Correct selfadjoint and positive extensions of nondensely defined minimal symmetric operators

Abstract

Let $A_0$ be a closed, minimal symmetric operator from a Hilbert space $ℍ$ into $ℍ$ with domain not dense in $ℍ$. Let $\widehat{A}$ also be a correct selfadjoint extension of $A_0$. The purpose of this paper is (1) to characterize, with the help of $\widehat{A}$, all the correct selfadjoint extensions $B$ of $A_0$ with domain equal to $D\left(\widehat{A}\right)$, (2) to give the solution of their corresponding problems, (3) to find sufficient conditions for $B$ to be positive (definite) when $\widehat{A}$ is positive (definite).