On generating functions of Volterra integral operators (Q1081792)

We consider the Volterra operator \[ Mf=\int^{x}_{0}M(x,t)f(t)dt\quad (0\leq x\leq 1), \] where \(M(x,t)=(x-t)^{n-1}/(n-1)!+o((x-t)^ n)\). We indicate sufficient conditions on the smoothness of the kernel M(x,t) and the function g(x), under which g(x) is the generating function of M in \(L^ 2[0,1]\). We give an example showing that if the smoothness conditions for the kernel M(x,t) are not satisfied then there exists a noncyclical Volterra operator M. We establish the completeness of the systems of eigen- and associated functions of a finite-dimensional perturbation of the operator M and a convolution operator.