Spectral multiplicity of selfadjoint dilations

Abstract

Let A be a closed symmetric operator in a Hilbert space H. If A+ is a selfadjoint operator in a Hilbert space H+ such that HCH+ and A CA+, then we shall call A+ a dilation of A. In the particular case that H= H+, A+ is also called an extension of A. A dilation A+ is called minimal if A+ is not reduced by H+eH nor by any of its nontrivial subspaces. For further information on selfadjoint extensions and dilations of symmetric operators, we refer the reader to Achieser and Glasmann [1]. Suppose that A has deficiency indices (1, 1). It is the purpose of this article to show that if pto is a point of regular type for A, then there is a neighborhood of /Lo in which every minimal selfadjoint dilation of A has spectral multiplicity not exceeding 1. More precisely, there exists an interval ('y, 5) containing /uo such that if A+ is a minimal selfadjoint dilation of A with spectral function E+(X), then the operator A+ [E+(5) -E+('y)] is unitarily equivalent to the inultiplication operator in a space L2(p) for some nondecreasing function p defined on the real line. This constitutes a generalization of Theorem 5.2 of McKelvey [4], which is in turn a generalization of Theorem 14 of Coddington [2 1. As a preliminary we prove two lemmas.