Compact linear operators on functional spaces with two norms (Q1381222)

The original version of this paper (in Ukrainian) was published in Sbirnik Praz' Instituta Matem. Akademii Nauk, Ukrainsk. SSR, No. 9, 104-129 (1947). Some statement from this paper were later rediscovered by P. D. Lax and by J. A. Dieudonne. In Section 1 some general facts of linear operator theory in a space with two norms are established. Important results, as for instance the well-known Hilbert-Schmidt theorem on expansions with respect to eigenfunctions, Mercer's theorem etc. are obtained in this general context. In Section 2, the author develops the theory of symmetrizable operators. Section 3 contains a theorem which can be used to investigate the asymptotics of eigenvalues of a compact operator, with applications to the theory of integral equations with a symmetrizable kernel. Although written more than fifty years ago, the paper has not only the value of an historical document but it remains as modern and fresh as a masterpiece can be. An excellent idea to offer this translation to a large audience.