Unbounded operator#Closed linear operators

The theory of unbounded operators developed in the late 1920s and early 1930s as part of developing a rigorous mathematical framework for quantum mechanics.{{harvnb|Reed|Simon|1980|loc=Notes to Chapter VIII, page 305}} The theory's development is due to John von Neumann{{harvnb | von Neumann | 1930 | pp=49–131}} and Marshall Stone.{{ harvnb | Stone | 1932 }} Von Neumann introduced using graphs to analyze unbounded operators in 1932.{{ harvnb | von Neumann | 1932 | pp = 294–310 }}

Let {{math|X, Y}} be Banach spaces. An unbounded operator (or simply operator) {{math|T : D(T) → Y}} is a linear map {{mvar|T}} from a linear subspace {{math|D(T) ⊆ X}}—the domain of {{mvar|T}}—to the space {{math|Y}}.{{harvnb|Pedersen|1989|loc=5.1.1}} 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.{{ harvnb |Pedersen|1989| loc=5.1.4 }} (Here, the graph {{math|Γ(T)}} is a linear subspace of the 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}} .) Explicitly, this means that for every sequence {{math|{x_n} }} of points from the domain of {{mvar|T}} such that {{math|x_n → x}} and {{math|Tx_n → y}}, it holds that {{mvar|x}} belongs to the domain of {{mvar|T}} and {{math|Tx {{=}} y}}. 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:{{ harvnb |Berezansky|Sheftel|Us|1996| loc=page 5 }}

: $\backslash |x\backslash |\_T\; =\; \backslash sqrt\{\; \backslash |x\backslash |^2\; +\; \backslash |Tx\backslash |^2\; \}.$

An operator {{mvar|T}} is said to be densely defined if its domain is dense in {{mvar|X}}. 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 : D(T) → Y}} is closed, densely defined and continuous on its domain, then its domain is all of {{mvar|X}}.Suppose f_j 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_j is Cauchy in {{mvar|Y}}. Thus, {{math|( f_j , T f_j )}} is Cauchy and so converges to some {{math|( f , T f )}} since the graph of {{mvar|T}} is closed. Hence, {{math| f  {{=}} g}}, and the domain of {{mvar|T}} is closed.

A densely defined symmetric{{clarify|At this point, symmetric isn’t defined. Maybe we should move the paragraph down.|date=July 2024}} 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{{!!}}²}} for all {{mvar|x}} in the domain of {{mvar|T}} (or alternatively {{math|⟨Tx{{!}}x⟩ ≥ a {{!!}}x{{!!}}²}} since {{math|a}} is arbitrary). If both {{mvar|T}} and {{math|−T}} are bounded from below then {{mvar|T}} is bounded.

Let {{math|C([0, 1])}} denote the space of continuous functions on the unit interval, and let {{math|C¹([0, 1])}} denote the space of continuously differentiable functions. We equip $C([0,1])$ with the supremum norm, $\backslash |\backslash cdot\backslash |\_\{\backslash infty\}$, making it a Banach space. Define the classical differentiation operator {{math|{{sfrac|d|dx}} : C¹([0, 1]) → C([0, 1])}} by the usual formula:

: $\backslash left\; (\backslash frac\{d\}\{dx\}f\; \backslash right\; )(x)\; =\; \backslash lim\_\{h\; \backslash to\; 0\}\; \backslash frac\{f(x+h)\; -\; f(x)\}\{h\},\; \backslash qquad\; \backslash forall\; x\; \backslash in\; [0,\; 1].$

Every differentiable function is continuous, so {{math|C¹([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|C¹([0, 1])}}. For this, we need to show that $\backslash frac\{d\}\{dx\}$ is linear and then, for example, exhibit some $\backslash \{f\_n\backslash \}\_n\; \backslash subset\; C^1([0,1])$ such that $\backslash |f\_n\backslash |\_\backslash infty=1$ and $\backslash sup\_n\; \backslash |\backslash frac\{d\}\{dx\}\; f\_n\backslash |\_\backslash infty=+\backslash infty$.

