Trace formulas for a class of Toeplitz-like operators

LetP _T denote projection onto the space of entire functions of exponential type ≦T which are square summable on the line relative to a measuredΔ and letG denote multiplication by a suitably restricted complex valued function,g. For a reasonably large class of measuresdΔ, which includes Lebesgue measuredγ, it is shown that trace {(P _TGP_T)ⁿ−P_TGⁿP_T} tends boundedly to a limit asT↑∞ and that the limit isindependent of the choice ofdΔ within the permitted class. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special casedΔ=dγ using a different formalism.