Symmetric operators with singular spectral functions

In a previous article [3] it was shown that if A is a closed symmetric operator with deficiency indices (1, 1) in the Hilbert space H and if ,Ao is a point of regular type for A, then there is a neighborhood of /Ao in which every minimal selfadjoint dilation A+ of A has spectral multiplicity not more than 1. This shows that in Theorem 14 of Coddington [2] and Theorem 5.2 of McKelvey [6] it is the hypothesis that every point is of regular type which determines the spectral multiplicity of the dilation; the hypothesis that the contraction F(X) is continuous down to the real axis and has norm less than 1 there serves to make the spectral function absolutely continuous. (See Remark 2.) The procedure used in [31 depended on the fact that if .o is a point of regular type for A, then there exists a selfadjoint extension A o of A for which ,Ao is in the resolvent set. This means that for different points IO one might have to choose different operators Ao. In the present note this necessity is eliminated. It is shown that if A has a selfadjoint extension Ao with a pure point spectrum with no finite limit points, then for each minimal selfadjoint extension or dilation A+ of A there is defined on the real axis a nondecreasing function p(a) (which depends solely on A+ and on the single selfadjoint extension Ao) such that A+ is unitarily equivalent to the multiplication operator in L,. Kre1n [5] has shown that if A is simple, then there exists a selfadjoint extension A0 of the type described above if and only if every point is of regular type for A. (See Remark 1 also.) In the case that A is a singular Sturm-Liouville operator, the results of the paper are extended to the case that there exists a selfadjoint extension Ao with a singular spectral function. For terminology used we refer the reader to [3] and to Achieser and Glasmann [1]. We note first of all the following lemma due to Straus [8].