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

:$\backslash left\; (\backslash tfrac\{d\}\{dx\}\; \backslash right\; )(af+bg)=\; a\; \backslash left\; (\backslash tfrac\{d\}\{dx\}\; f\; \backslash right\; )\; +\; b\; \backslash left\; (\backslash tfrac\{d\}\{dx\}\; g\; \backslash right\; ).$

The operator is not bounded. For example,

:$\backslash begin\{cases\}\; f\_n\; :\; [0,\; 1]\; \backslash to\; [-1,\; 1]\; \backslash \backslash \; f\_n(x)\; =\; \backslash sin\; (2\backslash pi\; n\; x)\; \backslash end\{cases\}$

satisfy

:$\backslash left\; \backslash |f\_n\; \backslash right\; \backslash |\_\{\backslash infty\}\; =\; 1,$

but

:$\backslash left\; \backslash |\; \backslash left\; (\backslash tfrac\{d\}\{dx\}\; f\_n\; \backslash right\; )\; \backslash right\; \backslash |\_\{\backslash infty\}\; =\; 2\backslash pi\; n\; \backslash to\; \backslash infty$

as $n\backslash to\backslash infty$.

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

:$\backslash frac\{d\}\{dx\}\; :\; \backslash left\; (C^1\; (I),\; \backslash |\backslash cdot\; \backslash |\_\{C^1\}\; \backslash right\; )\; \backslash to\; \backslash left\; (\; C\; (I),\; \backslash |\; \backslash cdot\; \backslash |\_\{\backslash infty\}\; \backslash right),$

where:

:$\backslash |\; f\; \backslash |\_\{C^1\}\; =\; \backslash |\; f\; \backslash |\_\{\backslash infty\}\; +\; \backslash |\; f\text{'}\; \backslash |\_\{\backslash infty\}.$

The adjoint of an unbounded operator can be defined in two equivalent ways. Let $T\; :\; D(T)\; \backslash subseteq\; H\_1\; \backslash to\; H\_2$ 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 $T^*\; :\; D\backslash left(T^*\backslash right)\; \backslash subseteq\; H\_2\; \backslash to\; H\_1$ of {{mvar|T}} is defined as an operator with the property:

$$\backslash langle\; Tx\; \backslash mid\; y\; \backslash rangle\_2\; =\; \backslash left\; \backslash langle\; x\; \backslash mid\; T^*y\; \backslash right\; \backslash rangle\_1,\; \backslash qquad\; x\; \backslash in\; D(T).$$

More precisely, $T^*\; y$ is defined in the following way. If $y\; \backslash in\; H\_2$ is such that $x\; \backslash mapsto\; \backslash langle\; Tx\; \backslash mid\; y\; \backslash rangle$ is a continuous linear functional on the domain of {{mvar|T}}, then $y$ is declared to be an element of $D\backslash left(T^*\backslash right),$ and after extending the linear functional to the whole space via the Hahn–Banach theorem, it is possible to find some $z$ in $H\_1$ such that

$$\backslash langle\; Tx\; \backslash mid\; y\; \backslash rangle\_2\; =\; \backslash langle\; x\; \backslash mid\; z\; \backslash rangle\_1,\; \backslash qquad\; x\; \backslash in\; D(T),$$

since Riesz representation theorem allows the continuous dual of the Hilbert space $H\_1$ to be identified with the set of linear functionals given by the inner product. This vector $z$ is uniquely determined by $y$ if and only if the linear functional $x\; \backslash mapsto\; \backslash langle\; Tx\; \backslash mid\; y\; \backslash rangle$ is densely defined; or equivalently, if {{mvar|T}} is densely defined. Finally, letting $T^*\; y\; =\; z$ completes the construction of $T^*,$ which is necessarily a linear map. The adjoint $T^*\; y$ exists if and only if {{mvar|T}} is densely defined.

By definition, the domain of $T^*$ consists of elements $y$ in $H\_2$ such that $x\; \backslash mapsto\; \backslash langle\; Tx\; \backslash mid\; y\; \backslash rangle$ is continuous on the domain of {{mvar|T}}. Consequently, the domain of $T^*$ could be anything; it could be trivial (that is, contains only zero).{{ harvnb |Berezansky|Sheftel|Us|1996| loc=Example 3.2 on page 16 }} It may happen that the domain of $T^*$ is a closed hyperplane and $T^*$ vanishes everywhere on the domain.{{ harvnb |Reed|Simon|1980| loc=page 252 }}{{harvnb|Berezansky|Sheftel|Us|1996|loc=Example 3.1 on page 15 }} Thus, boundedness of $T^*$ on its domain does not imply boundedness of {{mvar|T}}. On the other hand, if $T^*$ 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.Proof: being closed, the everywhere defined $T^*$ is bounded, which implies boundedness of $T^\{**\},$ 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}}. If the domain of $T^*$ is dense, then it has its adjoint $T^\{**\}.$ A closed densely defined operator {{mvar|T}} is bounded if and only if $T^*$ is bounded.Proof: $T^\{**\}\; =\; T.$ So if $T^*$ is bounded then its adjoint {{mvar|T}} is bounded.

