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{{Short description|Operation on self-adjoint operators}}
{{No footnotes|date=June 2020}}
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 [[observable]]s in [[quantum mechanics]]. Other applications of solutions to this problem can be seen in various [[moment problem]]s.

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 operator]]s is equivalent to finding unitary extensions of suitable [[partial isometry|partial isometries]].

== Symmetric operators ==
Let <math>H</math> be a Hilbert space. A [[linear operator]] <math>A</math> acting on <math>H</math> with [[dense set|dense]] [[Domain of a function|domain]] <math>\operatorname{dom}(A)</math> is '''symmetric''' if
:<math>\langle Ax, y\rangle = \langle x, A y\rangle, \quad \forall x,y\in\operatorname{dom}(A).</math>

If <math>\operatorname{dom}(A) = H</math>, the [[Hellinger-Toeplitz theorem]] says that <math>A</math> is a [[bounded operator]], in which case <math>A</math> is [[self-adjoint operator|self-adjoint]] and the extension problem is trivial. In general, a symmetric operator is self-adjoint if the domain of its adjoint, <math>\operatorname{dom}(A^*)</math>, lies in <math>\operatorname{dom}(A)</math>.

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

In the next section, a symmetric operator will be assumed to be [[densely defined]] and closed.

=== Self-adjoint extensions of symmetric operators ===
If an operator <math>A</math> on the Hilbert space <math>H</math> is symmetric, when does it have self-adjoint extensions?  An operator that has a unique self-adjoint extension is said to be '''essentially self-adjoint'''; equivalently, an operator is essentially self-adjoint if its closure (the operator whose graph is the closure of the [[graph of a function|graph]] of <math>A</math>) is self-adjoint. In general, a symmetric operator could have many self-adjoint extensions or none at all. Thus, we would like a classification of its self-adjoint extensions.

The first basic criterion for essential self-adjointness is the following:<ref>{{harvnb|Hall|2013}} Theorem 9.21</ref>
{{math theorem| If <math>A</math> is a symmetric operator on <math>H</math>, then <math>A</math> is essentially self-adjoint if and only if the [[Image_(mathematics)#Image_of_a_function|range]] of the operators <math>A-i</math> and <math>A+i</math> are dense in <math>H</math>. }}
Equivalently, <math>A</math> is essentially self-adjoint if and only if the operators <math>A^* \pm i</math> have trivial [[Kernel_(linear_algebra)|kernels]].<ref>{{harvnb|Hall|2013}} Corollary 9.22</ref> That is to say, <math>A</math> ''fails to be'' self-adjoint if and only if <math>A^*</math> has an eigenvector with [[complex number|complex]] eigenvalues <math>\pm i</math>.

Another way of looking at the issue is provided by the '''[[Cayley transform]]''' of a self-adjoint operator and the deficiency indices.{{sfn | Rudin | 1991 | p=356-357 §13.17}} 

{{math theorem|Suppose <math>A</math> is a symmetric operator. Then there is a unique densely defined linear operator
<math display="block">W(A) : \operatorname{ran}(A + i) \to \operatorname{ran}(A - i)</math>
such that
<math display="block"> W(A)(Ax + ix) = Ax - ix, \quad  x \in \operatorname{dom}(A). </math>}}

<math>W(A)</math> is [[isometry|isometric]] on its domain.  Moreover, <math>\operatorname{ran}(1-W(A))</math> is dense in <math>A</math>.

Conversely, given any densely defined operator <math>U</math> which is isometric on its (not necessarily closed) domain and such that <math>1-U</math> is dense, then there is a (unique) densely defined symmetric operator
: <math> S(U) : \operatorname{ran}(1 - U) \to \operatorname{ran}(1 + U)</math>

such that
: <math> S(U)(x - Ux) = i(x + U x), \quad x \in \operatorname{dom}(U).</math>

The mappings <math>W</math> and <math>S</math> are inverses of each other, i.e., <math>S(W(A))=A</math>. 

The mapping <math>A \mapsto W(A)</math> is called the '''Cayley transform'''.  It associates a [[partial isometry|partially defined isometry]] to any symmetric densely defined operator.  Note that the mappings <math>W</math> and <math>S</math> are [[monotone convergence theorem|monotone]]: This means that if <math>B</math> is a symmetric operator that extends the densely defined symmetric operator <math>A</math>, then <math>W(B)</math> extends <math>W(A)</math>, and similarly for <math>S</math>.

