Szegö limit theorems for the harmonic oscillator

Let (FORMULA PRESENTED) be the harmonic oscillator Hamiltonian on L²(R), and let A be a selfadjoint DO of order O in the Beals-Fefferman class with weights (FORMULA PRESENTED). Form the measure (FORMULA PRESENTED) where, (FORMULA PRESENTED) is the compression of A onto the span of the Hermite functions with eigenvalue less than or equal to ⋌. Then one has the following Szego limit theorem: (FORMULA PRESENTED) For the special case where f(x) = x, this will be proved for a considerably wider class of operators by employing the Weyl correspondence. Moreover, by using estimates on Wigner functions of Hermite functions we are able to prove the full Szego theorem for a fairly general class of multiplication operators.