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{{Short description|Linear operator defined on a dense linear subspace}}
In [[Mathematics|mathematics]], more specifically [[Functional analysis|functional analysis]] and [[Operator theory|operator theory]], the notion of '''unbounded operator''' provides an abstract framework for dealing with [[Differential operator|differential operator]]s, unbounded [[Physics:Observable|observable]]s in [[Physics:Quantum mechanics|quantum mechanics]], and other cases.

The term "unbounded operator" can be misleading, since
* "unbounded" should sometimes be understood as "not necessarily bounded";
* "operator" should be understood as "linear operator" (as in the case of "bounded operator");
* the domain of the operator is a [[Linear subspace|linear subspace]], not necessarily the whole space;
* this linear subspace is not necessarily [[Closed set|closed]]; often (but not always) it is assumed to be dense;
* in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.

In contrast to [[Bounded operator|bounded operator]]s, unbounded operators on a given space do not form an [[Algebra over a field|algebra]], nor even a linear space, because each one is defined on its own domain.

The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a [[Hilbert space]].{{clarify|reason=This restriction is not adhered to in the article.|date=May 2015}} Some generalizations to [[Banach space]]s and more general [[Topological vector space|topological vector space]]s are possible.

==Short history==
The theory of unbounded operators developed in the late 1920s and early 1930s as part of developing a rigorous mathematical framework for [[Hilbert space#quantum mechanics|quantum mechanics]].<ref>{{harvnb|Reed|Simon|1980|loc=Notes to Chapter VIII, page 305}}</ref> The theory's development is due to [[Biography:John von Neumann|John von Neumann]]<ref>{{harvnb | von Neumann | 1930 | pp=49&ndash;131}}</ref> and Marshall Stone.<ref name="Stone1932">{{ harvnb | Stone | 1932 }}</ref> Von Neumann introduced using [[Graph of a function|graphs]] to analyze unbounded operators in 1932.<ref>{{ harvnb | von Neumann | 1932 | pp = 294&ndash;310 }}</ref>

== Definitions and basic properties ==
Let {{math|''X'', ''Y''}} be [[Banach space]]s.  An '''unbounded operator''' (or simply ''operator'') {{math|''T'' : ''D''(''T'') → ''Y''}} is a [[Linear map|linear map]] {{mvar|T}} from a linear subspace {{math|''D''(''T'') ⊆ ''X''}}—the domain of {{mvar|T}}—to the space {{math|''Y''}}.<ref name="Pedersen-5.1.1">{{harvnb|Pedersen|1989|loc=5.1.1}}</ref> Contrary to the usual convention, {{mvar|T}} may not be defined on the whole space {{mvar|X}}.

An operator {{mvar|T}} is said to be '''closed''' if its graph {{math|Γ(''T'')}} is a [[Closed set|closed set]].<ref name="Pedersen-5.1.4">{{ harvnb |Pedersen|1989| loc=5.1.4 }}</ref> (Here, the graph {{math|Γ(''T'')}} is a linear subspace of the [[Direct sum of modules#Direct sum of Hilbert spaces|direct sum]] {{math|''X'' ⊕ ''Y''}}, defined as the set of all pairs {{math|(''x'', ''Tx'')}}, where {{mvar|x}} runs over the domain of {{mvar|T}}&thinsp;.) Explicitly, this means that for every sequence {{math|{''x<sub>n</sub>''} }} of points from the domain of {{mvar|T}} such that {{math|''x<sub>n</sub>'' → ''x''}} and {{math|''Tx<sub>n</sub>'' → ''y''}}, it holds that {{mvar|x}} belongs to the domain of {{mvar|T}} and {{math|''Tx'' {{=}} ''y''}}.<ref name="Pedersen-5.1.4"/> The closedness can also be formulated in terms of the ''graph norm'': an operator {{mvar|T}} is closed if and only if its domain {{math|''D''(''T'')}} is a complete space with respect to the norm:<ref name="BSU-5">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=page 5 }}</ref>

: <math>\|x\|_T = \sqrt{ \|x\|^2 + \|Tx\|^2 }.</math>

An operator {{mvar|T}} is said to be '''[[Densely defined operator|densely defined]]''' if its domain is [[Dense set|dense]] in {{mvar|X}}.<ref name="Pedersen-5.1.1" /> This also includes operators defined on the entire space {{mvar|X}}, since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint (if {{math|X}} and {{math|Y}} are Hilbert spaces) and the transpose; see the sections below. 

