# Extensions of symmetric operators

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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. More specifically, finding self-adjoint extensions, with various requirements, of symmetric operators is equivalent to finding unitary extensions of suitable partial isometries.

## Symmetric operators[edit]

Let *H* be a Hilbert space. A linear operator *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 densely defined and closed.

**Problem** *Given a densely defined closed symmetric operator A, find its self-adjoint extensions.*

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

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

on *Ran*(*A* + *i*), the range of *A* + *i*. The operator *U _{A}* is in fact an isometry between closed subspaces that takes (

*A*+

*i*)

*x*to (

*A*-

*i*)

*x*for

*x*in Dom(

*A*). The map

is also called the **Cayley transform** of the symmetric operator *A*. Given *U _{A}*,

*A*can be recovered by

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

is an isometric extension of *U _{A}*, the operator

acting on

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 *U _{A}*.

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* ± *i*) = *H*, i.e. when its Cayley transform *U _{A}* is a 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 *U _{A}*.

Define the **deficiency subspaces** of *A* by

and

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 *U _{A}* has unitary extensions to

*H*, i.e. the deficiency subspaces

*K*

_{+}and

*K*

_{−}have the same dimension.

### An example[edit]

Consider the Hilbert space *L*^{2}[0,1]. On the subspace of absolutely continuous function that vanish on the boundary, define the operator *A* by

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

and

where *a* is a normalizing constant. So the self-adjoint extensions of *A* are parametrized by the unit circle in the complex plane, {|*α*| = 1}. For each unitary *U _{α}* :

*K*

_{−}→

*K*

_{+}, defined by

*U*(

_{α}*φ*

_{−}) =

*αφ*

_{+}, there corresponds an extension

*A*

_{α}with domain

If *f* ∈ Dom(*A*_{α}), then *f* is absolutely continuous and

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

The self-adjoint operators { *A*_{α} } are instances of the momentum operator in quantum mechanics.

## Self-adjoint extension on a larger space[edit]

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

## Positive symmetric operators[edit]

A symmetric operator *A* is called **positive** if 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.

For two positive operators *A* and *B*, we put *A* ≤ *B* if

in the sense of bounded operators.

### Structure of 2 × 2 matrix contractions[edit]

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 × 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 Γ, acting on *H*, we define its *defect operators* by

The *defect spaces* of Γ are

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

where each Γ_{i} is a contraction.

### Extensions of Positive symmetric operators[edit]

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

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

defined by

which have matrix representation

It is easily verified that the Γ_{1} entry, *C _{A}* projected onto

*Ran*(

*A*+ 1) =

*Dom*(

*C*), is self-adjoint. The operator

_{A}*A*can be written as

with *Dom*(*A*) = *Ran*(*C _{A}* - 1). If

is a contraction that extends *C _{A}* and its projection onto its domain is self-adjoint, then it is clear that its inverse Cayley transform

defined on

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*) be self-adjoint.

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

**Theorem** A symmetric positive operator *A* 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". Specifically, we need to embed the column contraction *C _{A}* into a 2 × 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 *C _{A}* takes the form

So the self-adjoint positive extensions of *A* are in bijective correspondence with the self-adjoint contractions Γ_{4} on the defect space

of Γ_{3}. The contractions

give rise to positive extensions

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

for any positive self-adjoint extension *B* of *A*. The operator *A*_{∞} is the **Friedrichs extension** of *A* and *A*_{0} is the **von Neumann-Krein extension** of *A*.

Similar results can be obtained for accretive operators.

## References[edit]

- A. Alonso and B. Simon, The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators.
*J. Operator Theory***4**(1980), 251-270. - Gr. Arsene and A. Gheondea, Completing matrix contractions,
*J. Operator Theory***7**(1982), 179-189. - N. Dunford and J.T. Schwartz,
*Linear Operators*, Part II, Interscience, 1958. - B.C. Hall,
*Quantum Theory for Mathematicians*, Chapter 9, Springer, 2013. - M. Reed and B. Simon,
*Methods of Modern Mathematical Physics*, vol. I and II, Academic Press, 1975.