On the Schatten-von Neumann properties of some pseudo-differential operators

We obtain a number of explicit estimates for quasi-norms of pseudo-differential operators in the Schatten-von Neumann classes $S_q$ with $0<q\le 1$. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.Comment: 22 page

Decompositions of Gelfand-Shilov kernels into kernels of similar class

We prove that any linear operator with kernel in a Gelfand-Shilov space is a composition of two operators with kernels in the same Gelfand-Shilov space. We also give links on numerical approximations for such compositions. We apply these composition rules to establish Schatten-von Neumann properties for such operators.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1102.033

A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behaviour of the $s$-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain $\Omega$ with smooth boundary $\partial\Omega$. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on $\partial\Omega$. It will be shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order $p$ for which $p>(dim\Omega-1)/3$. Moreover, we also give a simple sufficient condition for the resolvent difference of two generalized Robin Laplacians to belong to a Schatten-von Neumann class of arbitrary small order. Our results extend and complement classical theorems due to M.Sh.Birman on Schatten-von Neumann properties of the resolvent differences of Dirichlet, Neumann and self-adjoint Robin Laplacians