If {{math|''T'' : ''X'' → ''Y''}} is closed, densely defined and continuous on its domain, then its domain is all of {{mvar|X}}.<ref group="nb">Suppose ''f<sub>j</sub>'' is a sequence in the domain of {{mvar|T}} that converges to {{math|''g'' ∈ ''X''}}. Since {{mvar|T}} is uniformly continuous on its domain, ''Tf<sub>j</sub>'' is [[Cauchy sequence|Cauchy]] in {{mvar|Y}}. Thus, {{math|(&thinsp;''f<sub>j</sub>''&thinsp;, ''T&thinsp;f<sub>j</sub>''&thinsp;)}} is Cauchy and so converges to some {{math|(&thinsp;''f''&thinsp;, ''T&thinsp;f''&thinsp;)}} since the graph of {{mvar|T}} is closed. Hence, {{math|&thinsp;''f''&thinsp; {{=}} ''g''}}, and the domain of {{mvar|T}} is closed.</ref>

A densely defined operator {{mvar|T}} on a [[Hilbert space]] {{mvar|H}} is called '''bounded from below''' if {{math|''T'' + ''a''}} is a positive operator for some real number {{mvar|a}}. That is, {{math|⟨''Tx''{{!}}''x''⟩ ≥ −''a'' {{!!}}''x''{{!!}}<sup>2</sup>}} for all {{mvar|x}} in the domain of {{mvar|T}} (or alternatively {{math|⟨''Tx''{{!}}''x''⟩ ≥ ''a'' {{!!}}''x''{{!!}}<sup>2</sup>}} since {{math|''a''}} is arbitrary).<ref name="Pedersen-5.1.12" /> If both {{mvar|T}} and {{math|−''T''}} are bounded from below then {{mvar|T}} is bounded.<ref name="Pedersen-5.1.12" />


== Example ==
Let {{math|''C''([0, 1])}} denote the space of continuous functions on the unit interval, and let {{math|''C''<sup>1</sup>([0, 1])}} denote the space of continuously differentiable functions. We equip <math>C([0,1])</math> with the supremum norm, <math>\|\cdot\|_{\infty}</math>, making it a Banach space. Define the classical differentiation operator {{math|{{sfrac|''d''|''dx''}} : ''C''<sup>1</sup>([0, 1]) → ''C''([0, 1])}} by the usual formula:

: <math> \left (\frac{d}{dx}f \right )(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, \qquad \forall x \in [0, 1].</math>

Every differentiable function is continuous, so {{math|''C''<sup>1</sup>([0, 1]) ⊆ ''C''([0, 1])}}. We claim that {{math|{{sfrac|''d''|''dx''}} : ''C''([0, 1]) → ''C''([0, 1])}} is a well-defined unbounded operator, with domain {{math|''C''<sup>1</sup>([0, 1])}}. For this, we need to show that <math>\frac{d}{dx}</math> is linear and then, for example, exhibit some <math>\{f_n\}_n \subset C^1([0,1])</math> such that <math>\|f_n\|_\infty=1</math> and <math>\sup_n \|\frac{d}{dx} f_n\|_\infty=+\infty</math>.

This is a linear operator, since a linear combination {{math|''a&thinsp;f&thinsp;'' + ''bg''}} of two continuously differentiable functions {{math|&thinsp;''f''&thinsp;, ''g''}} is also continuously differentiable, and

:<math>\left (\tfrac{d}{dx} \right )(af+bg)= a \left (\tfrac{d}{dx}  f \right ) + b \left (\tfrac{d}{dx} g \right ).</math>

The operator is not bounded. For example,

:<math>\begin{cases} f_n : [0, 1] \to [-1, 1] \\ f_n(x) = \sin (2\pi n x) \end{cases}</math>

satisfy

:<math> \left \|f_n \right \|_{\infty} = 1,</math>

but

:<math> \left \| \left (\tfrac{d}{dx}  f_n \right ) \right \|_{\infty} = 2\pi n \to \infty</math>
as <math>n\to\infty</math>.

The operator is densely defined, and closed.

The same operator can be treated as an operator {{math|''Z'' → ''Z''}} for many choices of Banach space {{mvar|Z}} and not be bounded between any of them. At the same time, it can be bounded as an operator {{math|''X'' → ''Y''}} for other pairs of Banach spaces {{math|''X'', ''Y''}}, and also as operator {{math|''Z'' → ''Z''}} for some topological vector spaces {{mvar|Z}}.{{clarify|reason=Why the shift from Banach spaces to topological vector spaces? What is a bounded operator between topological vector spaces?|date=May 2015}} As an example let {{math|''I'' ⊂ '''R'''}} be an open interval and consider

:<math>\frac{d}{dx} : \left (C^1 (I), \|\cdot \|_{C^1} \right ) \to \left ( C (I), \| \cdot \|_{\infty} \right),</math>

where:

:<math>\| f \|_{C^1} = \| f \|_{\infty} + \| f' \|_{\infty}.</math>

== Adjoint ==
The adjoint of an unbounded operator can be defined in two equivalent ways. Let <math>T : D(T) \subseteq H_1 \to H_2</math> be an unbounded operator between Hilbert spaces.

