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CLR-estimate for the generators of positivity preserving and positively dominated semi-groups. (English. Russian original) Zbl 0911.47025
St. Petersbg. Math. J. 9, No. 6, 1195-1211 (1998); translation from Algebra Anal. 9, No. 6, 214-236 (1997).
Let $$\Omega$$ be a space with $$\sigma$$-finite measure $$\mu$$. Given a selfadjoint nonnegatively definite operator $$B$$ in $$L_2(\Omega,\mu)$$ such that the semigroup $$Q(t)= e^{-tB}$$ $$(0\leq t<\infty)$$ is a positivity preserving in $$L_2(\Omega,\mu)$$ contractive semigroup and given a measurable function $$V\geq 0$$, the authors give some upper bounds for the number of negative eigenvalues of the operator $$B-V$$, under only one condition that $$Q(t)$$ for $$t>0$$ is a bounded operator from $$L_2(\Omega,\mu)$$ into $$L_\infty(\Omega, \mu)$$. The results are given in term of $M_B(t):= \| e^{-{t\over 2}B}\|^2_{L_2\to L_\infty}= \underset {x}{\text{ess sup }} \int_\Omega | Q_B(t/2; x,y)|^2d\mu(y),$ where $$Q_B(t; x,y)$$ is the kernel of $$e^{-tB}$$. The main tool for the proofs is Trotter’s well-known formula. The proofs are not “probabilistic”, and so the authors need not assume as in various previous papers that $$e^{-tB}$$ is Markovian (i.e. has the above properties in all $$L_p(\Omega,\mu)$$, $$1\leq p\leq\infty$$).
From their abstract theorems, they deduce all previous results of this type in 1972-1997 and also new ones for the relativistic magnetic Schrödinger operator and for the sub-Laplacian on a nilpotent Lie group.

##### MSC:
 47B38 Linear operators on function spaces (general) 47D06 One-parameter semigroups and linear evolution equations 47D07 Markov semigroups and applications to diffusion processes 47B65 Positive linear operators and order-bounded operators