Is there a natural partial order and/or lattice structure on the set of closed symmetric or self-adjoint extensions of a densely defined, unbounded, symmetric operator on a Hilbert space? Any reference where such order structures are discussed?
One sometimes encounters references to the "minimal" or "maximal" self-adjoint extension. It'd be nice to see this terminology fit in a more general order structure context.
Update: Since the answer appears to be Yes (cf. Rafe Mazzeo's answer below), an immediate extension to the original question comes to mind. Is there a simple relation between the spectra and spectral projections of comparable self-adjoint extensions?
Update: Just a clarifying remark, since part of the relevant information is hidden in the comments to Rafe Mazzeo's answer below. Given an operator $A$ on $H$, the partial order on the closed extensions is the same as the inclusion relation $\subseteq$ on closed subspaces of $H\times H$, applied to the graphs of the closed extensions of $A$. The lattice operations are the usual lattice operations on closed linear subspaces of $H\times H$. The relation to boundary conditions of elliptic PDEs are discussed at length in the book "Distributions and Operators" by Gerd Grubb and the references therein.