First, it can be defined in a way analogous to how one defines the adjoint of a bounded operator. Namely, the adjoint <math>T^* : D\left(T^*\right) \subseteq H_2 \to H_1</math> of {{mvar|T}} is defined as an operator with the property:
<math display=block>\langle Tx \mid y \rangle_2 = \left \langle x \mid T^*y \right \rangle_1, \qquad x \in D(T).</math>
More precisely, <math>T^* y</math> is defined in the following way. If <math>y \in H_2</math> is such that <math>x \mapsto \langle Tx \mid y \rangle</math> is a continuous linear functional on the domain of {{mvar|T}}, then <math>y</math> is declared to be an element of <math>D\left(T^*\right),</math> and after extending the linear functional to the whole space via the [[Hahn–Banach theorem]], it is possible to find some <math>z</math> in <math>H_1</math> such that
<math display=block>\langle Tx \mid y \rangle_2 = \langle x \mid z \rangle_1, \qquad x \in D(T),</math>
since [[Riesz representation theorem]] allows the continuous dual of the Hilbert space <math>H_1</math> to be identified with the set of linear functionals given by the inner product. This vector <math>z</math> is uniquely determined by <math>y</math> if and only if the linear functional <math>x \mapsto \langle Tx \mid y \rangle</math> is densely defined; or equivalently, if {{mvar|T}} is densely defined. Finally, letting <math>T^* y = z</math> completes the construction of <math>T^*,</math> which is necessarily a linear map. The adjoint <math>T^* y</math> exists if and only if {{mvar|T}} is densely defined.

By definition, the domain of <math>T^*</math> consists of elements <math>y</math> in <math>H_2</math> such that <math>x \mapsto \langle Tx \mid y \rangle</math> is continuous on the domain of {{mvar|T}}. Consequently, the domain of <math>T^*</math> could be anything; it could be trivial (that is, contains only zero).<ref name="BSU-3.2">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=Example 3.2 on page 16 }}</ref> It may happen that the domain of <math>T^*</math> is a closed [[Hyperplane|hyperplane]] and <math>T^*</math> vanishes everywhere on the domain.<ref name="RS-252">{{ harvnb |Reed|Simon|1980| loc=page 252 }}</ref><ref name="BSU-3.1">{{harvnb|Berezansky|Sheftel|Us|1996|loc=Example 3.1 on page 15 }}</ref> Thus, boundedness of <math>T^*</math> on its domain does not imply boundedness of {{mvar|T}}. On the other hand, if <math>T^*</math> is defined on the whole space then {{mvar|T}} is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space.<ref group="nb">Proof: being closed, the everywhere defined <math>T^*</math> is bounded, which implies boundedness of <math>T^{**},</math> the latter being the closure of {{mvar|T}}. See also {{harv |Pedersen|1989| loc=2.3.11 }} for the case of everywhere defined {{mvar|T}}.</ref> If the domain of <math>T^*</math> is dense, then it has its adjoint <math>T^{**}.</math><ref name="Pedersen-5.1.5" /> A closed densely defined operator {{mvar|T}} is bounded if and only if <math>T^*</math> is bounded.<ref group="nb">Proof: <math>T^{**} = T.</math> So if <math>T^*</math> is bounded then its adjoint {{mvar|T}} is bounded.</ref>

The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator <math>J</math> as follows:<ref name="Pedersen-5.1.5">{{ harvnb |Pedersen|1989| loc=5.1.5 }}</ref>
<math display=block>\begin{cases} J: H_1 \oplus H_2 \to H_2 \oplus H_1 \\ J(x \oplus y) = -y \oplus x \end{cases}</math>
Since <math>J</math> is an isometric surjection, it is unitary. Hence: <math>J(\Gamma(T))^{\bot}</math> is the graph of some operator <math>S</math> if and only if {{mvar|T}} is densely defined.<ref name="BSU-12">{{harvnb|Berezansky|Sheftel|Us|1996| loc=page 12}}</ref> A simple calculation shows that this "some" <math>S</math> satisfies:
<math display=block>\langle Tx \mid y \rangle_2 = \langle x \mid Sy \rangle_1,</math>
for every {{mvar|x}} in the domain of {{mvar|T}}. Thus <math>S</math> is the adjoint of {{mvar|T}}.

