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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
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