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Relative boundedness and relative compactness for linear operators in Banach spaces

Authors: P. Binding and R. Hryniv
Journal: Proc. Amer. Math. Soc. 128 (2000), 2287-2290
MSC (2000): Primary 47A55, 47B07
Published electronically: March 29, 2000
MathSciNet review: 1756088
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Abstract | References | Similar Articles | Additional Information


If $A$ and $B$ are linear operators acting between Banach spaces, we show that compactness of $B$ relative to $A$ does not in general imply that $B$has $A$-bound zero. We do, however, give conditions under which the above implication is valid.

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

  • 1. Kato, T., Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR 53:11389
  • 2. Schechter, M., Spectra of Partial Differential Operators, North-Holland Publ. Company, Amsterdam-London, 1971. MR 56:6144
  • 3. Birman, M. S., On the spectrum of singular boundary-value problems (Russian), Mat. Sb. (N.S.), 55(97)(1961), no. 2, 125-174. MR 26:463
  • 4. Lancaster, P., Shkalikov, A., Damped vibration of beams and related spectral problems, Canad. Appl. Math. Quart., 2(1994), no. 1, 45-90. MR 95m:47090
  • 5. Binding, P., Hryniv, R., Langer, H., Najman, B. Elliptic eigenvalue problems with eigenparameter dependent boundary conditions, to appear.
  • 6. Diestel, J., Uhl, J. J., Vector Measures, Mathematical Surveys, Vol. 15, AMS, Providence-Rhode Island, 1977. MR 56:12216
  • 7. Glazman, I. M. Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Daniel Davey & Co., New York, 1966. MR 32:8210

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Additional Information

P. Binding
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

R. Hryniv
Affiliation: Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 290601 Lviv, Ukraine

Keywords: Relatively bounded operators, relatively compact operators
Received by editor(s): July 24, 1998
Published electronically: March 29, 2000
Additional Notes: The first author’s research was supported by NSERC of Canada.
The second author acknowledges appointment as a Post Doctoral Fellow of the Pacific Institute for the Mathematical Sciences at the University of Calgary.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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