Correct selfadjoint and positive extensions of nondensely defined minimal symmetric operators

Abstract

Let A0 be a closed, minimal symmetric operator from a Hilbert space ℍ into ℍ with domain not dense in ℍ. Let A^ also be a correct selfadjoint extension of A0. The purpose of this paper is (1) to characterize, with the help of A^, all the correct selfadjoint extensions B of A0 with domain equal to D(A^), (2) to give the solution of their corresponding problems, (3) to find sufficient conditions for B to be positive (definite) when A^ is positive (definite).