Symmetric operators with twice continuously differentiable spectral functions

Abstract

In a previous article [2] it was shown that if A is a simple closed symmetric operator with deficiency indices (1, 1) in a Hilbert space H and if A has a selfadjoint extension Ao in H with a discrete spectrum, then every selfadjoint extension and every minimal selfadjoint dilation A+ of A has spectral multiplicity 1. Further, it was possible to make certain statements about the spectrum of A+. In the present article it is shown that if A has a self adjoint extension A 0 with a twice continuously differentiable spectral function, whose derivative is always positive, then there are minimal selfadjoint dilations A+ of A for which the spectral multiplicity is two; indeed, there are minimal selfadjoint dilations A+ which are unitarily equivalent to the multiplication operator in L2(o, co)EDL2(o, oc). As in [2] the proof of these facts depends upon obtaining an expansion theorem which is valid for each of the selfadjoint extensions or dilations A+. This expansion theorem is the analog of the well-known expansion theorems for linear ordinary differential operators; however, it is necessary in deriving it to overcome the fact that, unlike a linear ordinary differential operator, there does not necessarily exist a basis of entire functions for the solutions of the equation Af =Xf. After the expansion theorem is obtained, it is used to define a unitary map of H+ (the space in which A+ acts) onto L2(oc, oc) fL2(o , oc). The same map takes A+ into the multiplication operator in L2(oc, oc) EL2(-oco o). Throughout this article we shall adhere to the notation of [2], except that we shall use the symbol Q(X) in place of Q,(X). We begin with a lemma on the limit of a function analytic in the upper half-plane as one approaches the real axis.