It follows immediately from the above definition that the adjoint <math>T^*</math> is closed.<ref name="Pedersen-5.1.5" /> In particular, a self-adjoint operator (meaning <math>T = T^*</math>) is closed. An operator {{mvar|T}} is closed and densely defined if and only if <math>T^{**} = T.</math><ref group="nb">Proof: If {{mvar|T}} is closed densely defined then <math>T^*</math> exists and is densely defined. Thus <math>T^{**}</math> exists. The graph of {{mvar|T}} is dense in the graph of <math>T^{**};</math> hence <math>T = T^{**}.</math> Conversely, since the existence of <math>T^{**}</math> implies that that of <math>T^*,</math> which in turn implies {{mvar|T}} is densely defined. Since <math>T^{**}</math> is closed, {{mvar|T}} is densely defined and closed.</ref>

Some well-known properties for bounded operators generalize to closed densely defined operators. The kernel of a closed operator is closed. Moreover, the kernel of a closed densely defined operator <math>T : H_1 \to H_2</math> coincides with the orthogonal complement of the range of the adjoint. That is,<ref>{{ harvnb | Brezis | 1983|p=28}}</ref>
<math display=block>\operatorname{ker}(T) = \operatorname{ran}(T^*)^\bot.</math>
[[Von Neumann's theorem|von Neumann's theorem]] states that <math>T^* T</math> and <math>T T^*</math> are self-adjoint, and that <math>I + T^* T</math> and <math>I + T T^*</math> both have bounded inverses.<ref>{{harvnb | Yoshida | 1980| p=200 }}</ref> If <math>T^*</math> has trivial kernel, {{mvar|T}} has dense range (by the above identity.) Moreover:

:{{mvar|T}} is surjective if and only if there is a <math>K > 0</math> such that <math>\|f\|_2 \leq K \left\|T^* f\right\|_1</math> for all <math>f</math> in <math>D\left(T^*\right).</math><ref group="nb">If <math>T</math> is surjective then <math>T : (\ker T)^{\bot} \to H_2</math> has bounded inverse, denoted by <math>S.</math> The estimate then follows since
<math display="block">\|f\|_2^2 = \left |\langle TSf \mid f \rangle_2 \right | \leq \|S\| \|f\|_2 \left \|T^*f \right \|_1</math>
Conversely, suppose the estimate holds. Since <math>T^*</math> has closed range, it is the case that <math>\operatorname{ran}(T) = \operatorname{ran}\left(T T^*\right).</math> Since <math>\operatorname{ran}(T)</math> is dense, it suffices to show that <math>T T^*</math> has closed range. If <math>T T^* f_j</math> is convergent then <math> f_j</math> is convergent by the estimate since
<math display="block">\|T^*f_j\|_1^2 = | \langle T^*f_j \mid T^*f_j \rangle_1| \leq \|TT^*f_j\|_2 \|f_j\|_2.</math>

Say, <math>f_j \to g.</math> Since <math>T T^*</math> is self-adjoint; thus, closed, (von Neumann's theorem), <math>T T^* f_j \to T T^* g.</math> QED</ref> (This is essentially a variant of the so-called [[Closed range theorem|closed range theorem]].) In particular, {{mvar|T}} has closed range if and only if <math>T^*</math> has closed range.

In contrast to the bounded case, it is not necessary that <math>(T S)^* = S^* T^*,</math> since, for example, it is even possible that <math>(T S)^*</math> does not exist.{{Citation needed|date=July 2009}}<!-- Need a concrete example.--> This is, however, the case if, for example, {{mvar|T}} is bounded.<ref>{{harvnb | Yoshida|1980| p= 195}}.</ref>

A densely defined, closed operator {{mvar|T}} is called ''[[Normal operator|normal]]'' if it satisfies the following equivalent conditions:<ref name="Pedersen-5.1.11">{{ harvnb |Pedersen|1989| loc=5.1.11 }}</ref>

* <math>T^* T = T T^*</math>;
* the domain of {{mvar|T}} is equal to the domain of <math>T^*,</math> and <math>\|T x\| = \left\|T^* x\right\|</math> for every {{mvar|x}} in this domain;
* there exist self-adjoint operators <math>A, B</math> such that <math>T = A + i B,</math><math>T^* = A - i B,</math> and <math>\|T x\|^2 = \|A x\|^2 + \|B x\|^2</math> for every {{mvar|x}} in the domain of {{mvar|T}}.

