# Unbounded operator

In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in 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, not necessarily the whole space;
• this linear subspace is not necessarily 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 operators, unbounded operators on a given space do not form an 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. Some generalizations to Banach spaces and more general topological vector spaces 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 quantum mechanics.[1] The theory's development is due to John von Neumann[2] and Marshall Stone.[3] Von Neumann introduced using graphs to analyze unbounded operators in 1936.[4]

## Definitions and basic properties

Let X, Y be Banach spaces. An unbounded operator (or simply operator) T : XY is a linear map Template:Mvar from a linear subspace D(T) ⊆ X — the domain of Template:Mvar — to the space Y.[5] Contrary to the usual convention, Template:Mvar may not be defined on the whole space Template:Mvar. Two operators are equal if they have the common domain and they coincide on that common domain.[5]

An operator Template:Mvar is said to be closed if its graph Γ(T) is a closed set.[6] (Here, the graph Γ(T) is a linear subspace of the direct sum XY, defined as the set of all pairs (x, Tx), where Template:Mvar runs over the domain of Template:Mvar ). Explicitly, this means that for every sequence {xn} of points from the domain of Template:Mvar such that xnx and Txny, it holds that Template:Mvar belongs to the domain of Template:Mvar and Tx = y.[6] The closedness can also be formulated in terms of the graph norm: an operator Template:Mvar is closed if and only if its domain D(T) is a complete space with respect to the norm:[7]

${\displaystyle \|x\|_{T}={\sqrt {\|x\|^{2}+\|Tx\|^{2}}}.}$

An operator Template:Mvar is said to be densely defined if its domain is dense in Template:Mvar.[5] This also includes operators defined on the entire space Template:Mvar, since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint and the transpose; see the next section.

If T : XY is closed, densely defined and continuous on its domain, then it is defined on Template:Mvar.[8]

A densely defined operator Template:Mvar on a Hilbert space Template:Mvar is called bounded from below if T + a is a positive operator for some real number Template:Mvar. That is, Tx|x⟩ ≥ −a Template:!!xTemplate:!!2 for all Template:Mvar in the domain of Template:Mvar.[9] If both Template:Mvar and T are bounded from below then Template:Mvar is bounded.[9]

## Example

Let C([0, 1]) denote the space of continuous functions on the interval, and let C1([0, 1]) denote the space of continuously differentiable functions. Define the classical differentiation operator {{ safesubst:#invoke:Unsubst||$B=d/dx}} : C1([0, 1]) → C([0, 1]) by the usual formula: ${\displaystyle \left({\frac {d}{dx}}f\right)(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}},\qquad \forall x\in [0,1].}$ Every differentiable function is continuous, so C1([0, 1]) ⊆ C([0, 1]). Consequently, {{ safesubst:#invoke:Unsubst||$B=d/dx}} : C([0, 1]) → C([0, 1]) is a well-defined unbounded operator, with domain C1([0, 1]).

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

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

The operator is not bounded. For example,

${\displaystyle {\begin{cases}f_{n}:[0,1]\to [-1,1]\\f_{n}(x)=\sin(2\pi nx)\end{cases}}}$

satisfy

${\displaystyle \left\|f_{n}\right\|_{2}={\frac {1}{\sqrt {2}}},}$

but

${\displaystyle \left\|\left({\tfrac {d}{dx}}f_{n}\right)\right\|_{2}={\frac {2\pi n}{\sqrt {2}}}\to \infty .}$

The operator is densely defined, and closed.

The same operator can be treated as an operator ZZ for many Banach spaces Template:Mvar and is still not bounded. However, it is bounded as an operator XY for some pairs of Banach spaces X, Y, and also as operator ZZ for some topological vector spaces Template:Mvar. As an example let IR be an open interval and consider

${\displaystyle {\frac {d}{dx}}:\left(C^{1}(I),\|\cdot \|_{C^{1}}\right)\to \left(C^{0}(I),\|\cdot \|_{\infty }\right),}$

where:

${\displaystyle \|f\|_{C^{1}}=\|f\|_{\infty }+\|f'\|_{\infty }.}$

The adjoint of an unbounded operator can be defined in two equivalent ways. First, it can be defined in a way analogous to how we define the adjoint of a bounded operator. Namely, the adjoint T : H2H1 of Template:Mvar is defined as an operator with the property:

${\displaystyle \langle Tx\mid y\rangle _{2}=\left\langle x\mid T^{*}y\right\rangle _{1},\qquad x\in D(T).}$

More precisely, T is defined in the following way. If Template:Mvar is such that ${\displaystyle x\mapsto \langle Tx\mid y\rangle }$ is a continuous linear functional on the domain of Template:Mvar, then, after extending it to the whole space via the Hahn–Banach theorem, we can find a z such that

${\displaystyle \langle Tx\mid y\rangle _{2}=\langle x\mid z\rangle _{1},\qquad x\in D(T),}$

since the dual of a Hilbert space can be identified with the set of linear functionals given by the inner product. For each y, z is uniquely determined if and only if the linear functional is densely defined; i.e., Template:Mvar is densely defined. Finally, we let Ty = z, completing the construction of T.[10] Note that T exists if and only if Template:Mvar is densely defined.

