Lp Spectral Independence of Elliptic Operators via Commutator Estimates

Let fT p : q 1 p q 2 g be a family of consistent C 0 semigroups on L p ((), with q 1 ; q 2 2 1; 1) and R n open. We show that certain commutator conditions on T p and on the resolvent of its generator A p ensure the p independence of the spectrum of A p for p 2 q 1 ; q 2 ]. Applications include the case of Petrovskij correct systems with HH older continuous coeecients, Schrr odinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coeecients. Under what circumstances is the L p-spectrum of an elliptic operator acting in L p ((), where R n is an open set, independent of p 2 1; 1)? This question attracted new attention when B. Simon conjectured the p-independence of the spectrum of Schrr odinger operators with Kato class potentials 25] in L p (R n). An aarmative answer was given by Hempel and Voigt in 1986 18]. On the other hand, it is well-known that, in general, the L p-spectra of elliptic operators depend on p. For various aspects in this direction see e. Spectral properties of elliptic operators A are intimately related to questions about the proper deenition of the domain of A in L p ((). Here we adopt the view that A should be the generator a strongly continuous semigroup T on L 2 (() which can be extended consistently to L p ((). More precisely, we assume that there exists a C 0-semigroup T p on L p ((), 1 p < 1, such that T p (t)f = T(t)f; t 0; f 2 L 2 \ L p : Denote by A p the generator of T p. In particular, A p always is a closed operator. The case which is best understood is when A is a second order selfadjoint diierential operator in L 2 ((), R n open, with a heat-kernel bound for T(t), i.e. T(t) may be represented by an integral kernel k(t; ;) satisfying jk(t; x; y)j c t ?n=2 e !t e ?a jx?yj 2 t ; t > 0; x; y 2 ; (0.1) for suitable a; c > 0, ! 0. It was shown independently by Davies 12] and Arendt 3] that then the spectrum of A p is independent of p …