Every self-adjoint operator is normal.

== Transpose ==

Let <math>T : B_1 \to B_2</math> be an operator between Banach spaces. Then the ''[[Transpose|transpose]]'' (or ''dual'') <math>{}^t T: {B_2}^* \to {B_1}^*</math> of <math>T</math> is the linear operator satisfying:
<math display=block>\langle T x, y' \rangle = \langle x, \left({}^t T\right) y' \rangle</math>
for all <math>x \in B_1</math> and <math>y \in B_2^*.</math> Here, we used the notation: <math>\langle x, x' \rangle = x'(x).</math><ref>{{harvnb | Yoshida|1980 | p= 193}}</ref>

The necessary and sufficient condition for the transpose of <math>T</math> to exist is that <math>T</math> is densely defined (for essentially the same reason as to adjoints, as discussed above.)

For any Hilbert space <math>H,</math> there is the anti-linear isomorphism:
<math display=block>J: H^* \to H</math>
given by <math>J f = y</math> where <math>f(x) = \langle x \mid y \rangle_H, (x \in H).</math> 
Through this isomorphism, the transpose <math>{}^t T</math> relates to the adjoint <math>T^*</math> in the following way:<ref>{{harvnb | Yoshida | 1980 | p = 196}}</ref>
<math display=block>T^* = J_1 \left({}^t T\right) J_2^{-1},</math>
where <math>J_j: H_j^* \to H_j</math>. (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.) Note that this gives the definition of adjoint in terms of a transpose.

== Closed linear operators ==

Closed linear operators are a class of linear operators on [[Banach space]]s. They are more general than [[Bounded operator|bounded operator]]s, and therefore not necessarily [[Continuous function|continuous]], but they still retain nice enough properties that one can define the [[Spectrum (functional analysis)|spectrum]] and  (with certain assumptions) [[Functional calculus|functional calculus]] for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the [[Derivative|derivative]] and a large class of [[Differential operator|differential operator]]s. 

Let {{math|''X'', ''Y''}} be two [[Banach space]]s. A linear operator {{math|''A'' : ''D''(''A'') ⊆ ''X'' → ''Y''}} is '''closed''' if for every [[Sequence|sequence]] {{math|{''x''<sub>''n''</sub>} }} in {{math|''D''(''A'')}} [[Limit of a sequence|converging]] to {{mvar|x}} in {{mvar|X}} such that {{math|''Ax<sub>n</sub>'' → ''y'' ∈ ''Y''}} as {{math|''n'' → ∞}} one has {{math|''x'' ∈ ''D''(''A'')}} and {{math|1=''Ax'' = ''y''}}. 
Equivalently, {{mvar|A}} is closed if its graph is [[Closed set|closed]] in the direct sum {{math|''X'' ⊕ ''Y''}}.

Given a linear operator {{mvar|A}}, not necessarily closed, if the closure of its graph in {{math|''X'' ⊕ ''Y''}} happens to be the graph of some operator, that operator is called the '''closure''' of {{mvar|A}}, and we say that {{mvar|A}} is '''closable'''. Denote the closure of {{mvar|A}} by {{math|{{overline|''A''}}}}. It follows that {{mvar|A}} is the [[Function (mathematics)|restriction]] of {{math|{{overline|''A''}}}} to {{math|''D''(''A'')}}.

A '''core''' (or '''essential domain''') of a closable operator is a [[Subset|subset]] {{mvar|C}} of {{math|''D''(''A'')}} such that the closure of the restriction of {{mvar|A}} to {{mvar|C}} is {{math|{{overline|''A''}}}}.

=== Example ===

Consider the [[Derivative|derivative]] operator {{math|1=''A'' = {{sfrac|''d''|''dx''}}}} where {{math|1=''X'' = ''Y'' = ''C''([''a'', ''b''])}} is the Banach space of all [[Continuous function|continuous function]]s on an [[Interval (mathematics)|interval]] {{math|[''a'', ''b'']}}. 
If one takes its domain {{math|''D''(''A'')}} to be {{math|''C''<sup>1</sup>([''a'', ''b''])}}, then {{mvar|A}} is a closed operator which is not bounded.<ref>{{ harvnb | Kreyszig | 1978 | p = 294}}</ref> 
On the other hand if {{math|1=''D''(''A'') = smooth function{{!}}''C''<sup>∞</sup>([''a'', ''b''])}}, then {{mvar|A}} will no longer be closed, but it will be closable, with the closure being its extension defined on {{math|''C''<sup>1</sup>([''a'', ''b''])}}.

