Point-spectrum of semibounded operator extensions

Abstract:

Let $\tilde H$ denote the Friedrichs extension of a given semibounded operator $H$ in a Hilbert space. Assume $\lambda I \leqslant H$, and $\lambda \in \sigma (\tilde H)$. If for a finite-dimensional projection $P$ in the Hubert space we have $I - P \leqslant$ Const. $(H - \lambda I)$, then it follows that $\lambda$ is an eigenvalue of $\tilde H$, and the corresponding eigenspace is contained in the range of $P$. Using this, together with the known order structure on the family of selfadjoint extensions, with given lower bound 0, of minus the Laplace-Beltrami operator, we establish the identity ${U_g}(1) = 1$ for all $g \in G$ for the following problem. $U$ is a unitary representation of a Lie group $G$, and acts on the Hilbert space ${L^2}(\Omega )$ for some Nikodym-domain $\Omega \subset G$. Moreover $U$ is obtained as a certain normalized integral for the left-$G$-in variant vector fields on $\Omega$, that is, for each such vector field $X$, the skew-adjoint operator $dU(X)$ is an extension of $X$ when regarded as a skew-symmetric operator in ${L^2}(\Omega )$ with domain $C_0^\infty (\Omega )$.