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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 [[observable]]s in [[quantum mechanic]]s. Other applications of solutions to this problem can be seen in various [[moment problem]]s.
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| 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 isometry|partial isometries]].
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| == Symmetric operators ==
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| Let ''H'' be a Hilbert space. A linear operator ''A'' acting on ''H'' with dense domain Dom(''A'') is '''symmetric''' if
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| :<''Ax'', ''y''> = <''x'', ''Ay''>, for all ''x'', ''y'' in Dom(''A'').
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| If Dom(''A'') = ''H'', the [[Hellinger-Toeplitz theorem]] says that ''A'' is a bounded operator, in which case ''A'' 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, Dom(''A*''), lies in Dom(''A'').
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| When dealing with unbounded operators, 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 symmetric operator ''A'' is
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| [[closable operator|closable]]. That is, ''A'' has a smallest closed extension, called the ''closure'' of ''A''. This can
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| 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.
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| In the sequel, a symmetric operator will be assumed to be densely defined and closed.
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| '''Problem''' ''Given a densely defined closed symmetric operator A, find its self-adjoint extensions.''
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| 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
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| :<math>z \mapsto \frac{z-i}{z+i}</math>
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| maps the real line to the unit circle. This suggests one define, for a symmetric operator ''A'',
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| :<math>U_A = (A - i)(A + i)^{-1}\,</math>
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| on ''Ran''(''A'' + ''i''), the range of ''A'' + ''i''. The operator ''U<sub>A</sub>'' is in fact an isometry between closed subspaces that takes (''A'' + ''i'')''x'' to (''A'' - ''i'')''x'' for ''x'' in Dom(''A''). The map
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| :<math>A \mapsto U_A</math>
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| is also called the '''Cayley transform''' of the symmetric operator ''A''. Given ''U<sub>A</sub>'', ''A'' can be recovered by
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| :<math>A = - i(U + 1)(U - 1)^{-1} ,\,</math>
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| defined on ''Dom''(''A'') = ''Ran''(''U'' - 1). Now if
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| :<math> \tilde{U} </math>
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|
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| is an isometric extension of ''U<sub>A</sub>'', the operator
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| :<math>\tilde{A} = - i( \tilde{U} + 1)( \tilde{U} - 1 )^{-1} </math>
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| acting on
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| :<math> Ran (- \frac{i}{2} ( \tilde{U} - 1)) = Ran ( \tilde{U} - 1) </math>
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| is a symmetric extension of ''A''.
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| '''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<sub>A</sub>''.
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| Of more interest is the existence of ''self-adjoint'' extensions. The following is true.
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| '''Theorem''' A closed symmetric operator ''A'' is self-adjoint if and only if Ran (''A'' ± ''i'') = ''H'', i.e. when its Cayley transform ''U<sub>A</sub>'' is a unitary operator on ''H''.
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| '''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<sub>A</sub>''.
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| Define the '''deficiency subspaces''' of ''A'' by
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| :<math>K_+ = Ran(A+i)^{\perp}</math>
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| and
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| :<math>K_- = Ran(A-i)^{\perp}.</math>
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| 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<sub>A</sub>'' has unitary extensions to ''H'', i.e. the deficiency subspaces ''K''<sub>+</sub> and ''K''<sub>-</sub> have the same dimension.
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| === An example ===
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| Consider the Hilbert space ''L''<sup>2</sup>[0,1]. On the subspace of absolutely continuous function that vanish on the boundary, define the operator ''A'' by
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| :<math>A f = i \frac{d}{dx} f.</math> | |
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| 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.
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| Direct calculation shows that ''K''<sub>+</sub> and ''K''<sub>-</sub> are one dimensional subspaces given by
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| :<math>K_+ = span \{\phi_+ = a \cdot e^x \}</math>
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| and
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| :<math>K_- = span\{ \phi_- = a \cdot e^{-x} \}</math>
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| 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<sub>α</sub>'' : ''K''<sub>-</sub> → ''K''<sub>+</sub>, defined by ''U<sub>α</sub>''(''φ''<sub>-</sub>) = ''αφ''<sub>+</sub>, there corresponds an extension ''A''<sub>''α''</sub> with domain
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| :<math>Dom(A_{\alpha}) = \{ f + \beta (\alpha \phi_{-} - \phi_+) | f \in Dom(A) , \; \beta \in \mathbb{C} \}.</math>
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| If ''f'' ∈ Dom(''A''<sub>''α''</sub>), then ''f'' is absolutely continuous and
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| :<math>\left|\frac{f(0)}{f(1)}\right| = \left|\frac{e\alpha -1}{\alpha - e}\right| = 1.</math>
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| 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.
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| The self-adjoint operators { ''A''<sub>''α''</sub> } are instances of the [[momentum operator]] in quantum mechanics.
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| == Self-adjoint extension on a larger space ==
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| {{Expand section|date=June 2008}}
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| 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.
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| == Positive symmetric operators ==
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| 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''<sub>+</sub>) = dim(''K''<sub>-</sub>). Therefore every positive symmetric operator has self-adjoint extensions. The more interesting question in this direction is whether ''A'' has positive self-adjoint extensions.
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| For two positive operators ''A'' and ''B'', we put ''A'' ≤ ''B'' if
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| :<math>(A + 1)^{-1} \ge (B + 1)^{-1}</math>
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| in the sense of bounded operators.
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| === Structure of 2 × 2 matrix contractions ===
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| 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 × 2 self-adjoint contraction, we obtain the positive self-adjoint extensions of a positive symmetric operator.
