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Multiparameter definiteness conditions II

Published online by Cambridge University Press:  14 November 2011

Paul Binding
Paul Binding
Affiliation:
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada T2N 1N4
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Synopsis

A previous attempt to systematize various conditions from multiparameter spectral theory is extended to weaker forms of definiteness. The latter not only occur in the established literature but are also under active investigation at present. Several algebraic and geometrical formulations exist, and questions concerning their equivalence are approached in a unified fashion where possible.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 93 , Issue 1-2 , 1982 , pp. 47 - 61
Copyright
Copyright © Royal Society of Edinburgh 1982

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References

1Arscott, F. M.. Periodic Differential Equations: an Introduction to Mathieu, Lamé and Allied Functions (Macmillan Press, 1964).Google Scholar
2Atkinson, F. V.. Multiparameter Eigenvalue Problems, Vol. 1 (New York: Academic Press, 1972).Google Scholar
3Binding, P. A.. Another positivity result for determinantal operators. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 333–337.CrossRefGoogle Scholar
4Binding, P. A.. On generalised and quadratic eigenvalue problems. Applicable Anal. 12 (1981), 27–45.Google Scholar
5Binding, P. A.. Multiparameter definiteness conditions. Proc. Roy. Soc. Edinburgh 89 (1981), 319–332.Google Scholar
6Binding, P. A.. On a problem of B. D. Sleeman. J. Math. Anal. Appl. 85 (1982), 291–307.Google Scholar
7Binding, P. A.. Multiparameter variational principles. SLAM J. Math. Anal, (to appear).Google Scholar
8Binding, P. A. and Browne, P. J.. Positivity results for determinantal operators. Proc. Roy. Soc. Edinburgh Sect. A 81 (1978), 267–271.Google Scholar
9Binding, P. A. and Browne, P. J.. Comparison cones for multiparameter eigenvalue problems. J. Math. Anal. Appl. 77 (1980), 132–149.Google Scholar
10Binding, P. A. and Browne, P. J.. Spectral properties of two-parameter eigenvalue problems. Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 157–173.Google Scholar
11Binding, P. A., Browne, P. J. and Turyn, L.. Existence conditions for two-parameter eigenvalue problems. Proc. Roy. Soc. Edinburgh Sect A 91 (1982), 15–30.CrossRefGoogle Scholar
12Dunford, N. and Schwartz, J. T.. Linear Operators Part II (New York: Interscience, 1963).Google Scholar
13Greguš, M., Neuman, F. and Arscott, F.. Three-point boundary value problems in differential equations. J. London Math. Soc. 3 (1971), 429–436.CrossRefGoogle Scholar
14Hammer, P. C.. Maximal convex sets. Duke Math. J. 22 (1955), 103–106.Google Scholar
15Hilbert, D.. Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen (1912, reprinted Chelsea, 1953).Google Scholar
16Källström, 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
17Klee, V. L.. The structure of semispaces. Math. Scand. 4 (1956), 54–64.CrossRefGoogle Scholar
18Pell, A.. Linear equations with two parameters. Trans. Amer. Math. Soc. 23 (1922), 198–211.CrossRefGoogle Scholar
19Rockafellar, R. T.. Convex Analysis (Princeton University Press, 1970).CrossRefGoogle Scholar
20Sleeman, B. D.. Multiparameter Spectral Theory in Hilbert Space (Bath; Pitman, 1978).CrossRefGoogle Scholar
21Sleeman, B. D.. Multiparameter spectral theory in Hilbert space. J. Math. Anal. Appl. 65 (1978), 511–530.Google Scholar
22Sleeman, B. D.. Klein oscillation theorems for multiparameter eigenvalue problems in ordinary differential equations. Nieuw Arch. Wisk. 27 (1979), 341–362.Google Scholar
23Uhlig, F.. A recurring theorem about pairs of quadratic forms: a survey. Linear Algebra Appl. 25 (1979), 219–237.CrossRefGoogle Scholar