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Eigenvalue asymptotics of perturbed selfadjoint operators. (English) Zbl 1243.47032

The author studies perturbations of a selfadjoint positive operator \(T\) satisfying the \(\alpha\)-non-condenseness condition (\(\alpha >0\)): \[ n(t^{1/\alpha}+0,T)-n((t-1)^{1/\alpha},T)\leq l\quad \text{for some } l\in \mathbb N, \] where \(n(r,T)\) is the number of eigenvalues of \(T\) on \((0,r)\) including their multiplicity.
Conditions on a perturbation \(B\) are found under which \[ | n(r,T)-n(r,T+B)| \leq C[n(r+ar^\gamma ,T)-n(r-ar^\gamma ,T)]+C_1 \] for some positive constants \(C,C_1,a\), and \(\gamma \in [0,1)\).

MSC:

47A55 Perturbation theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
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