Kyungpook Mathematical Journal

  • 제47권2호
  • /
  • Pages.221-226
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  • 2007
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  • 1225-6951(pISSN)
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  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
권호 표지

Every Operator Almost Commutes with a Compact Operator

  • Jung, Il Bong (Department of Mathematics, Kyungpook National University) ;
  • Ko, Eungil (Department of Mathematics, Ewha Women's University) ;
  • Pearcy, Carl (Department of Mathematics, Texas A&M University)
  • 투고 : 2006.01.16
  • 발행 : 2007.06.23
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초록

In this note we set forth three possible definitions of the property of "almost commuting with a compact operator" and discuss an old result of W. Arveson that says that every operator on Hilbert space has the weakest of the three properties. Finally, we discuss some recent progress on the hyperinvariant subspace problem (see the bibliography), and relate it to the concept of almost commuting with a compact operator.

키워드

참고문헌

  1. W. Arveson, Notes on extensions of C-algebras, Duke Math. J., 44(1977), 329-355. https://doi.org/10.1215/S0012-7094-77-04414-3
  2. H. Bercovici, C. Foias and C. Pearcy, On the hyperinvariant subspace problem, IV, in preparation.
  3. B. Chevreau, W. Li, and C. Pearcy, A new Lomonosov lemma, J. Operator Theory, 40(1998), 409-417.
  4. D. Deckard, R. G. Douglas, and C. Pearcy, On invariant subspaces of quasitriangular operators, Amer. J. Math., 9(1969), 637-647.
  5. C. Foias, S. Hamid, C. Onica and C. Pearcy, On the hyperinvariant subspace problem, III, J. Funct. Anal., 222(2005), 129-142. https://doi.org/10.1016/j.jfa.2004.07.002
  6. C. Foias and C. Pearcy, On the hyperinvariant subspace problem, J. Funct. Anal., 219(2005), 134-142. https://doi.org/10.1016/j.jfa.2004.04.003
  7. D. Herrero, Quasidiagonality, similarity and approximation by nilpotent operators, Indiana Univ. Math. J., 30(1981), 199-233. https://doi.org/10.1512/iumj.1981.30.30017
  8. D. Hadwin, An operator still not satisfying Lomonosov's hypothesis, Proc. Amer. Math. Soc., 123(1995), 3039-3041.
  9. P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc., 76(1970), 887-933. https://doi.org/10.1090/S0002-9904-1970-12502-2
  10. S. Hamid, C. Onica and C. Pearcy, On the hyperinvariant subspace problem, II, Indiana Univ. Math. J., 54(2005), 743-754. https://doi.org/10.1512/iumj.2005.54.2639
  11. V. Lomonosov, An extension of Burnside's theorem to infinite dimensional spaces, Israel J. Math., 75(1991), 329-339. https://doi.org/10.1007/BF02776031
  12. C. Pearcy and N. Salinas, An invariant subspace theorem, Michigan Math. J., 20(1973), 21-31. https://doi.org/10.1307/mmj/1029001007