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Relative boundedness and relative compactness for linear operators in Banach spaces
Author(s):
P.
Binding;
R.
Hryniv
Journal:
Proc. Amer. Math. Soc.
128
(2000),
2287-2290.
MSC (2000):
Primary 47A55, 47B07
Posted:
March 29, 2000
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Abstract:
If  and  are linear operators acting between Banach spaces, we show that compactness of  relative to  does not in general imply that  has  -bound zero. We do, however, give conditions under which the above implication is valid.
References:
-
- 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
Email:
binding@ucalgary.ca
R.
Hryniv
Affiliation:
Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 290601 Lviv, Ukraine
Email:
hryniv@mebm.lviv.ua
DOI:
10.1090/S0002-9939-00-05729-4
PII:
S 0002-9939(00)05729-4
Keywords:
Relatively bounded operators,
relatively compact operators
Received by editor(s):
July 24, 1998
Posted:
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
Copyright of article:
Copyright
2000,
American Mathematical Society
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