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  • Research Article
  • Open Access

Essential spectra of quasisimilar -quasihyponormal operators

Journal of Inequalities and Applications20062006:72641

  • Received: 1 July 2005
  • Accepted: 20 September 2005
  • Published:


It is shown that if is an upper-triangular operator matrix acting on the Hilbert space and if denotes the essential spectrum, then the passage from to is accomplished by removing certain open subsets of from the former. Using this result we establish that quasisimilar -quasihyponormal operators have equal spectra and essential spectra.


  • Hilbert Space
  • Open Subset
  • Operator Matrix
  • Essential Spectrum
  • Equal Spectrum


Authors’ Affiliations

Department of Mathematics, Changwon National University, Changwon, 641–773, Korea
Department of Mathematics, Seoul National University, Seoul, 151-742, Korea


  1. Clary S: Equality of spectra of quasi-similar hyponormal operators. Proceedings of the American Mathematical Society 1975,53(1):88–90. 10.1090/S0002-9939-1975-0390824-7MathSciNetView ArticleMATHGoogle Scholar
  2. Douglas RG: On the operator equationand related topics. Acta Scientiarum Mathematicarum (Szeged) 1969, 30: 19–32.MathSciNetMATHGoogle Scholar
  3. Gohberg I, Goldberg S, Kaashoek MA: Classes of Linear Operators. Vol. I, Operator Theory: Advances and Applications. Volume 49. Birkhäuser, Basel; 1990:xiv+468.View ArticleMATHGoogle Scholar
  4. Gupta BC: Quasisimilarity and-quasihyponormal operators. Mathematics Today 1985, 3: 49–54.MathSciNetMATHGoogle Scholar
  5. Han JK, Lee HY, Lee WY: Invertible completions ofupper triangular operator matrices. Proceedings of the American Mathematical Society 2000,128(1):119–123. 10.1090/S0002-9939-99-04965-5MathSciNetView ArticleMATHGoogle Scholar
  6. Harte RE: Fredholm, Weyl and Browder theory. Proceedings of the Royal Irish Academy. Section A 1985,85(2):151–176.MathSciNetMATHGoogle Scholar
  7. Harte RE: Invertibility and Singularity for Bounded Linear Operators, Monographs and Textbooks in Pure and Applied Mathematics. Volume 109. Marcel Dekker, New York; 1988:xii+590.MATHGoogle Scholar
  8. Jeon IH, Lee JI, Uchiyama A: On-quasihyponormal operators and quasisimilarity. Mathematical Inequalities & Applications 2003,6(2):309–315.MathSciNetView ArticleMATHGoogle Scholar
  9. Kim IH: On-quasihyponormal operators. Mathematical Inequalities & Applications 2004,7(4):629–638.MathSciNetView ArticleMATHGoogle Scholar
  10. Lee WY: Weyl spectra of operator matrices. Proceedings of the American Mathematical Society 2001,129(1):131–138. 10.1090/S0002-9939-00-05846-9MathSciNetView ArticleMATHGoogle Scholar
  11. Williams LR: Equality of essential spectra of certain quasisimilar seminormal operators. Proceedings of the American Mathematical Society 1980,78(2):203–209. 10.1090/S0002-9939-1980-0550494-3MathSciNetView ArticleMATHGoogle Scholar
  12. Williams LR: Equality of essential spectra of quasisimilar quasinormal operators. Journal of Operator Theory 1980,3(1):57–69.MathSciNetMATHGoogle Scholar
  13. Yang LM: Quasisimilarity of hyponormal and subdecomposable operators. Journal of Functional Analysis 1993,112(1):204–217. 10.1006/jfan.1993.1030MathSciNetView ArticleMATHGoogle Scholar
  14. Yingbin R, Zikun Y: Spectral structure and subdecomposability of-hyponormal operators. Proceedings of the American Mathematical Society 2000,128(7):2069–2074. 10.1090/S0002-9939-99-05257-0MathSciNetView ArticleMATHGoogle Scholar


© A.-H. Kim and I. H. Kim 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.