A formula for the resolvent of $(-\Delta )^ m+M^ {2m}_ q$ with applications to trace class

Abstract:

We derive a formula for the resolvent of the elliptic operator $H = {( - \Delta )^m} + M_q^{2m}$ on ${L_2}({\mathbb {R}^N})$ in terms of bounded integral operators ${S_\lambda }$ and ${T_\lambda }$ whose kernels we know explicitly. We use this formula to specify the domain of the operator ${A_\lambda } = (H + \lambda I){M_p}$ on ${L_2}({\mathbb {R}^N})$, and to estimate the Hilbert-Schmidt norm of its inverse $A_\lambda ^{ - 1}$, for $\lambda \geqslant 0$. Finally we exploit the last two results to prove a trace class criterion for an integral operator $K$ on ${L_2}({\mathbb {R}^N})$.