By definition, the domain of T consists of elements Template:Mvar in H2 such that ${\displaystyle x\mapsto \langle Tx\mid y\rangle }$ is continuous on the domain of Template:Mvar. Consequently, the domain of T could be anything; it could be trivial (i.e., contains only zero).[11] It may happen that the domain of T is a closed hyperplane and T vanishes everywhere on the domain.[12][13] Thus, boundedness of T on its domain does not imply boundedness of Template:Mvar. On the other hand, if T is defined on the whole space then Template:Mvar is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space.[14] If the domain of T is dense, then it has its adjoint T∗∗.[15] A closed densely defined operator Template:Mvar is bounded if and only if T is bounded.[16]

The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator Template:Mvar as follows:[15]

${\displaystyle {\begin{cases}J:H_{1}\oplus H_{2}\to H_{2}\oplus H_{1}\\J(x\oplus y)=-y\oplus x\end{cases}}}$

Since Template:Mvar is an isometric surjection, it is unitary. We then have: J(Γ(T)) is the graph of some operator Template:Mvar if and only if Template:Mvar is densely defined.[17] A simple calculation shows that this "some" Template:Mvar satisfies:

${\displaystyle \langle Tx\mid y\rangle _{2}=\langle x\mid Sy\rangle _{1},}$

for every Template:Mvar in the domain of Template:Mvar. Thus, Template:Mvar is the adjoint of Template:Mvar.

It follows immediately from the above definition that the adjoint T is closed.[15] In particular, a self-adjoint operator (i.e., T = T) is closed. An operator Template:Mvar is closed and densely defined if and only if T∗∗ = T.[18]

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 : H1H2 coincides with the orthogonal complement of the range of the adjoint. That is,[19]

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

von Neumann's theorem states that TT and TT are self-adjoint, and that I + TT and I + TT both have bounded inverses.[20] If T has trivial kernel, Template:Mvar has dense range (by the above identity.) Moreover:

Template:Mvar is surjective if and only if there is a K > 0 such that for all f in D(T).[21] (This is essentially a variant of the so-called closed range theorem.) In particular, Template:Mvar has closed range if and only if T has closed range.

## Symmetric operators and self-adjoint operators

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An operator T on a Hilbert space is symmetric if and only if for each x and y in the domain of Template:Mvar we have ${\displaystyle \langle Tx\mid y\rangle =\langle x\mid Ty\rangle }$. A densely defined operator Template:Mvar 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 Template:Mvar.[26]

In general, the domain of the adjoint T need not equal the domain of T. If the domain of T and the domain of the adjoint coincide, then we say that T is self-adjoint.[27] Note that, when T is self-adjoint, the existence of the adjoint implies that T is dense and since T is necessarily closed, T is closed.

A densely defined operator T is symmetric, if the subspace Γ(T) is orthogonal to its image J(Γ(T)) under J (where J(x,y):=(y,-x)).[28]

Equivalently, an operator T is self-adjoint if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators Ti, 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 Tyiy = x and Tz + iz = x.[29]

An operator T is self-adjoint, if the two subspaces Γ(T), J(Γ(T)) are orthogonal and their sum is the whole space ${\displaystyle H\oplus H.}$[15]

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 ${\displaystyle \langle Tx\mid x\rangle }$ is real for all x in the domain of T.[26]

A densely defined closed symmetric operator T is self-adjoint if and only if T is symmetric.[30] It may happen that it is not.[31][32]

A densely defined operator T is called positive[9] (or nonnegative[33]) if its quadratic form is nonnegative, that is, ${\displaystyle \langle Tx\mid x\rangle \geq 0}$ for all x in the domain of T. Such operator is necessarily symmetric.

The operator TT is self-adjoint[34] and positive[9] for every densely defined, closed T.

The spectral theorem applies to self-adjoint operators [35] and moreover, to normal operators,[36][37] but not to densely defined, closed operators in general, since in this case the spectrum can be empty.[38][39]

A symmetric operator defined everywhere is closed, therefore bounded,[6] which is the Hellinger–Toeplitz theorem.[40]

## Extension-related

By definition, an operator T is an extension of an operator S if Γ(S) ⊆ Γ(T).[41] An equivalent direct definition: for every x in the domain of S, x belongs to the domain of T and Sx = Tx.[5][41]

Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at 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:[6][41][42]

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

Not all operators are closable.[43]

A closable operator T has the least closed extension ${\displaystyle {\overline {T}}}$ called the closure of T. The closure of the graph of T is equal to the graph of ${\displaystyle {\overline {T}}.}$[6][41]