The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator $J$ as follows:{{ harvnb |Pedersen|1989| loc=5.1.5 }}

$$\backslash begin\{cases\}\; J:\; H\_1\; \backslash oplus\; H\_2\; \backslash to\; H\_2\; \backslash oplus\; H\_1\; \backslash \backslash \; J(x\; \backslash oplus\; y)\; =\; -y\; \backslash oplus\; x\; \backslash end\{cases\}$$

Since $J$ is an isometric surjection, it is unitary. Hence: $J(\backslash Gamma(T))^\{\backslash bot\}$ is the graph of some operator $S$ if and only if {{mvar|T}} is densely defined.{{harvnb|Berezansky|Sheftel|Us|1996| loc=page 12}} A simple calculation shows that this "some" $S$ satisfies:

$$\backslash langle\; Tx\; \backslash mid\; y\; \backslash rangle\_2\; =\; \backslash langle\; x\; \backslash mid\; Sy\; \backslash rangle\_1,$$

for every {{mvar|x}} in the domain of {{mvar|T}}. Thus $S$ is the adjoint of {{mvar|T}}.

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

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 $T\; :\; H\_1\; \backslash to\; H\_2$ coincides with the orthogonal complement of the range of the adjoint. That is,{{ harvnb | Brezis | 1983|p=28}}

$$\backslash operatorname\{ker\}(T)\; =\; \backslash operatorname\{ran\}(T^*)^\backslash bot.$$

von Neumann's theorem states that $T^*\; T$ and $T\; T^*$ are self-adjoint, and that $I\; +\; T^*\; T$ and $I\; +\; T\; T^*$ both have bounded inverses.{{harvnb | Yoshida | 1980| p=200 }} If $T^*$ has trivial kernel, {{mvar|T}} has dense range (by the above identity.) Moreover:

:{{mvar|T}} is surjective if and only if there is a $K\; >\; 0$ such that $\backslash |f\backslash |\_2\; \backslash leq\; K\; \backslash left\backslash |T^*\; f\backslash right\backslash |\_1$ for all $f$ in $D\backslash left(T^*\backslash right).$If $T$ is surjective then $T\; :\; (\backslash ker\; T)^\{\backslash bot\}\; \backslash to\; H\_2$ has bounded inverse, denoted by $S.$ The estimate then follows since

$$\backslash |f\backslash |\_2^2\; =\; \backslash left\; |\backslash langle\; TSf\; \backslash mid\; f\; \backslash rangle\_2\; \backslash right\; |\; \backslash leq\; \backslash |S\backslash |\; \backslash |f\backslash |\_2\; \backslash left\; \backslash |T^*f\; \backslash right\; \backslash |\_1$$

Conversely, suppose the estimate holds. Since $T^*$ has closed range, it is the case that $\backslash operatorname\{ran\}(T)\; =\; \backslash operatorname\{ran\}\backslash left(T\; T^*\backslash right).$ Since $\backslash operatorname\{ran\}(T)$ is dense, it suffices to show that $T\; T^*$ has closed range. If $T\; T^*\; f\_j$ is convergent then $f\_j$ is convergent by the estimate since

$$\backslash |T^*f\_j\backslash |\_1^2\; =\; |\; \backslash langle\; T^*f\_j\; \backslash mid\; T^*f\_j\; \backslash rangle\_1|\; \backslash leq\; \backslash |TT^*f\_j\backslash |\_2\; \backslash |f\_j\backslash |\_2.$$

Say, $f\_j\; \backslash to\; g.$ Since $T\; T^*$ is self-adjoint; thus, closed, (von Neumann's theorem), $T\; T^*\; f\_j\; \backslash to\; T\; T^*\; g.$ QED (This is essentially a variant of the so-called closed range theorem.) In particular, {{mvar|T}} has closed range if and only if $T^*$ has closed range.

In contrast to the bounded case, it is not necessary that $(T\; S)^*\; =\; S^*\; T^*,$ since, for example, it is even possible that $(T\; S)^*$ does not exist.{{Citation needed|date=July 2009}} This is, however, the case if, for example, {{mvar|T}} is bounded.{{harvnb | Yoshida|1980| p= 195}}.

