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Estimates of the number of eigenvalues of self-adjoint operator functions. (English. Russian original) Zbl 1131.47014

Math. Notes 74, No. 6, 794-802 (2003); translation from Mat. Zametki 74, No. 6, 838-847 (2003).
Let \(F\) be an operator function defined on the segment \([\sigma,\tau]\subset{\mathbb{R}}\), with values \(F(\cdot)\) being selfadjoint operators with compact resolvent on a Hilbert space \({\mathcal H}\).
Theorem. If the operators \(F(\lambda)\) are uniformly bounded from below for all \(\lambda\in[\sigma,\tau]\) and the operator function \(\lambda\mapsto(F(\lambda)-iI)^{-1}\) is uniformly continuous, then \[ {\mathcal N}_F\geq\nu_F(\tau)-\nu_F(\sigma), \] where \({\mathcal N}_F\) denotes the number of eigenvalues (counted with their multiplicities) for the operator function \(F\) on \([\sigma,\tau)\), and \(\nu_F(\lambda)\) means the number of negative eigenvalues of the operator \(F(\lambda)\) for an arbitrary \(\lambda\in[\sigma,\tau]\), again counted with multiplicities.
Sufficient conditions in terms of the closures of the quadratic forms \(\langle F(\lambda) \cdot,\cdot\rangle_{{\mathcal H}}\) that provide the equality \({\mathcal N}_F=\nu_F(\tau)-\nu_F(\sigma)\) are also obtained. These results are applied to ordinary differential operators on a finite interval.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A10 Spectrum, resolvent
47E05 General theory of ordinary differential operators
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