Normal Extensions of Linear Operators

Abstract

Let \( L_0 \) be a densely defined minimal linear operator in a Hilbert space \( H \). We prove that if there exists at least one correct extension \( L_S \) of \( L_0 \) with the property \( D(L_S) = D(L_S^*) \), then we can describe all correct extensions \( L \) with the property \( D(L) = D(L^*) \). We also prove that if \( L_0 \) is formally normal and there exists at least one correct normal extension \( L_N \), then we can describe all correct normal extensions \( L \) of \( L_0 \). As an example, the Cauchy-Riemann operator is considered.