Semi-group Domination and Eigenvalue Estimates

For a class of integral operators it is shown that if the integral kernel of one operator majorates the kernel of the other one then certain forms of eigenvalue estimates are inherited. The operators must be related to a positivity preserving semi-group and, respectively, a positively dominated semi-group. It follows, in particular, that any, suuciently regular, eigenvalue estimate for the Schrr odinger operator is carried over automatically to the magnetic Schrr odinger operator, regardless of the method of obtaining the estimate. 1. There are two natural notions of positivity for operators acting in L 2 (X) where X is a space with a-nite measure. On the one hand, it is the positivity in the operator sense: K is non-negative if (Ku; u) 0 for any u in the domain D(K) of the operator K. Another notion of positivity is related to the lattice structure of the L 2 space: K is positivity preserving (P.P.) if (Ku)(x) 0 almost everywhere (a.e) for any a.e. positive function u 2 D(K). For particular operators, interaction of these notions proves to be a rather eeective tool in the analysis, see, e.g. RSi, v.2,4] and CFKirSi]. Having a positivity preserving operator K, we say that the operator L is dominated by K (L 4 K) if j(Lu)(x)j (Kjuj)(x); u 2 L 2 ; for almost all x (the term 'majorated' is also used in the literature). For integral operators, with kernels K(x; y); L(x; y) domination is equivalent to the inequality jL(x; y)j K(x; y) for almost all (x; y). A natural question is which properties of the operator K are inherited by L. It is obvious that boundedness is inherited. It was shown in P] that compactness is inherited as well. Now, the 'quality' of a compact operator can be described by the rate of decay of its s-numbers (or eigenvalues, if operators are self-adjoint). But the simple two-dimensional example L = 1 0 0 1 , K = 1 1 1 1 , L 4 K shows that while the naturally expected inequality for the largest eigen-values of L and K holds, it may be violated for other eigenvalues. Thus one must