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In functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for formal expressions of observables in quantum mechanics. Other applications of solutions to this problem can be seen in various moment problems.

This article discusses a few related problems of this type. The unifying theme is that each problem has an operator-theoretic characterization which gives a corresponding parametrization of solutions.

Symmetric operators
Let H be a Hilbert space. A linear opeartor A acting on H with dense domain Dom(A) is symmetric if


 *  = , for all x, y in Dom(A).

If Dom(A) = H, the Hellinger-Toeplitz theorem says that A is a bounded operator, in which case A is self-adjoint and the extension problem is trivial. In general, a symmetric operator is self-adjoint if the domain of its adjoint, Dom(A*), lies in Dom(A).

When dealing with unbounded operators, it is often desirable to be able to assume that the operator in question is closed. In the present context, it is a convenient fact that every symmetric operator A is closable. That is, A has a smallest closed extension, called the closure of A. This can be shown by invoking the symmetric assumption and Riesz representation theorem. Since A and its closure have the same closed extensions, it can always be assumed that the symmetric operator of interest is closed.

In the sequel, a symmetric operator will be assumed to be a densely defined and closed.

Extensions of symmetric operators
Problem Given a densely defined closed symmetric operator A, does it always have extension?

This question can be translated to an operator-theoretic one. As a heuristic motivation, notice that the Cayley transform on the complex plane, defined by


 * $$z \mapsto \frac{z-i}{z+i}$$

maps the real line to the unit circle. This suggests one define, for a symmetric operator A,


 * $$U_A = (A - i)(A + i)^{-1}\,$$

on Ran(A + i), the range of A + i. The operator UA is in fact an isometry between closed subspaces that takes (A + i)x to (A - i)x for x in Dom(A).

The map


 * $$A \mapsto U_A$$

is also called the Cayley transform of the symmetric operator A. Given UA, A can be recovered by


 * $$A = - i(U + 1)(U - 1)^{-1} ,\,$$

defined on Dom(A) = Ran(U - 1).

Now if


 * $$ \tilde{U} $$

is an isometric extension of UA, the operator


 * $$\tilde{A} = - i( \tilde{U} + 1)( \tilde{U} - 1 )^{-1} $$

acting on


 * $$ Ran (- \frac{i}{2} ( \tilde{U} - 1)) = Ran ( \tilde{U} - 1) $$

is a symmetric extension of A.

Theorem The symmetric extensions of a closed symmetric operator A is in one-to-one correspondence with the isometric extensions of its Cayley transform UA.

Of more interest is the existence of self-adjoint extensions. The following is true.

Theorem A closed symmetric operator A is self-adjoint if and only if Ran(A &plusmn; i) = H, i.e. when its Cayley transform UA is an unitary operator on H.

Corollary The self-adjoint extensions of a closed symmetric operator A is in one-to-one correspondence with the unitary extensions of its Cayley transform UA.

Define the deficiency subspaces of A by


 * $$K_+ = Ran(A+i)^{\perp}$$

and


 * $$K_- = Ran(A-i)^{\perp}$$

In this language, The description of the self-adjoint extension problem given by the corollary can be restated as follows: a symmetric operator A has self-adjoint extensions if and only if its Cayley transform UA has unitary extensions to H, i.e. the deficiency subspaces K+ and K- have the same dimension.

An example
Consider the Hilbert space L2[0,1]. On the subspace of absolutely continuous function that vanish on the boundary, define the operator A by


 * $$A f = i \frac{d}{dx} f.$$

Integration by parts shows A is symmetric. Its adjoint A* is the same operator with Dom(A*) being the absolutely continuous functions with no boundary condition. We will see that extending A amounts to modifying the boundary conditions, thereby enlarging Dom(A) and reducing Dom(A*) until the two coincide.

Direct calculation shows that K+ and K- are one dimensional subspaces given by


 * $$K_+ = span \{\phi_+ = a \cdot e^x \}$$

and


 * $$K_- = span\{ \phi_- = a \cdot e^{-x} \}$$

where a is a normalizing constant. So the self-adjoint extensions of A are parametrized by the unit circle in the complex plane, {|&alpha;| = 1}. For each unitary U&alpha; : K- &rarr; K+, defined by U&alpha;(&phi;-) = &alpha;&phi;+), there corresponds an extension A&alpha; with domain


 * $$Dom(A_{\alpha}) = \{ f + \beta (\alpha \phi_{-} - \phi_+) | f \in Dom(A) \}.$$

If f &isin; Dom(A&alpha;), then f is absolutely continuous and


 * $$|\frac{f(0)}{f(1)}| = |\frac{e\alpha -1}{\alpha - e^2}| = 1.$$

Conversely, if f is absolutely continuous and f(0) = &gamma;f(1) for some complex &gamma; with |&gamma;| = 1, then f lies in the above domain.

The self-adjoint operators { A&alpha; } are instances of the momentum operator in quantum mechanics.

Self adjoint extension on a larger space
Every partial isometry can be extended, on a possibly larger space, to an unitary operator. Consequently, every symmetric operator has a self-adjoint extension, on a possibly larger space.

Positive symmetric operators
A symmetric operator A is called positive if  &ge; 0 for all x in Dom(A). It is known that for every such A, one has dim(K+) = dim(K-). Therefore every positive symmetric operator has self-adjoint extensions. The more interesting question in this direction is whether A has positive self-adjoint extensions.

