On a representation of the kernels of resolvents of Volterra operators and its applications

By using an integral representation for the kernel $M(x,t,\lambda)$ of the operator $(E-\nobreak\lambda^n M)^{-1}M$, where $E$ is the identity operator, and $Mf(x)=\int_0^xM(x,t)f(t)\,dt$, formulas are obtained for transformation operators of the solutions of integro-differential equations which generalize results of Ju. N. Valitskii (RZhMat., 1966, 4Б285); results of L. A. Sahnovich (RZhMat., 1960, 5409) on the linear equivalence of Volterra operators are generalized; and the question of the expansion in eigenfunctions of one-dimensional perturbations of Volterra operators is studied.
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