Wiener - Hopf Operators and Absolutely Continuous Spectra

CONTINUOUS SPECTRA. II BY C. R. PUTNAM Communicated by Maurice Heins, November 1, 1967 1. This paper is a continuation of [4]. It may be recalled that if A is a self-adjoint operator on a Hubert space § with spectral resolution A=zf\dE\, then the set of elements x in § for which ||-Ex#|| is an absolutely continuous function of X is a subspace, &a(A), of § (see, e.g., Halmos [l, p. 104]). The operator A is said to be absolutely continuous if §0(^4)~§. As in [4], both spaces i (0, oo) and i(— oo, oo) will be considered, but the underlying Hilbert space for the integral operators T and A occurring below will be §=Z (0, oo). As in [4], let k(t) on •— 00 < / < 00 satisfy (1) kEL(-oo9 00) HZ, 2 ( -00, 00) and * ( 0 J E ( 0 , and let KÇK) denote the (real-valued) function ƒ 00 k(f)e**dt, 0 0 < \ < 00.