Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T10:21:46.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiparameter definiteness conditions

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
Get access Rights & Permissions [Opens in a new window]

Synopsis

Several definiteness conditions in the multiparameter spectral literature are discussed. It is shown that some of these conditions permit simplifying transformations of the eigenvalues, leading to further definiteness properties. Geometrical equivalents for the algebraic conditions are established in terms of separation of convex cones. As a result, the relationship between the standard left and right definiteness conditions is clarified.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 89 , Issue 3-4 , 1981 , pp. 319 - 332
Copyright
Copyright © Royal Society of Edinburgh 1981

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
3Binding, P. A.. On generalised and quadratic eigenvalue problems. Applicable Anal., to appear.Google Scholar
4Binding, P.. Left definite multiparameter eigenvalue problems, to appear.Google Scholar
5Binding, P. and Browne, P.. A variational approach to multiparameter eigenvalue problems in Hilbert space. SIAM J. Math. Anal. 9 (1978), 1054–1067.CrossRefGoogle Scholar
6Binding, P. and Browne, P.. Positivity results for determinantal operators. Proc. Roy. Soc. Edinburgh Sect. A 81 (1978), 267–271.CrossRefGoogle Scholar
7Binding, P. and Browne, P.. Comparison cones for multiparameter eigenvalue problems. J. Math. Anal. Appl. 77 (1980), 132–149.CrossRefGoogle Scholar
8Binding, P. and Browne, P.. Spectral properties of two-parameter eigenvalue problems. Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 157–173.CrossRefGoogle Scholar
9Browne, P.. A singular multiparameter eigenvalue problem in second order ordinary differential equations. J. Differential Equations 12 (1972), 81–94.CrossRefGoogle Scholar
10Browne, P. J.. Abstract multiparameter theory I. J. Math. Anal. Appl. 60 (1977), 259–273.CrossRefGoogle Scholar
11Ince, E.. Ordinary Differential Equations, 1926 (Dover reprint, 1956).Google Scholar
12Källström, 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
13Källström, A. and Sleeman, B. D.. Multiparameter spectral theory. Ark. Mat. 15 (1977), 93–99.CrossRefGoogle Scholar
14Meixner, J. and Schäfke, F.. Mathieusche Funktionen und Sphäroidfunktion (Berlin: Springer, 1954).CrossRefGoogle Scholar
15Pell, A.. Linear equations with two parameters. Trans. Amer. Math. Soc. 23 (1922), 198–211.CrossRefGoogle Scholar
16Richardson, R.. Theorems of oscillation for two linear differential equations of the second order with two parameters. Trans. Amer. Math. Soc. 13 (1912), 22–34.CrossRefGoogle Scholar
17Sleeman, B. D.. Multiparameter Spectral Theory in Hilbert Space (Bath: Pitman, 1978).CrossRefGoogle Scholar
18Sleeman, B. D.. Multiparameter spectral theory in Hilbert space. J. Math. Anal. Appl. 65 (1978), 511–530.CrossRefGoogle Scholar
19Sleeman, B. D.. Klein oscillation theorems for multiparameter eigenvalue problems in ordinary differential equations. Nieuw Arch. Wisk. 27 (1979), 341–362.Google Scholar
20Turner, R.. Some variational principles for a nonlinear eigenvalue problem. J. Math. Anal. Appl. 17 (1967), 151–160.CrossRefGoogle Scholar
21Turyn, L.. Strum–Liouville problems with several parameters. J. Differential Equations 38 (1980), 239–259.CrossRefGoogle Scholar