{{math theorem|A necessary and sufficient condition for <math>A</math> to be self-adjoint is that its Cayley transform <math>W(A)</math> be [[Unitary operator|unitary]] on <math>H</math>.}}

This immediately gives us a necessary and sufficient condition for <math>A</math> to have a self-adjoint extension, as follows:

{{math theorem|A necessary and sufficient condition for <math>A</math> to have a self-adjoint extension is that <math>W(A)</math> have a unitary extension to <math>H</math>.}}

A partially defined isometric operator <math>V</math> on a Hilbert space <math>H</math> has a unique isometric extension to the norm closure of <math>\operatorname{dom}(V)</math>. A partially defined isometric operator with closed domain is called a [[partial isometry]].

Define the '''deficiency subspaces''' of ''A'' by
:<math>\begin{align}
K_+ &= \operatorname{ran}(A+i)^{\perp}\\
K_- &= \operatorname{ran}(A-i)^{\perp}
\end{align}</math>
In this language, the description of the self-adjoint extension problem given by the theorem can be restated as follows: a symmetric operator <math>A</math> has self-adjoint extensions if and only if the deficiency subspaces <math>K_{+}</math> and <math>K_{-}</math> have the same dimension.{{sfn | Jørgensen | Kornelson | Shuman | 2011 | p=85}}

The '''deficiency indices''' of a partial isometry <math>V</math> are defined as the dimension of the [[orthogonal complement]]s of the domain and range:

:<math>\begin{align}
  n_+(V) &= \dim \operatorname{dom}(V)^\perp \\
  n_-(V) &= \dim \operatorname{ran}(V)^\perp
\end{align}</math>

{{math theorem|A partial isometry <math>V</math> has a unitary extension if and only if the deficiency indices are identical. Moreover, <math>V</math> has a ''unique'' unitary extension if and only if the deficiency indices are both zero.}}

We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. The symmetric extension is self-adjoint if and only if the corresponding isometric extension is unitary.

A symmetric operator has a unique self-adjoint extension if and only if both its deficiency indices are zero.  Such an operator is said to be '''essentially self-adjoint'''. Symmetric operators which are not essentially self-adjoint may still have a [[canonical form|canonical]] self-adjoint extension.  Such is the case for ''non-negative'' symmetric operators (or more generally, operators which are bounded below).  These operators always have a canonically defined [[Friedrichs extension]] and for these operators we can define a canonical functional calculus. Many operators that occur in  analysis are bounded below (such as the negative of the [[Laplacian]] operator), so the issue of essential adjointness for these operators is less critical.

Suppose <math>A</math> is symmetric densely defined. Then any symmetric extension of <math>A</math> is a restriction of <math>A^*</math>.  Indeed, <math>A\subseteq B</math> and <math>B</math> symmetric yields <math>B \subseteq A^*</math> by applying the definition of <math>\operatorname{dom}(A^*)</math>. This notion leads to the '''von Neumann formulae''':{{sfn | Akhiezer | 1981 | p=354}}

{{math theorem| Suppose <math>A</math> is a densely defined symmetric operator, with domain <math>\operatorname{dom}(A)</math>. Let
<math display="block"> N_\pm = \operatorname{ran}(A \pm i)^\perp,</math>
be any pair of its deficiency subspaces. Then
<math display="block"> N_\pm = \operatorname{ker}(A^* \mp i),</math>
and
<math display="block"> \operatorname{dom}\left(A^*\right) = \operatorname{dom}\left(\overline{A}\right) \oplus N_+ \oplus N_-,</math>
where the decomposition is orthogonal relative to the graph inner product of <math>\operatorname{dom}(A^*)</math>:
<math display="block">\langle \xi \mid \eta \rangle_\text{graph} = \langle \xi \mid \eta \rangle + \left\langle A^* \xi \mid A^* \eta \right\rangle .</math>}}

=== Example ===
{{unreferenced section|date=January 2024}}
Consider the Hilbert space <math>L^2([0,1])</math>. On the subspace of absolutely continuous function that vanish on the boundary, define the operator <math>A</math> by

:<math>A f = i \frac{d}{dx} f.</math>

Integration by parts shows <math>A</math> is symmetric. Its adjoint <math>A^*</math> is the same operator with <math>\operatorname{dom}(A^*)</math> being the [[absolutely continuous function]]s with no boundary condition. We will see that extending ''A'' amounts to modifying the boundary conditions, thereby enlarging <math>\operatorname{dom}(A)</math> and reducing <math>\operatorname{dom}(A^*)</math>, until the two coincide.

