Simultaneous unitary equivalence to Carleman operators with arbitrarily   smooth kernels
  In this paper, we describe families of those bounded linear operators on a separable Hilbert space that are simultaneously unitarily equivalent to integral operators on $L_2(R)$ with bounded and arbitrarily smooth Carleman kernels. The main result is a qualitative sharpening of an earlier result of [7]. 
Igor M. Novitskii