Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T04:51:42.641Z Has data issue: false hasContentIssue false
  • English
  • Français

An abstract multiparameter spectral theory

Published online by Cambridge University Press:  14 November 2011

P. A. Binding ,
A. Källström  and
B. D. Sleeman
P. A. Binding
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, CanadaT2N 1N4
A. Källström
Affiliation:
Department of Mathematics, Uppsala University, S-75238 Uppsala, Sweden
B. D. Sleeman
Affiliation:
Department of Mathematical Sciences, University of Dundee, Dundee DD1 4HN, Scotland
Get access Rights & Permissions [Opens in a new window]

Synopsis

We consider the eigenvalue problem

for self-adjoint operators Ai and Bij on separable Hilbert spaces Hi. It is assumed that and Bij are bounded with compact. Various properties of the eigentuples λi, and xi are deduced under a “definiteness condition” weaker than those used by previous authors, at least in infinite dimensions. In particular, a Parseval relation and eigenvector expansion are derived in a suitably constructed tensor product space.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 3-4 , 1982 , pp. 193 - 204
Copyright
Copyright © Royal Society of Edinburgh 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Atkinson, F. V.. Multiparameter Eigenvalue Problems Vol. 1(New York: Academic, 1972).Google Scholar
2Binding, P. A.. Another positivity result for determinantal operators. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 333–337.CrossRefGoogle Scholar
3On a problem of B. D. Sleeman. J. Math. Anal. Appl, to appear.Google Scholar
4Binding, P. A.. Left definite multiparameter eigenvalue problems, to appear.Google Scholar
5Binding, P. A.. Multiparameter definiteness conditions II, to appear.Google Scholar
6Browne, P. J.. A multiparameter eigenvalue problem. j. Math. Anal. Appl. 38 (1972), 553–568.CrossRefGoogle Scholar
7Browne, P. J.. Abstract multiparameter theory I. J. Math. Anal. Appl. 60 (1977), 259–273.CrossRefGoogle Scholar
8Browne, P. J.. Abstract multiparameter theory II. J. Math. Anal. Appl. 60 (1977), 274–279.CrossRefGoogle Scholar
9Dunford, N. and Schwartz, J. T.. Linear Operators, Part II (Interscience, New York: 1963).Google Scholar
10Faierman, M.. The completeness and expansion theorems associated with the multiparameter eigenvalue problem in ordinary differential equations. j. Differential Equations 5 (1969), 197–213.CrossRefGoogle Scholar
11Källström, A. and Sleeman, B. D.. A left definite multiparameter eigenvalue problem in ordinary differential equations. Proc. Roy. Soc. Edinburgh Sect. A 74 (1976), 145–155.CrossRefGoogle Scholar
12Källström, A. and Sleeman, B. D.. Solvability of a linear operator system. J. Math. Anal. Appl. 55 (1976), 785–793.CrossRefGoogle Scholar
13Källström, A and Sleeman, B. D.. An abstract multiparameter spectral theory. Univ. of Dundee Math. Report 75: 2 (1975).Google Scholar
14Källström, A. and Sleeman, B. D.. Multiparameter spectral theory. Ark. Mat. 15 (1977), 93–99.CrossRefGoogle Scholar
15Pell, A.. Linear equations with two parameters. Trans. Amer. Math. Soc. 23 (1922), 198–211.CrossRefGoogle Scholar
16Sleeman, B. D.. Multiparameter Spectral Theory in Hilbert Space (Bath: Pitman, 1978).CrossRefGoogle Scholar
17Sleeman, B. D.. Multiparameter spectral theory in Hilbert space. J. Math. Anal. Appl. 65 (1978), 511–530.CrossRefGoogle Scholar
18Volkmer, P.. On multiparameter theory. j. Math. Anal. Appl., to appear.Google Scholar