We study the adjointness problem for the closed extensions of a general
b-elliptic operator A in x^{-\nu}Diff^m_b(M;E), \nu>0, initially defined as an
unbounded operator A:C_c^\infty(M;E)\subset x^\mu L^2_b(M;E)\to x^\mu
L^2_b(M;E), \mu \in \R. 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.Comment: 40 pages, LaTeX, preliminary versio