Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

Abstract

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $Ω\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in $L^2(Ω)$ is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space $L^2(\partialΩ)$.