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- Title
Estimates of the Number of Eigenvalues of Self-Adjoint Operator Functions.
- Authors
Vladimirov, A. A.
- Abstract
We consider an operator function <MATH>F</MATH> defined on the interval <MATH>[\sigma,\tau]\subset\Bbb R</MATH> whose values are semibounded self-adjoint operators in the Hilbert space <MATH>\frak H</MATH>. To the operator function <MATH>F</MATH> we assign quantities <MATH>\Cal N_F</MATH> and <MATH>\nu_F(\lambda)</MATH> that are, respectively, the number of eigenvalues of the operator function <MATH>F</MATH> on the half-interval <MATH>[\sigma,\tau)</MATH> and the number of negative eigenvalues of the operator <MATH>F(\lambda)</MATH> for an arbitrary <MATH>\lambda\in[\sigma,\tau]</MATH>. We present conditions under which the estimate <MATH>\Cal N_F\geq\nu_F(\tau)-\nu_F(\sigma)</MATH> holds. We also establish conditions for the relation <MATH>\Cal N_F=\nu_F(\tau)-\nu_F(\sigma)</MATH> to hold. The results obtained are applied to ordinary differential operator functions on a finite interval.
- Subjects
OPERATOR functions; HILBERT space; BANACH spaces; HYPERSPACE
- Publication
Mathematical Notes, 2003, Vol 74, Issue 5/6, p794
- ISSN
0001-4346
- Publication type
Article
- DOI
10.1023/B:MATN.0000009015.40046.63