== Symmetric operators and self-adjoint operators ==
{{main|Self-adjoint operator}}

An operator ''T'' on a Hilbert space is ''symmetric'' if and only if for each ''x'' and ''y'' in the domain of {{mvar|T}} we have <math>\langle Tx \mid y \rangle = \lang x \mid Ty \rang</math>. A densely defined operator {{mvar|T}} is symmetric if and only if it agrees with its adjoint ''T''<sup>∗</sup> restricted to the domain of ''T'', in other words when ''T''<sup>∗</sup> is an extension of {{mvar|T}}.<ref name="Pedersen-5.1.3">{{ harvnb |Pedersen|1989| loc=5.1.3 }}</ref>

In general, if ''T'' is densely defined and symmetric, the domain of the adjoint ''T''<sup>∗</sup> need not equal the domain of ''T''. If ''T'' is symmetric and the domain of ''T'' and the domain of the adjoint coincide, then we say that ''T'' is ''self-adjoint''.<ref>{{ harvnb |Kato|1995| loc=5.3.3 }}</ref>  Note that, when ''T'' is self-adjoint, the existence of the adjoint implies that ''T'' is densely defined and since ''T''<sup>∗</sup> is necessarily closed, ''T'' is closed.

A densely defined operator ''T'' is ''symmetric'', if the subspace {{math|Γ(''T'')}} (defined in a previous section) is orthogonal to its image {{math|''J''(Γ(''T''))}} under ''J'' (where ''J''(''x'',''y''):=(''y'',-''x'')).<ref group="nb">Follows from {{ harv |Pedersen|1989| loc=5.1.5 }} and the definition via adjoint operators.</ref>

Equivalently, an operator ''T'' is ''self-adjoint'' if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators {{math|''T'' – ''i''}}, {{math|''T'' + ''i''}} are surjective, that is, map the domain of ''T'' onto the whole space ''H''. In other words: for every ''x'' in ''H'' there exist ''y'' and ''z'' in the domain of ''T'' such that {{math|''Ty'' – ''iy'' {{=}} ''x''}} and {{math|''Tz'' + ''iz'' {{=}} ''x''}}.<ref name="Pedersen-5.2.5">{{ harvnb |Pedersen|1989| loc=5.2.5 }}</ref>

An operator ''T'' is ''self-adjoint'', if the two subspaces {{math|Γ(''T'')}}, {{math|''J''(Γ(''T''))}} are orthogonal and their sum is the whole space <math> H \oplus H .</math><ref name="Pedersen-5.1.5" />

This approach does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators.

A symmetric operator is often studied via its [[Cayley transform]].

An operator ''T'' on a complex Hilbert space is symmetric if and only if its quadratic form is real, that is, the number <math> \langle Tx \mid x \rangle </math> is real for all ''x'' in the domain of ''T''.<ref name="Pedersen-5.1.3" />

A densely defined closed symmetric operator ''T'' is self-adjoint if and only if ''T''<sup>∗</sup> is symmetric.<ref name="RS-256">{{ harvnb |Reed|Simon|1980| loc=page 256 }}</ref> It may happen that it is not.<ref name="Pedersen-5.1.16">{{ harvnb |Pedersen|1989| loc=5.1.16 }}</ref><ref name="RS-257-9">{{ harvnb |Reed|Simon|1980| loc=Example on pages 257-259 }}</ref>

A densely defined operator ''T'' is called ''positive''<ref name="Pedersen-5.1.12">{{ harvnb |Pedersen|1989| loc=5.1.12 }}</ref> (or ''nonnegative''<ref name="BSU-25">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=page 25 }}</ref>) if its quadratic form is nonnegative, that is, <math>\langle Tx \mid x \rangle \ge 0 </math> for all ''x'' in the domain of ''T''. Such operator is necessarily symmetric.

The operator ''T''<sup>∗</sup>''T'' is self-adjoint<ref name="Pedersen-5.1.9">{{ harvnb |Pedersen|1989| loc=5.1.9 }}</ref> and positive<ref name="Pedersen-5.1.12" /> for every densely defined, closed ''T''.

