Schatten classes and nuclearity of boundary value problems

Julio Delgado, Michael Ruzhansky, Niyaz Tokmagambetov

Published 2015-05-09Version 1

Given a bounded open set $\Omega$, in this paper we analyse Schatten classes and nuclearity of operators in $\Omega$ satisfying some boundary conditions on the boundary of $\Omega$. Our analysis relies on the global symbolic calculus in terms of the biorthogonal expansions in eigenfunctions of a fixed differential operator with the same boundary conditions. Several criteria for the membership in Schatten classes on $L^2(\Omega)$ and r-nuclearity on $L^p(\Omega)$ are obtained, with applications (and a new addition) to the Grothendieck-Lidskii formula and asymptotic behaviour of eigenvalues. Examples and applications are given to operators on $\Omega=(0,1)^n$ with non-periodic boundary conditions, and of operators with non-local boundary conditions.