Other, non-minimal closed extensions may exist.[31][32]

A densely defined operator T is closable if and only if T is densely defined. In this case ${\displaystyle {\overline {T}}=T^{**}}$ and ${\displaystyle ({\overline {T}})^{*}=T^{*}.}$[15][44]

If S is densely defined and T is an extension of S then S is an extension of T.[45]

Every symmetric operator is closable.[46]

A symmetric operator is called maximal symmetric if it has no symmetric extensions, except for itself.[26]

Every self-adjoint operator is maximal symmetric.[26] The converse is wrong.[47]

An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension.[30]

An operator may have more than one self-adjoint extension, and even a continuum of them.[32]

A densely defined, symmetric operator T is essentially self-adjoint if and only if both operators Ti, T + i have dense range.[48]

Let T be a densely defined operator. Denoting the relation "T is an extension of S" by ST (a conventional abbreviation for Γ(S) ⊆ Γ(T)) one has the following.[49]

• If T is symmetric then TT∗∗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 TT∗∗ = 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 Self-adjoint operator#Self adjoint extensions in quantum mechanics. Such unitary groups are especially important for describing time evolution in classical and quantum mechanics.

## Notes

1. Template:Harvnb
2. {{#invoke:citation/CS1|citation |CitationClass=citation }}
3. {{#invoke:citation/CS1|citation |CitationClass=book }}
4. {{#invoke:citation/CS1|citation |CitationClass=citation }}
5. Template:Harvnb
6. Template:Harvnb
7. Template:Harvnb
8. Suppose fj is a sequence in the domain of Template:Mvar that converges to gX. Since Template:Mvar is uniformly continuous on its domain, Tfj is Cauchy in Template:Mvar. Thus, ( fj , T fj ) is Cauchy and so converges to some ( f , T f ) since the graph of Template:Mvar is closed. Hence, f  = g, and the domain of Template:Mvar is closed.
9. Template:Harvnb
10. Verifying that T is linear trivial.
11. Template:Harvnb
12. Template:Harvnb
13. Template:Harvnb
14. Proof: being closed, the everywhere defined T is bounded, which implies boundedness of T∗∗, the latter being the closure of Template:Mvar. See also Template:Harv for the case of everywhere defined Template:Mvar.
15. Template:Harvnb
16. Proof: We have: T∗∗ = T. So, if T is bounded, then its adjoint Template:Mvar is bounded.
17. Template:Harvnb
18. Proof: If Template:Mvar is closed densely defined, then T exists and is densely defined. Thus, T∗∗ exists. The graph of Template:Mvar 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 Template:Mvar is densely defined. Since T∗∗ is closed, Template:Mvar is densely defined and closed.
19. Brezis, pp. 28.
20. Yoshida, pp. 200.
21. If Template:Mvar is surjective, then T : (ker T)H2 has bounded inverse, which we denote by Template:Mvar. The estimate then follows since
${\displaystyle \|f\|_{2}^{2}=\left|\langle TSf\mid f\rangle _{2}\right|\leq \|S\|\|f\|_{2}\left\|T^{*}f\right\|_{1}}$
Conversely, suppose the estimate holds. Since T has closed range then, we have: ran(T) = ran(TT*). Since ran(T) is dense, it suffices to show that TT has closed range. If TTfj is convergent, then fj is convergent by the estimate since
${\displaystyle \|T^{*}f_{j}\|_{1}^{2}=|\langle T^{*}f_{j}\mid T^{*}f_{j}\rangle _{1}|\leq \|TT^{*}f_{j}\|_{2}\|f_{j}\|_{2}.}$
Say, fj  → g. Since TT is self-adjoint; thus, closed, (von Neumann's theorem), TTfj  → TTg. QED
22. Yoshida, pp. 195.
23. Template:Harvnb
24. Yoshida, pp. 193.
25. Yoshida, pp. 196.
26. Template:Harvnb
27. Template:Harvnb
28. Follows from Template:Harv and the definition via adjoint operators.
29. Template:Harvnb
30. Template:Harvnb
31. Template:Harvnb
32. Template:Harvnb
33. Template:Harvnb
34. Template:Harvnb
35. Template:Harvnb
36. Template:Harvnb
37. Template:Harvnb
38. Template:Harvnb
39. Template:Harvnb
40. Template:Harvnb
41. Template:Harvnb
42. Template:Harvnb
43. Template:Harvnb
44. Template:Harvnb
45. Template:Harvnb
46. Template:Harvnb
47. Template:Harvnb
48. Template:Harvnb
49. Template:Harvnb

## References

• {{#invoke:citation/CS1|citation

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|CitationClass=citation }} (see Chapter 5 "Unbounded operators").

• {{#invoke:citation/CS1|citation

|CitationClass=citation }} (see Chapter 8 "Unbounded operators").

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