The [[Self-adjoint operator#Spectral theorem|spectral theorem]] applies to self-adjoint operators <ref name="Pedersen-5.3.8">{{ harvnb|Pedersen|1989|loc=5.3.8}}</ref> and moreover, to normal operators,<ref name="BSU-89">{{harvnb |Berezansky|Sheftel|Us|1996|loc=page 89}}</ref><ref name="Pedersen-5.3.19">{{ harvnb |Pedersen|1989| loc=5.3.19 }}</ref> but not to densely defined, closed operators in general, since in this case the spectrum can be empty.<ref name="RS-254-E5">{{ harvnb |Reed|Simon|1980| loc=Example 5 on page 254 }}</ref><ref name="Pedersen-5.2.12">{{ harvnb |Pedersen|1989| loc=5.2.12 }}</ref>

A symmetric operator defined everywhere is closed, therefore bounded,<ref name="Pedersen-5.1.4" /> which is the [[Hellinger–Toeplitz theorem]].<ref name="RS-84">{{ harvnb |Reed|Simon|1980| loc=page 84 }}</ref>

==Extension-related==

By definition, an operator ''T'' is an ''extension'' of an operator ''S'' if {{math|Γ(''S'') ⊆ Γ(''T'')}}.<ref name="RS-250">{{ harvnb |Reed|Simon|1980| loc=page 250 }}</ref> An equivalent direct definition: for every ''x'' in the domain of ''S'', ''x'' belongs to the domain of ''T'' and {{math|''Sx'' {{=}} ''Tx''}}.<ref name="Pedersen-5.1.1" /><ref name="RS-250" />

Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at {{slink|Discontinuous linear map#General existence theorem}} and based on the [[Axiom of choice|axiom of choice]]. If the given operator is not bounded then the extension is a [[Discontinuous linear map|discontinuous linear map]]. It is of little use since it cannot preserve important properties of the given operator (see below), and usually is highly non-unique.

An operator ''T'' is called ''closable'' if it satisfies the following equivalent conditions:<ref name="Pedersen-5.1.4" /><ref name="RS-250"/><ref name="BSU-6,7">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=pages 6,7 }}</ref>
* ''T'' has a closed extension;
* the closure of the graph of ''T'' is the graph of some operator;
* for every sequence (''x<sub>n</sub>'') of points from the domain of ''T'' such that ''x<sub>n</sub>'' → 0 and also ''Tx<sub>n</sub>'' → ''y'' it holds that {{math|''y'' {{=}} 0}}.

Not all operators are closable.<ref name="BSU-7">{{ harvnb |Berezansky|Sheftel|Us|1996| loc=page 7 }}</ref>

A closable operator ''T'' has the least closed extension <math> \overline T </math> called the ''closure'' of ''T''. The closure of the graph of ''T'' is equal to the graph of <math> \overline T. </math><ref name="Pedersen-5.1.4" /><ref name="RS-250" /> Other, non-minimal closed extensions may exist.<ref name="Pedersen-5.1.16" /><ref name="RS-257-9" />

A densely defined operator ''T'' is closable if and only if ''T''<sup>∗</sup> is densely defined. In this case <math>\overline T = T^{**} </math> and <math> (\overline T)^* = T^*. </math><ref name="Pedersen-5.1.5" /><ref name="RS-253">{{ harvnb |Reed|Simon|1980| loc=page 253 }}</ref>

If ''S'' is densely defined and ''T'' is an extension of ''S'' then ''S''<sup>∗</sup> is an extension of ''T''<sup>∗</sup>.<ref name="Pedersen-5.1.2">{{ harvnb |Pedersen|1989| loc=5.1.2 }}</ref>

Every symmetric operator is closable.<ref name="Pedersen-5.1.6">{{ harvnb |Pedersen|1989| loc=5.1.6 }}</ref>

A symmetric operator is called ''maximal symmetric'' if it has no symmetric extensions, except for itself.<ref name="Pedersen-5.1.3" /> Every self-adjoint operator is maximal symmetric.<ref name="Pedersen-5.1.3" /> The converse is wrong.<ref name="Pedersen-5.2.6">{{ harvnb |Pedersen|1989| loc=5.2.6 }}</ref>

An operator is called ''essentially self-adjoint'' if its closure is self-adjoint.<ref name="Pedersen-5.1.6" /> An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension.<ref name="RS-256" />

A symmetric operator may have more than one self-adjoint extension, and even a continuum of them.<ref name="RS-257-9" />

A densely defined, symmetric operator ''T'' is essentially self-adjoint if and only if both operators {{math|''T'' – ''i''}}, {{math|''T'' + ''i''}} have dense range.<ref name="RS-257">{{ harvnb |Reed|Simon|1980| loc=page 257 }}</ref>

Let ''T'' be a densely defined operator. Denoting the relation "''T'' is an extension of ''S''" by ''S'' ⊂ ''T'' (a conventional abbreviation for Γ(''S'') ⊆ Γ(''T'')) one has the following.<ref name="RS-255-6">{{ harvnb |Reed|Simon|1980| loc=pages 255, 256 }}</ref>
* If ''T'' is symmetric then ''T'' ⊂ ''T''<sup>∗∗</sup> ⊂ ''T''<sup>∗</sup>.
* If ''T'' is closed and symmetric then ''T'' = ''T''<sup>∗∗</sup> ⊂ ''T''<sup>∗</sup>.
* If ''T'' is self-adjoint then ''T'' = ''T''<sup>∗∗</sup> = ''T''<sup>∗</sup>.
* If ''T'' is essentially self-adjoint then ''T'' ⊂ ''T''<sup>∗∗</sup> = ''T''<sup>∗</sup>.