Direct calculation shows that <math>K_+</math> and <math>K_-</math> are one-dimensional subspaces given by

:<math>\begin{align}
K_+ &= \operatorname{span} \{\phi_+ = c \cdot e^x \}\\
K_- &= \operatorname{span}\{ \phi_- = c \cdot e^{-x} \}
\end{align}</math>

where <math>c</math> is a normalizing constant. The self-adjoint extensions <math>A_\alpha</math> of <math>A</math> are parametrized by the [[circle group]] <math>\mathbb T = \{\alpha \in \mathbb C : |\alpha| = 1 \}</math>. For each [[unitary transformation]] <math>U_\alpha : K_- \to K_+</math> defined by 
:<math>U_\alpha (\phi_-) =\alpha \phi_+</math>

there corresponds an extension <math>A_\alpha</math> with domain
:<math> \operatorname{dom}(A_{\alpha}) = \{ f + \beta (\alpha \phi_{-} - \phi_+) | f \in \operatorname{dom}(A) , \; \beta \in \mathbb{C} \}.</math>

If <math>f \in \operatorname{dom}(A_\alpha)</math>, then <math>f</math> is absolutely continuous and

:<math>\left|\frac{f(0)}{f(1)}\right| = \left|\frac{e\alpha -1}{\alpha - e}\right| = 1.</math>

Conversely, if <math>f</math> is absolutely continuous and <math>f(0)=\gamma f(1)</math> for some <math>\gamma \in \mathbb{T}</math>, then <math>f</math> lies in the above domain.

The self-adjoint operators <math>A_\alpha</math> are instances of the [[momentum operator]] in quantum mechanics.

== Self-adjoint extension on a larger space ==
{{Expand section|date=June 2008}}
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 ==
{{unreferenced section|date=January 2024}}
A symmetric operator <math>A</math> is called '''positive''' if
:<math>\langle A x, x\rangle\ge 0, \quad \forall x\in \operatorname{dom}(A).</math>
It is known that for every such <math>A</math>, one has <math>\operatorname{dim}K_+ = \operatorname{dim}K_-</math>. Therefore, every positive symmetric operator has self-adjoint extensions. The more interesting question in this direction is whether <math>A</math> has positive self-adjoint extensions.

For two positive operators <math>A</math> and <math>B</math>, we put <math>A\leq B</math> if
:<math>(A + 1)^{-1} \ge (B + 1)^{-1}</math>

in the sense of bounded operators.

=== 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 [[Contraction (operator theory)|contraction]]s: 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 <math>\Gamma</math>, acting on <math>H</math>, we define its '''defect operators''' by

:<math>\begin{align} 
&D_{ \Gamma }\; = (1 - \Gamma^*\Gamma  )^{\frac{1}{2}}\\
&D_{\Gamma^*} = (1 - \Gamma \Gamma^*)^{\frac{1}{2}}
\end{align}</math>

The '''defect spaces''' of <math>\Gamma</math> are

:<math>\begin{align} 
&\mathcal{D}_{\Gamma}\; = \operatorname{ran}(  D_{\Gamma} )\\ 
&\mathcal{D}_{\Gamma^*} = \operatorname{ran}(  D_{\Gamma^*})
\end{align}</math>

The defect operators indicate the non-unitarity of <math>\Gamma</math>, 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 &times; 2 case. Every 2 &times; 2 contraction <math>\Gamma</math> can be uniquely expressed as

:<math>
\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}
</math>

where each <math>\Gamma_i</math> 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 <math>a</math>,

:<math>\left|\frac{a-1}{a+1}\right| \le 1.</math>

This suggests we assign to every positive symmetric operator <math>A</math> a contraction

:<math>C_A : \operatorname{ran}(A + 1) \rightarrow \operatorname{ran}(A-1) \subset H </math>

defined by

:<math>C_A (A+1)x = (A-1)x. \quad \mbox{i.e.} \quad C_A = (A-1)(A+1)^{-1}.\,</math>

which have matrix representation{{clarify|date=January 2024|reason = What is this supposed to look like?}}

:<math>
C_A =
\begin{bmatrix}
\Gamma_1 \\
\Gamma_3 D_{\Gamma_1}
\end{bmatrix}
: \operatorname{ran}(A+1) \rightarrow
\begin{matrix}
\operatorname{ran}(A+1) \\
\oplus \\
\operatorname{ran}(A+1)^{\perp}
\end{matrix}.
</math>

