The Absolute Continuity of Phase Operators

Abstract

This paper studies the spectral properties of a class of operators known as phase operators which originated in the study of harmonic oscillator phase. Ifantis conjectured that such operators had no point spectrum. It was later shown that certain phase operators were, in fact, absolutely continuous and that all phase operators at least had an absolutely continuous part. The present work completes the discussion by showing that all phase operators are absolutely continuous. Introduction. Let H be an infinite dimensional separable Hilbert space with orthonormal basis {0 }n= 1. For any bounded linear operator A on H, let Sp(A) denote the spectrum of A. If A is selfadjoint with spectral resolution A = fAdEx, denote by Hfa(A) the set of elements x in H for which IIExxII2 is an absolutely continuous function of X. It can be shown [2, p. 104] that tfa(A) is a subspace of H which reduces A. The restriction of A to Ht(a(A) is called the absolutely continuous part of A, and, if Hfla(A) = H, the operator A is said to be absolutely continuous. Let V denote the unilateral shift operator on H, so that VOn = On+ 1. Furthermore, let {an}n'=1 be any of positive real numbers converging monotonically to 1 and satisfying the following chain sequence condition: (1) l/4a2 = (1 -gn.l)gn whereO 0). Define the operator A by AOn = an-l,n with ao = 0. Consider now the following operators on H-: C = (V *A + A V)/2, S = (V *A -A P)/2i. Such operators, called phase operators, have been studied in conjunction with the phase of the harmonic oscillator. Note that C and S are the real and imaginary parts, respectively, of T= V*A and, in particular, are selfadjoint. The conditions on the {a,n}n'=1 guarantee that Sp(C) = Sp(S) = [-1, I]. [For a more detailed discussion of the origin of such operators and their properties see [3], Presented to the Society, January 25, 1975; received by the editors November 15, 1974. AMS (MOS) subject classifications (1970). Primary 47B15, 47B47.