On the inverse of integral operators with Kernel operators

We study the boundedness of integral operators whose kernels are functions of operators V f(x):= f(x) + ∫ k(x, t, L)f(t) dµ(t), where k(x, t, λ) is an entire function of λ and L is an unbounded self-adjoint operator in L²_dμ(t). By using Korotkov’s theorem we derive a simple necessary condition for V to be a Carleman type operator. We are particularly interested in the cases when the inverse operator exists and has the same form as V. This study provides a new method for the inversion of integral equation of Carleman type.