## 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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