Structure of 2 &times; 2 matrix contractions
While the extension problem for general symmetric operators is essentially that of extending partial isometries to unitaries, for positive symmetric operators the question becomes one of extending contractions: by "filling out" certain unknown entries of a 2 &times; 2 self-adjoint contraction, we obtain the positive self-adjoint extensions of a positive symmetric operator.

Before stating the relevant result, we first fix some terminology. For a contraction &Gamma;, acting on H, we define its defect operators by


 * $$ D_{\Gamma} = (1 - D^*D)^{\frac{1}{2}} \quad \mbox{and} \quad D_{\Gamma^*} = (1 - DD^*)^{\frac{1}{2}}.$$

The defect spaces of &Gamma; are


 * $$\mathcal{D}_{\Gamma} = Ran( D_{\Gamma} ) \quad \mbox{and} \quad \mathcal{D}_{\Gamma^*} = Ran(  D_{\Gamma^*} ).$$

The defect operators indicate the non-unitarity of &Gamma;, while the defect spaces ensure uniqueness in some parametrizations. Using this machinery, one can explicitly describe the structure of general matrix contractions. We will only need the 2 &times; 2 case. Every 2 &times; 2 contraction &Gamma; can be uniquely expressed as



\Gamma = \begin{bmatrix} \Gamma_1 & D_{\Gamma_1 ^*} \Gamma_2\\ \Gamma_3 D_{\Gamma_1} & - \Gamma_3 \Gamma_1^* \Gamma_2 + D_{\Gamma_3 ^*} \Gamma_4 D_{\Gamma_2} \end{bmatrix} $$

where each &Gamma;i is a contraction.

Extensions of Positive symmetric operators
The Cayley transform for general symmetric operators can be adapted to this special case. For every non-negative number a,


 * $$|\frac{a-1}{a+1}| \le 1.$$

This suggests we assign to every positive symmetric operator A a contraction


 * $$C_A : Ran(A + 1) \rightarrow Ran(A-1) \subset \mathcal{H} $$

defined by


 * $$C_A (A+1)x = (A-1)x. \quad \mbox{i.e.} \quad V_A = (A-1)(A+1)^{-1}.\,$$

which have matrix representation



C_A = \begin{bmatrix} \Gamma_1 \\ \Gamma_3 D_{\Gamma_1} \end{bmatrix}
 * Ran(A+1) \rightarrow

\begin{matrix} Ran(A+1) \\ \oplus \\ Ran(A+1)^{\perp} \end{matrix}. $$

It is easily verified that the &Gamma;1 entry, CA projected onto Ran(A + 1) = Dom(CA), is self-adjoint.

The operator A can be written as


 * $$A = (1+ C_A)(1 - C_A)^{-1} \,$$

with Dom(A) = Ran(CA - 1). If


 * $$ \tilde{C} $$

is a contraction that extends CA and its projection onto its domain is self-adjoint, then it is clear that its inverse Cayley transform


 * $$\tilde{A} = ( 1 + \tilde{C} +1) ( 1 - \tilde{C} )^{-1}  $$

defined on


 * $$Ran ( 1 - \tilde{C} )$$

is a positive symmetric extension of A. The symmetric property follows from its projection onto its own domain being self-adjoint and positivity follows from contractivity. The converse is also true: given a positive symmetric extension of A, its Cayley transform is a contraction satisfying the stated "partial" self-adjoint property.

Theorem The positive symmetric extensions of A are in one-to-one correspondence with the extensions of its Cayley transform where if C is such an extension, we require C projected onto Dom(C) is self-adjoint.

The unitarity criterion of the Cayley transform is replaced by self-adjointness for positive operators.

Theorem A symmetric positive operator B is self-adjoint if and only if its Cayley transform is a self-adjoint contraction defined on all of H, i.e. when Ran(A + 1) = H.

Therefore finding self-adjoint extension for a positive symmetric operator becomes a matrix "completion problem". Specificly, we need to embed the column contraction CA into a 2 &times; 2 self-adjoint contraction. This can always be done and the structure of such contractions gives a parametrization of all possible extensions.

By the preceding subsection, all self-adjoint extensions of CA takes the form



\tilde{C}(\Gamma_4) = \begin{bmatrix} \Gamma_1 & D_{\Gamma_1} \Gamma_3 ^* \\ \Gamma_3 D_{\Gamma_1} & - \Gamma_3 \Gamma_1 \Gamma_3^* + D_{\Gamma_3^*} \Gamma_4 D_{\Gamma_3^*} \end{bmatrix}. $$

So the self-adjoint positive extensions of A are in bijective correspondence with the self-adjoint contractions &Gamma;4 on the defect space


 * $$\mathcal{D}_{\Gamma_3^*}$$

of &Gamma;3. The contractions


 * $$\tilde{C}(-1) \quad \mbox{and} \quad \tilde{C}(1)$$

give rise to positive extensions


 * $$A_0 \quad \mbox{and} \quad A_{\infty}$$

respectively. These are the smallest and largest positive extensions of A in the sense that


 * $$A_0 \leq B \leq A_{\infty}$$

for any positive extension B of A. The operator A0 is the Friedrichs extension of A and A&infin; is the von Neumann-Krein extension of A.

Similar results can be obtained for accretive operators.