It is easily verified that the <math>\Gamma_1</math> entry, <math>C_A</math> projected onto <math>\operatorname{ran}(A+1)=\operatorname{dom}(C_A)</math>, is self-adjoint. The operator <math>A</math> can be written as

:<math>A = (1+ C_A)(1 - C_A)^{-1} \,</math>

with <math>\operatorname{dom}(A)=\operatorname{ran}(C_A -1)</math>. If <math>\tilde{C}</math> is a contraction that extends <math>C_A</math> and its projection onto its domain is self-adjoint, then it is clear that its inverse Cayley transform
:<math>\tilde{A} = ( 1 + \tilde{C}  ) ( 1 - \tilde{C} )^{-1}  </math>

defined on <math>\operatorname{ran}( 1 - \tilde{C})</math> is a positive symmetric extension of <math>A</math>. 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 <math>A</math>, its Cayley transform is a contraction satisfying the stated "partial" self-adjoint property.

{{math_theorem|Theorem|The positive symmetric extensions of <math>A</math> are in one-to-one correspondence with the extensions of its Cayley transform where, if <math>C</math> is such an extension, we require <math>C</math> projected onto <math>\operatorname{dom}(C)</math> be self-adjoint.}}

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

{{math_theorem|Theorem|A symmetric positive operator <math>A</math> is self-adjoint if and only if its Cayley transform is a self-adjoint contraction defined on all of <math>H</math>, i.e. when <math>\operatorname{ran}(A +1) = H</math>.}}

Therefore, finding self-adjoint extension for a positive symmetric operator becomes a "[[matrix completion]] problem". Specifically, we need to embed the column contraction <math>C_A</math> 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 <math>C_A</math> takes the form

:<math>
\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}.
</math>

So the self-adjoint positive extensions of <math>A</math> are in bijective correspondence with the self-adjoint contractions <math>\Gamma_4</math> on the defect space <math>\mathcal{D}_{\Gamma_3^*}</math> of <math>\Gamma_3</math>. The contractions <math>\tilde{C}(-1)</math> and <math>\tilde{C}(1)</math> give rise to positive extensions <math>A_0</math> and <math>A_{\infty}</math> respectively. These are the ''smallest'' and ''largest'' positive extensions of <math>A</math> in the sense that
:<math>A_0 \leq B \leq A_{\infty}</math>

for any positive self-adjoint extension <math>B</math> of <math>A</math>. The operator <math>A_\infty</math> is the '''[[Friedrichs extension]]''' of <math>A</math> and <math>A_0</math> is the '''von Neumann-Krein extension''' of <math>A</math>.

Similar results can be obtained for [[accretive operator]]s.
== Notes == 
{{reflist}}
== References ==
* {{cite book | last=Akhiezer | first=Naum Ilʹich | title=Theory of Linear Operators in Hilbert Space | publisher=Pitman | publication-place=Boston | date=1981 | isbn=0-273-08496-8}}
*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.
* {{citation | last=Hall | first=B. C. | title=Quantum Theory for Mathematicians | publisher=Springer |series=Graduate Texts in Mathematics | volume=267 | year=2013 | bibcode=2013qtm..book.....H | isbn=978-1461471158}}
* {{cite book | last1=Jørgensen | first1=Palle E. T. | last2=Kornelson | first2=Keri A. | last3=Shuman | first3=Karen L. | title=Iterated Function Systems, Moments, and Transformations of Infinite Matrices | publisher=American Mathematical Soc. | publication-place=Providence, RI | date=2011 | isbn=978-0-8218-5248-4}}
* {{cite book | last1=Reed | first1=M. | last2=Simon | first2=B. | title=Methods of Modern Mathematical Physics: Vol 1: Functional analysis | publisher=Academic Press | year=1980 | isbn=978-0-12-585050-6}}
* {{citation | last1=Reed | first1=M. | author-link=Michael C. Reed | last2=Simon | first2=B. | author-link2=Barry Simon  | title=Methods of Mathematical Physics: Vol 2: Fourier Analysis, Self-Adjointness | publisher=Academic Press | year=1972 }}
* {{cite book | last=Rudin | first=Walter | title=Functional Analysis | publisher=McGraw-Hill Science, Engineering & Mathematics | publication-place=Boston, Mass. | date=1991 | isbn=978-0-07-054236-5}}

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