J. Behrndt, M. Langer and V. Lotoreichik,
Trace formulae and singular values of resolvent power differences of
self-adjoint elliptic operators,
J. London Math. Soc. (2) 88 (2013), 319–337
Abstract:
In this note, self-adjoint realizations of second-order elliptic differential expressions
with non-local Robin boundary conditions on a domain
Ω ⊂ ℝn
with smooth compact boundary are studied. A Schatten–von Neumann-type
estimate for the singular values of the difference of the mth powers of the
resolvents of two Robin realizations is obtained, and,
for m > 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
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 L2(Ω) is written
in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps
on the boundary space L2(∂Ω).