  In this paper we provide a quantitative comparison of two obstructions for a
given symmetric operator S with dense domain in Hilbert space ${\cal H}$ to be
selfadjoint. The first one is the pair of deficiency spaces of von Neumann, and
the second one is of more recent vintage: Let P be a projection in ${\cal H}$.
We say that it is smooth relative to S if its range is contained in the domain
of S. We say that smooth projections $\{P_i \}_{i=1}^{\infty}$ diagonalize S if
(a) $(I-P_{i})SP_i=0$ for all i, and (b) $\sup_{i}P_{i}=I$. If such projections
exist, then S has a selfadjoint closure (i.e., $\bar{S}$ has a spectral
resolution), and so our second obstruction to selfadjointness is defined from
smooth projections $P_i$ with $(I-P_i)SP_i \neq 0$. We prove results both in
the case of a single operator S and a system of operators.
