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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

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

<Ax, y> = <x, Ay>, 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

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

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 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

$A\mapsto U_A$

is also called the Cayley transform of the symmetric operator A. Given U_A, 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 U_A, 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 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

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

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 U_A 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

$Af=i\frac{d}{dx}f.$

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

$K_{+}=span\{{\varphi }_{+}=a\cdot e^x\}$

and

$K_{-}=span\{{\varphi }_{-}=a\cdot e^{-x}\}$

where a is a normalizing constant. So the self-adjoint extensions of A are parametrized by the unit circle in the complex plane, {|α| = 1}. For each unitary U_α : K_- → K₊, defined by U_α(φ_-) = αφ₊, there corresponds an extension A_α with domain

$Dom(A_{\alpha })=\{f+\beta (\alpha {\varphi }_{-}-{\varphi }_{+})|f\in Dom(A),\beta \in \mathbb{ℂ}\}.$

If f ∈ Dom(A_α), then f is absolutely continuous and

$\left|\frac{f(0)}{f(1)}\right|=\left|\frac{e\alpha -1}{\alpha -e}\right|=1.$

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

Template:Expand section 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

A symmetric operator A is called positive if <Ax, x> ≥ 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.

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

$(A+1{)}^{-1}\ge (B+1{)}^{-1}$

in the sense of bounded operators.

Structure of 2 × 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 × 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

$D_{\mathrm{\Gamma }}=(1-{\mathrm{\Gamma }}^{*}\mathrm{\Gamma }{)}^{\frac{1}{2}}\text{and}{D}_{{\mathrm{\Gamma }}^{*}}=(1-\mathrm{\Gamma }{\mathrm{\Gamma }}^{*}{)}^{\frac{1}{2}}.$

The defect spaces of Γ are

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

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

$\mathrm{\Gamma }=\left[\begin{array}{cc}{\mathrm{\Gamma }}_1& D_{{\mathrm{\Gamma }}_1^{*}}{\mathrm{\Gamma }}_2\\ {\mathrm{\Gamma }}_3{D}_{{\mathrm{\Gamma }}_1}& -{\mathrm{\Gamma }}_3{\mathrm{\Gamma }}_1^{*}{\mathrm{\Gamma }}_2+D_{{\mathrm{\Gamma }}_3^{*}}{\mathrm{\Gamma }}_4{D}_{{\mathrm{\Gamma }}_2}\end{array}\right]$

where each Γ_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,

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

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

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

defined by

$C_A(A+1)x=(A-1)x.\text{i.e.}{C}_A=(A-1)(A+1{)}^{-1}.$

which have matrix representation

$C_A=\left[\begin{array}{c}{\mathrm{\Gamma }}_1\\ {\mathrm{\Gamma }}_3{D}_{{\mathrm{\Gamma }}_1}\end{array}\right]:Ran(A+1)\to \begin{array}{c}Ran(A+1)\\ \oplus \\ Ran(A+1{)}^{\perp }\end{array}.$

It is easily verified that the Γ₁ entry, C_A projected onto Ran(A + 1) = Dom(C_A), is self-adjoint. The operator A can be written as

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

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

$\tilde{C}$

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

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

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) 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

$\tilde{C}({\mathrm{\Gamma }}_4)=\left[\begin{array}{cc}{\mathrm{\Gamma }}_1& D_{{\mathrm{\Gamma }}_1}{\mathrm{\Gamma }}_3^{*}\\ {\mathrm{\Gamma }}_3{D}_{{\mathrm{\Gamma }}_1}& -{\mathrm{\Gamma }}_3{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_3^{*}+D_{{\mathrm{\Gamma }}_3^{*}}{\mathrm{\Gamma }}_4{D}_{{\mathrm{\Gamma }}_3^{*}}\end{array}\right].$

So the self-adjoint positive extensions of A are in bijective correspondence with the self-adjoint contractions Γ₄ on the defect space

${\mathcal{𝒟}}_{{\mathrm{\Gamma }}_3^{*}}$

of Γ₃. The contractions

$\tilde{C}(-1)\text{and}\tilde{C}(1)$

give rise to positive extensions

$A_0\text{and}{A}_{\mathrm{\infty }}$

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

$A_0\le B\le A_{\mathrm{\infty }}$

for any positive self-adjoint extension B of A. The operator A_∞ is the Friedrichs extension of A and A₀ is the von Neumann-Krein extension of A.

Similar results can be obtained for accretive operators.

References

• 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.