Integral representations of unbounded operators by arbitrarily smooth   Carleman kernels
  In this paper, we give a characterization of all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral operator in $L_2(R)$ with bounded and arbitrarily smooth Carleman kernel on $R^2$. In addition, we give an explicit construction of corresponding unitary operators. 
Igor M. Novitskii