A densely defined, closed operator {{mvar|T}} is called normal if it satisfies the following equivalent conditions:{{ harvnb |Pedersen|1989| loc=5.1.11 }}

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

Every self-adjoint operator is normal.

{{See also|Transpose of a linear map}}

Let $T\; :\; B\_1\; \backslash to\; B\_2$ be an operator between Banach spaces. Then the transpose (or dual) $\{\}^t\; T:\; \{B\_2\}^*\; \backslash to\; \{B\_1\}^*$ of $T$ is the linear operator satisfying:

$$\backslash langle\; T\; x,\; y\text{'}\; \backslash rangle\; =\; \backslash langle\; x,\; \backslash left(\{\}^t\; T\backslash right)\; y\text{'}\; \backslash rangle$$

for all $x\; \backslash in\; B\_1$ and $y\; \backslash in\; B\_2^*.$ Here, we used the notation: $\backslash langle\; x,\; x\text{'}\; \backslash rangle\; =\; x\text{'}(x).${{harvnb | Yoshida|1980 | p= 193}}

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

For any Hilbert space $H,$ there is the anti-linear isomorphism:

$$J:\; H^*\; \backslash to\; H$$

given by $J\; f\; =\; y$ where $f(x)\; =\; \backslash langle\; x\; \backslash mid\; y\; \backslash rangle\_H,\; (x\; \backslash in\; H).$

Through this isomorphism, the transpose $\{\}^t\; T$ relates to the adjoint $T^*$ in the following way:{{harvnb | Yoshida | 1980 | p = 196}}

$$T^*\; =\; J\_1\; \backslash left(\{\}^t\; T\backslash right)\; J\_2^\{-1\},$$

where $J\_j:\; H\_j^*\; \backslash to\; H\_j$. (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.

{{Main|Closed linear operator}}

Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators.

Let {{math|X, Y}} be two Banach spaces. A linear operator {{math|A : D(A) ⊆ X → Y}} is closed if for every sequence {{math|{x_n} }} in {{math|D(A)}} converging to {{mvar|x}} in {{mvar|X}} such that {{math|Ax_n → 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 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 restriction of {{math|{{overline|A}}}} to {{math|D(A)}}.

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

Consider the derivative operator {{math|1=A = {{sfrac|d|dx}}}} where {{math|1=X = Y = C([a, b])}} is the Banach space of all continuous functions on an interval {{math|[a, b]}}.

If one takes its domain {{math|D(A)}} to be {{math|C¹([a, b])}}, then {{mvar|A}} is a closed operator which is not bounded.{{ harvnb | Kreyszig | 1978 | p = 294}}

On the other hand if {{math|1=D(A) = smooth function{{!}}C^∞([a, b])}}, then {{mvar|A}} will no longer be closed, but it will be closable, with the closure being its extension defined on {{math|C¹([a, b])}}.

{{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 $\backslash langle\; Tx\; \backslash mid\; y\; \backslash rangle\; =\; \backslash lang\; x\; \backslash mid\; Ty\; \backslash rang$. A densely defined operator {{mvar|T}} is symmetric if and only if it agrees with its adjoint T^∗ restricted to the domain of T, in other words when T^∗ is an extension of {{mvar|T}}.{{ harvnb |Pedersen|1989| loc=5.1.3 }}

In general, if T is densely defined and symmetric, the domain of the adjoint T^∗ 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.{{ harvnb |Kato|1995| loc=5.3.3 }} Note that, when T is self-adjoint, the existence of the adjoint implies that T is densely defined and since T^∗ 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)).Follows from {{ harv |Pedersen|1989| loc=5.1.5 }} and the definition via adjoint operators.

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}}.{{ harvnb |Pedersen|1989| loc=5.2.5 }}

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 $H\; \backslash oplus\; H\; .$

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 the number $\backslash langle\; Tx\; \backslash mid\; x\; \backslash rangle$ is real for all x in the domain of T.

