In this paper we study the resolution problem of an integral equation with operator valued kernel. We prove the equivalence between this equation and certain time varying linear operator system. Sufficient conditions for solving the problem and explicit expressions of the solutions are given.
We analyze some aspects of Mercer's theory when the integral operators act on L²(X,σ), where X is a first countable topological space and σ is a non-degenerate measure. We obtain results akin to the well-known Mercer's theorem and, under a positive definiteness assumption on the generating kernel of the operator, we also deduce series representations for the kernel, traceability of the operator and an integration formula to compute the trace. In this way, we upgrade considerably similar results...
In this paper, we prove that every unbounded linear operator satisfying the Korotkov-Weidmann characterization is unitarily equivalent to an integral operator in L 2(R), with a bounded and infinitely smooth Carleman kernel. The established unitary equivalence is implemented by explicitly definable unitary operators.