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.



Partial order on self-adjoint extensions?
Yes. First define $D\_{\max} = \{u \in H: Au \in H\}$ and $D\_{\min}$ as the graph closure
in $H \times H$ of the graph of $A$ over some "core domain". In the usual PDE examples, one should think of $D\_{\max}$ as consisting of all $L^2$ functions such that $Au$, defined as a distribution, happens to lie in $L^2$, and the core domain in this case is something like $\mathcal C^\infty\_0$. The basic theorem states that the closed extensions of $A$ (relative to the given core domain, or equivalently, $D\_{\min}$, are in one-to-one correspondence with $D\_{\max}/D\_{\min}$. The self-adjoint extensions are in one-to-one correspondence with the Lagrangian subspaces of this quotient (where, in the PDE setting, where say $A = \Delta$), the symplectic form is the natural one induced on the boundary value and normal derivative of a function by Green's formula. Thus there is a lattice structure on the space of all closed extensions. A good basic introduction to this point of view for elliptic boundary problems is in Gerry Folland's book `Partial Differential Equations'. Another very good source is the recent Springer book by Gerd Grubb "Distributions and Operators"