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| Before stating the relevant result, we first fix some terminology. For a contraction Γ, acting on ''H'', we define its ''defect operators'' by
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| :<math> D_{ \Gamma } = (1 - \Gamma^*\Gamma )^{\frac{1}{2}} \quad \mbox{and} \quad D_{\Gamma^*} = (1 - \Gamma \Gamma^*)^{\frac{1}{2}}.</math>
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| The ''defect spaces'' of Γ are
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| :<math>\mathcal{D}_{\Gamma} = Ran( D_{\Gamma} ) \quad \mbox{and} \quad \mathcal{D}_{\Gamma^*} = Ran( D_{\Gamma^*} ).</math>
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| The defect operators indicate the non-unitarity of Γ, while the defect spaces ensure uniqueness in some parameterizations.
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| 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
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| :<math>
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| \Gamma =
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| \begin{bmatrix}
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| \Gamma_1 & D_{\Gamma_1 ^*} \Gamma_2\\
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| \Gamma_3 D_{\Gamma_1} & - \Gamma_3 \Gamma_1^* \Gamma_2 + D_{\Gamma_3 ^*} \Gamma_4 D_{\Gamma_2}
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| \end{bmatrix}
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| </math>
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| where each Γ<sub>''i''</sub> is a contraction.
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| === Extensions of Positive symmetric operators ===
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| The Cayley transform for general symmetric operators can be adapted to this special case. For every non-negative number ''a'',
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| :<math>\left|\frac{a-1}{a+1}\right| \le 1.</math>
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| This suggests we assign to every positive symmetric operator ''A'' a contraction
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| :<math>C_A : Ran(A + 1) \rightarrow Ran(A-1) \subset \mathcal{H} </math>
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| defined by
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| :<math>C_A (A+1)x = (A-1)x. \quad \mbox{i.e.} \quad C_A = (A-1)(A+1)^{-1}.\,</math>
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| which have matrix representation
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| :<math>
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| C_A =
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| \begin{bmatrix}
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| \Gamma_1 \\
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| \Gamma_3 D_{\Gamma_1}
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| \end{bmatrix}
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| : Ran(A+1) \rightarrow
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| \begin{matrix}
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| Ran(A+1) \\
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| \oplus \\
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| Ran(A+1)^{\perp}
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| \end{matrix}.
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| </math>
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| It is easily verified that the Γ<sub>1</sub> entry, ''C<sub>A</sub>'' projected onto ''Ran''(''A'' + 1) = ''Dom''(''C<sub>A</sub>''), is self-adjoint. The operator ''A'' can be written as
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| :<math>A = (1+ C_A)(1 - C_A)^{-1} \,</math>
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| with ''Dom''(''A'') = ''Ran''(''C<sub>A</sub>'' - 1). If
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| :<math> \tilde{C} </math>
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| is a contraction that extends ''C<sub>A</sub>'' and its projection onto its domain is self-adjoint, then it is clear that its inverse Cayley transform
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| :<math>\tilde{A} = ( 1 + \tilde{C} ) ( 1 - \tilde{C} )^{-1} </math>
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| defined on
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| :<math>Ran ( 1 - \tilde{C} )</math>
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| 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.
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| '''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.
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| The unitarity criterion of the Cayley transform is replaced by self-adjointness for positive operators.
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| '''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''.
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| Therefore finding self-adjoint extension for a positive symmetric operator becomes a "[[matrix completion]] problem". Specifically, we need to embed the column contraction ''C<sub>A</sub>'' 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.
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| By the preceding subsection, all self-adjoint extensions of ''C<sub>A</sub>'' takes the form
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| :<math>
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| \tilde{C}(\Gamma_4) =
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| \begin{bmatrix}
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| \Gamma_1 & D_{\Gamma_1} \Gamma_3 ^* \\
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| \Gamma_3 D_{\Gamma_1} & - \Gamma_3 \Gamma_1 \Gamma_3^* + D_{\Gamma_3^*} \Gamma_4 D_{\Gamma_3^*}
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| \end{bmatrix}.
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| </math>
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| So the self-adjoint positive extensions of ''A'' are in bijective correspondence with the self-adjoint contractions Γ<sub>4</sub> on the defect space
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| :<math>\mathcal{D}_{\Gamma_3^*}</math>
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| of Γ<sub>3</sub>. The contractions
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| :<math>\tilde{C}(-1) \quad \mbox{and} \quad \tilde{C}(1)</math>
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|
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| give rise to positive extensions
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| :<math>A_0 \quad \mbox{and} \quad A_{\infty}</math>
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| respectively. These are the ''smallest'' and ''largest'' positive extensions of ''A'' in the sense that
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| :<math>A_0 \leq B \leq A_{\infty}</math>
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| for any positive self-adjoint extension ''B'' of ''A''. The operator ''A''<sub>∞</sub> is the '''[[Friedrichs extension]]''' of ''A'' and ''A''<sub>0</sub> is the '''von Neumann-Krein extension''' of ''A''.
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| Similar results can be obtained for [[accretive operator]]s.
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| == References ==
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| *A. Alonso and B. Simon, The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators. ''J. Operator Theory'' '''4''' (1980), 251-270.
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| *Gr. Arsene and A. Gheondea, Completing matrix contractions, ''J. Operator Theory'' '''7''' (1982), 179-189.
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| * N. Dunford and J.T. Schwartz, ''Linear Operators'', Part II, Interscience, 1958.
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| * B.C. Hall, ''Quantum Theory for Mathematicians'', Chapter 9, Springer, 2013.
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| *M. Reed and B. Simon, ''Methods of Modern Mathematical Physics'', vol. I and II, Academic Press, 1975.
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| {{DEFAULTSORT:Extensions Of Symmetric Operators}}
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| [[Category:Functional analysis]]
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| [[Category:Operator theory]]
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