Off-diagonal terms in symmetric operators

In this paper we provide a quantitative comparison of two obstructions for a given symmetric operator S with dense domain in Hilbert space H to be sel f-adjoint. The first one is the pair of deficiency spaces of von Neumann, a nd the second one is of more recent vintage; Let P be a projection in H. We say that it is smooth relative to S if its range is contained in the domai n of S. We say that smooth projections {P-i}(i = 1)(infinity) diagonalize S if (a) (I - P-i)SPi = 0 for all i, and (b) sup(i) P-i = I. If such project ions exist, then S has a self-adjoint closure (i.e., (S) over bar has a spe ctral resolution), and so our second obstruction to self-adjointness is def ined from smooth projections P-i with (I - P-i)SPi not equal 0. We prove re sults both in the case of a single operator S and a system of operators. (C ) 2000 American Institute of Physics. [S0022-2488(00)02604-9].