A class of formally normal operators

A densely closed operator N given in Hilbert space is called formally normal if D(N)D(N*)and Nf = N*f for allf D(N). In the present work the necessary and sufficient conditions for a formally normal operator, possessing a bounded inverse, to have a normal extension in the original Hilbert space are given. The result obtained is analogous to a result of M. I. Vishik [1], relating to the case of a symmetric operator [7 References].Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 605–614, December, 1967.