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