American Journal of Mathematics

We study the adjointness problem for the closed extensions of a general b-elliptic operator A ∈ x^-ν Diff^m_b (M; E), ν > 0, initially defined as an unbounded operator A: C^∞_c (M; E) ∈ x^μL²_b(M; E) → x^μL²_b(M; E), μ ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. The case where A is a symmetric semibounded operator is of particular interest, and we give a complete description of the domain of the Friedrichs extension of such an operator.