Non-negative perturbations of non-negative self-adjoint operators

Abstract

Let $A$ be a non-negative self-adjoint operator in a Hilbert space $\mathcal{H}$ and $A_0$ be some densely defined closed restriction of $A_0$, $A_0\subseteq A\ne A_0$. It is of interest to know whether $A$ is the unique non-negative self-adjoint extensions of $A_0$ in $\mathcal{H}$. We give a natural criterion that this is the case and if it fails, we describe all non-negative extensions of $A_0$. The obtained results are applied to investigation of non-negative singular point perturbations of the Laplace and poly-harmonic operators in ${\mathbb{L}}_2({\mathbf{R}}_n)$.

08 pages