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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.

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up vote 3 down vote accepted

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"

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Great answer, thanks! Since the answer is Yes, is there any chance that the spectra and spectral projections of comparable self-adjoint extensions obey a simple relationship as well? – Igor Khavkine Oct 29 '12 at 0:53
Could you give a page reference for the quotient space $D_\max/D_\min$? I'm having trouble parsing your definition of it, or finding it in the references provided. – Francois Ziegler Oct 29 '12 at 1:27
Look in the two references I cited for more about $D_{\max}/D_{\min}$. Also look in the old book by Matthias Lesch (on the arXiv in about '94) to see how this theory works out in the setting of Dirac operators on spaces with isolated conic singularities, where the quotient of maximal by minimal domains is finite dimensional. There is a lot of literature also about the ODE case (references in Grubb and Lesch I believe) which is also very explicit. There is very little relationship between the spectra for different self-adjoint extensions - again examples in the ODE case show this. – Rafe Mazzeo Oct 29 '12 at 3:31
@Rafe, I initially had some trouble (like Francois) parsing your statement about characterizing closed extensions via $D_{\max}/D_{\min}$. But then I realized... Duh! The partial order on extensions is just the inclusion $\subseteq$ of the graphs in $H\times H$. The closed extensions are then in correspondence with the usual lattice of closed subspaces of $D_{\max}/D_{\min}$. BTW, I had trouble finding a discussion of this point in Folland. However, Grubb does discuss extensions at length in his Chapter 13, with helpful references. – Igor Khavkine Oct 29 '12 at 14:08
Sorry that my initial answer was sloppily written and I didn't double-check the references to see that they actually said what I hoped they would ;) For the issue of how the spectrum depends on boundary conditions, just look at $d^2/dx^2$ on $L^2([0,1])$ with mixed boundary conditions $a_0 u(0) + b_0 u_x(0) = 0$ and $a_1 u(1) + b_1 u_x(1) = 0$. You can track how the eigenvalues vary as $a/b$ varies ``in the circle''; there are interesting monotonicity properties. – Rafe Mazzeo Oct 29 '12 at 23:09

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