==Importance of self-adjoint operators==
The class of '''self-adjoint operators''' is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. Self-adjointness is substantially more restricting than these three properties. The famous [[Self-adjoint operator#Spectral theorem|spectral theorem]] holds for self-adjoint operators. In combination with [[Stone's theorem on one-parameter unitary groups]] it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-parameter unitary groups, see {{slink|Self-adjoint operator#Self-adjoint extensions in quantum mechanics}}. Such unitary groups are especially important for describing [[Time evolution|time evolution]] in classical and quantum mechanics.

== See also ==
* {{slink|Hilbert space#Unbounded operators}}
* [[Stone–von Neumann theorem]]
* [[Bounded operator]]

==Notes==
{{reflist|group=nb}}

== References ==
=== Citations ===
{{reflist|22em}}
=== Bibliography ===
{{Refbegin}}
* {{ citation | last1=Berezansky| first1=Y.M. | last2=Sheftel| first2=Z.G. | last3=Us| first3=G.F.| title=Functional analysis | volume=II | year=1996| publisher=Birkhäuser }} (see Chapter 12 "General theory of unbounded operators in Hilbert spaces").
* {{ citation | last1=Brezis | first1=Haïm | title=Analyse fonctionnelle &mdash; Théorie et applications  | year=1983| publisher=Mason |place=Paris |language=fr}}
* {{springer|title=Unbounded operator|id=p/u095090}} <!--Hazewinkel, Michiel, ed. (2001) -->
* {{ citation | last=Hall | first=B.C. | title=Quantum Theory for Mathematicians | year=2013 | series=Graduate Texts in Mathematics
|volume=267 |chapter=Chapter 9. Unbounded Self-adjoint Operators  |publisher=Springer|isbn=978-1461471158}}
* {{ citation | last=Kato | first=Tosio | title=Perturbation theory for linear operators | year=1995 | series=Classics in Mathematics |chapter=Chapter 5. Operators in Hilbert Space  |publisher=Springer-Verlag |isbn=3-540-58661-X}}
* {{Cite book|title=Introductory Functional Analysis With Applications|last=Kreyszig|first=Erwin|publisher=John Wiley & Sons. Inc. | year=1978 | isbn=0-471-50731-8 | location=USA }}
* {{ citation | last=Pedersen | first=Gert K. | title=Analysis now | year=1989 | publisher=Springer }} (see Chapter 5 "Unbounded operators").
* {{ citation | last1=Reed | first1=Michael |  last2=Simon | first2=Barry |  title=Methods of Modern Mathematical Physics | edition=revised and enlarged | volume=1: Functional Analysis | year=1980 | publisher=Academic Press }} (see Chapter 8 "Unbounded operators").
* {{cite book|last=Stone|first=Marshall Harvey|title=Linear Transformations in Hilbert Space and Their Applications to Analysis. Reprint of the 1932 Ed|url=https://books.google.com/books?id=9n2CtOe9FLIC| year=1932| publisher=American Mathematical Society | isbn=978-0-8218-7452-3}}
* {{cite book| last = Teschl| given = Gerald| title=Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators| publisher=American Mathematical Society| place = Providence| year=2009 |url=https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ |isbn=978-0-8218-4660-5 }}
* {{citation|last=von Neumann |first =J. |year=1930|title=Allgemeine Eigenwerttheorie Hermitescher Functionaloperatoren (General Eigenvalue Theory of Hermitian Functional Operators) |journal=Mathematische Annalen |volume=102 |issue=1 |doi=10.1007/BF01782338|s2cid =121249803 }}
* {{citation|last=von Neumann |first=J. |year=1932 |title=Über Adjungierte Funktionaloperatore (On Adjoint Functional Operators) |journal=Annals of Mathematics |series=Second Series |volume=33 |doi=10.2307/1968331 |issue=2 |jstor=1968331}}
* {{ citation | last1=Yoshida| first1=Kôsaku | title=Functional Analysis | year=1980| publisher=Springer |edition=sixth}}
{{refend}}

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