Trace formulas for a class of Toeplitz-like operators II

Abstract

Let P T denote the orthogonal projection of L 2( R 1, dΔ) onto the space of entire functions of exponential type ⩽ T which are square summable on the line with respect to the measure dΔ(γ) = ¦ h(γ)¦ 2 dγ , and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if (γ 2 + 1) −1 log ¦ h(γ)¦ is summable, if ¦ h ¦ −2 is locally summable, and if h h # belongs to the span in L ∞ of e − iyT H ∞: T ⩾ 0, in which h is chosen to be an outer function and h #( γ) agrees with the complex conjugate of h( γ) on the line, then lim trace T↑∞ {(P TGP T) n − P TG nP T} exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections.