On \(\ell_ p\)-summability of the characteristic values of integral operators on \(L_ 2({\mathbb{R}}^ N)\).
Let \(B_ p(L_ 2)\) \((0<p<\infty)\) denote the space of all compact linear operators K on \(L_ 2({\mathbb{R}}^ N\)) with the following property: the sum of p-th powers of all positive eigenvalues (repeated accordng to their multiplicity) of the nonnegative compact operator (K \(*K)^{1/2}\) converges. We prove a \(B_ p\)-criterion for an integral operator K on \(L_ 2({\mathbb{R}}^ N\)), for \(0<p\leq 2\). We require that its kernel have sufficient fractional smoothness and decay at infinity. Our proof is based on a suitable factorization of the operator K which involves fractional powers \((H+\lambda I)^{-\omega}\), \(\lambda\geq 0\), \(0<\omega <1\), of the resolvent of the selfadjoint elliptic operator \(H=(- \Delta)^ m+M_ q^{2m}\) on \(L_ 2({\mathbb{R}}^ N\)).
\(\ell _ p\)-summability of characteristic values
\(B_ p\)-criterion for an integral operator
fractional smoothness and decay at infinity
resolvent of the selfadjoint elliptic operator
scientific article; zbMATH DE number 4038570
scientific article