P. Duggal, I. H. Jeon and I. H. Kim, On quasi-class A contractions, Linear Algebra Appl. 436 (2012), 3562-3567. https://doi.org/10.1016/j.laa.2011.12.026
  4. T. Furuta, Invitation to Linear Operators, Taylor and Francis, London (2001).
  5. F. Gilfeather, Operator valued roots of abelian analytic functions, Pacific J. Math. 55 (1974), 127-148. https://doi.org/10.2140/pjm.1974.55.127
  6. T. Furuta, M. Ito and T. Yamazaki, A subclass of paranormal operators including class of log-hyponormal and several related classes, Scientiae Mathematicae 1 (1998), 389-403.
  7. P. R. Halmos, A Hilbert Space Problem Book, 2nd edn., Graduate Texts in Mathematics Vol. 19, Springer, New York (1982).
  8. I. H. Jeon and I. H. Kim, On operators satisfying $T^*|T^2|T\;{\geq}\;T^*|T|^2T$, Linear Algebra Appl. 418 (2006), 854-862. https://doi.org/10.1016/j.laa.2006.02.040
  9. R.V. Kadison and I.M. Singer, Three test problems in operator theory, Pacific J. Math. 7 (1957), 1101-1106. https://doi.org/10.2140/pjm.1957.7.1101
  10. C.S. Kubrusly and N. Levan, Proper contractions and invariant subspaces, Internat. J. Math. Math. Sci. 28 (2001), 223-230. https://doi.org/10.1155/S0161171201006287
  11. B. Sz.-Nagy and C. Foias, Harmonis Analysis of Operators on Hilbert Space, North-Holland, Amsterdam (1970).
  12. H. Radjavi and P. Rosenthal, On roots of normal operators, J. Math. Anal. Appl. 34 (1971), 653-664. https://doi.org/10.1016/0022-247X(71)90105-3
  13. K.Takahashi, I.H. Jeon, I.H. Kim and M. Uchiyama, Quasinilpotent part of class A or (p, k)-quasihyponormal operators, J. Operator Theory: Advances and Applications 187 (2008), 199-210.
  14. 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.

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