Geodesic
On solution of the functional equations of the 1st and 2nd kind whose the transformations with orthogonal coordinats
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 31, Tome 271 (2000), pp. 56-62

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In this paper we introduce the notion of coordinats for a linear continuous transformation of Hilbert space $H$. We develop the complete solvability theory for the functional equations of the 1st and 2nd kind with kernels having orthogonal coordinats and we obtain all such solutions. In particular, this theory is applicable to equations with compact operator. 
@article{ZNSL_2000_271_a3,
     author = {G. Ya. Areshkin},
     title = {On solution of the functional equations of the 1st and 2nd kind whose the transformations with orthogonal coordinats},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {56--62},
     publisher = {mathdoc},
     volume = {271},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_271_a3/}
}
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G. Ya. Areshkin. On solution of the functional equations of the 1st and 2nd kind whose the transformations with orthogonal coordinats. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 31, Tome 271 (2000), pp. 56-62. http://geodesic.mathdoc.fr/item/ZNSL_2000_271_a3/