A densely defined closed symmetric operator T is self-adjoint if and only if T^∗ is symmetric.{{ harvnb |Reed|Simon|1980| loc=page 256 }} It may happen that it is not.{{ harvnb |Pedersen|1989| loc=5.1.16 }}{{ harvnb |Reed|Simon|1980| loc=Example on pages 257-259 }}

A densely defined operator T is called positive{{ harvnb |Pedersen|1989| loc=5.1.12 }} (or nonnegative{{ harvnb |Berezansky|Sheftel|Us|1996| loc=page 25 }}) if its quadratic form is nonnegative, that is, $\backslash langle\; Tx\; \backslash mid\; x\; \backslash rangle\; \backslash ge\; 0$ for all x in the domain of T. Such operator is necessarily symmetric.

The operator T^∗T is self-adjoint{{ harvnb |Pedersen|1989| loc=5.1.9 }} and positive for every densely defined, closed T.

The spectral theorem applies to self-adjoint operators {{ harvnb|Pedersen|1989|loc=5.3.8}} and moreover, to normal operators,{{harvnb |Berezansky|Sheftel|Us|1996|loc=page 89}}{{ harvnb |Pedersen|1989| loc=5.3.19 }} but not to densely defined, closed operators in general, since in this case the spectrum can be empty.{{ harvnb |Reed|Simon|1980| loc=Example 5 on page 254 }}{{ harvnb |Pedersen|1989| loc=5.2.12 }}

A symmetric operator defined everywhere is closed, therefore bounded, which is the Hellinger–Toeplitz theorem.{{ harvnb |Reed|Simon|1980| loc=page 84 }}

{{See also|Extensions of symmetric operators}}

By definition, an operator T is an extension of an operator S if {{math|Γ(S) ⊆ Γ(T)}}.{{ harvnb |Reed|Simon|1980| loc=page 250 }} An equivalent direct definition: for every x in the domain of S, x belongs to the domain of T and {{math|Sx {{=}} Tx}}.

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. If the given operator is not bounded then the extension is a 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:{{ harvnb |Berezansky|Sheftel|Us|1996| loc=pages 6,7 }}

• T has a closed extension;
• the closure of the graph of T is the graph of some operator;
• for every sequence (x_n) of points from the domain of T such that x_n → 0 and also Tx_n → y it holds that {{math|y {{=}} 0}}.

Not all operators are closable.{{ harvnb |Berezansky|Sheftel|Us|1996| loc=page 7 }}

A closable operator T has the least closed extension $\backslash overline\; T$ called the closure of T. The closure of the graph of T is equal to the graph of $\backslash overline\; T.$ Other, non-minimal closed extensions may exist.

A densely defined operator T is closable if and only if T^∗ is densely defined. In this case $\backslash overline\; T\; =\; T^\{**\}$ and $(\backslash overline\; T)^*\; =\; T^*.${{ harvnb |Reed|Simon|1980| loc=page 253 }}

If S is densely defined and T is an extension of S then S^∗ is an extension of T^∗.{{ harvnb |Pedersen|1989| loc=5.1.2 }}

Every symmetric operator is closable.{{ harvnb |Pedersen|1989| loc=5.1.6 }}

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

An operator is called essentially self-adjoint if its closure is self-adjoint. An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension.

A symmetric operator may have more than one self-adjoint extension, and even a continuum of them.

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.{{ harvnb |Reed|Simon|1980| loc=page 257 }}

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.{{ harvnb |Reed|Simon|1980| loc=pages 255, 256 }}

• If T is symmetric then T ⊂ T^∗∗ ⊂ T^∗.
• If T is closed and symmetric then T = T^∗∗ ⊂ T^∗.
• If T is self-adjoint then T = T^∗∗ = T^∗.
• If T is essentially self-adjoint then T ⊂ T^∗∗ = T^∗.

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 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 in classical and quantum mechanics.

• {{slink|Hilbert space#Unbounded operators}}
• Stone–von Neumann theorem
• Bounded operator

{{reflist|group=nb}}

{{reflist|22em}}

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• {{ citation | last=Pedersen | first=Gert K. | title=Analysis now | year=1989 | publisher=Springer }} (see Chapter 5 "Unbounded operators").
• {{ citation | last1=Reed | first1=Michael | author1-link=Michael C. Reed | last2=Simon | first2=Barry | author2-link=Barry Simon | 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|author-link=Gerald Teschl| 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}}

{{PlanetMath attribution|id=4526|title=Closed operator}}

{{Spectral theory}}

{{Hilbert space}}

{{Functional analysis}}

{{Boundedness and bornology}}

{{DEFAULTSORT:Unbounded Operator}}

Category:Linear operators

Category:Operator theory

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