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.
The theory of unbounded operators developed in the late 1920's and early 1930's as part of developing a rigorous mathematical framework for quantum mechanics. The theory's development is due to John von Neumann and Marshall Stone. Von Neumann introduced using graphs to analyze unbounded operators in 1936.
Definitions and basic properties
Let B1 and B2 be Banach spaces. An unbounded operator (or simply operator) T : B1 → B2 is a linear map T from a linear subspace D(T) of B1 — the domain of T — to the space B2. Contrary to the usual convention, T may not be defined on the whole space B1. Two operators are equal if they have the common domain and they coincide on that common domain.
An operator T is said to be closed if its graph Γ(T) is a closed set. (Here, the graph Γ(T) is a linear subspace of the direct sum B1 ⊕ B2, defined as the set of all pairs (x, Tx), where x runs over the domain of T). Explicitly, this means that for every sequence (xn) of points from the domain of T such that xn converge to some x and Txn converge to some y, it holds that x belongs to the domain of T and Tx = y. The closedness can also be formulated in terms of the graph norm: an operator T is closed if and only if its domain D(T) is a complete space with respect to the norm:
An operator T is said to be densely defined if its domain is dense in B1. This also includes operators defined on the entire space B1, 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.
A densely defined operator T on a Hilbert space H is called bounded from below if T + a is a positive operator for some real number a. That is, ⟨Tx|x⟩ ≥ −a·||x||2 for all x in the domain of T. If both T and (–T) are bounded from below then T is bounded.
Let denote the space of continuous functions on the interval, and let denote the space of continuously differentiable functions. Define the classical differentiation operator by the usual formula:
for all .
Every differentiable function is continuous, so . Consequently, is a well-defined unbounded operator, with domain .
This is a linear operator, since a linear combination af+bg of two continuously differentiable functions f, g is also continuously differentiable, and
The operator is not bounded. For example, the functions fn defined on [0,1] by satisfy but
The operator is densely defined, and closed.
The same operator can be treated as an operator B → B for many Banach spaces B and is still not bounded. However, it is bounded as an operator B1 → B2 for some pairs of Banach spaces B1, B2, and also as operator B → B for some topological vector spaces B. As an example consider , for some open interval and the norm being .
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∗ : H2 → H1 of T is defined as an operator with the property:
More precisely, T∗ is defined in the following way. If y is such that is a continuous linear functional on the domain of T, then, after extending it to the whole space via the Hahn–Banach theorem, we can find a z such that
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., T is densely defined. Finally, we let T∗y = z, completing the construction of T∗. Note that T∗ exists if and only if T is densely defined.
By definition, the domain of T∗ consists of elements such that is continuous on the domain of T. Consequently, the domain of T∗ could be anything; it could be trivial (i.e., contains only zero) It may happen that the domain of T∗ is a closed hyperplane and T∗ vanishes everywhere on the domain. Thus, boundedness of T∗ on its domain does not imply boundedness of T. On the other hand, if T∗ is defined on the whole space then T is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space. If the domain of T∗ is dense, then it has its adjoint T∗∗. A closed densely defined operator T is bounded if and only if T∗ is bounded.
The other equivalent definition of the adjoint can be obtained by noticing a general fact: define a linear operator by . (Since is an isometric surjection, it is unitary.) We then have: is the graph of some operator S if and only if is densely defined. A simple calculation shows that this "some" S satisfies: for every x in the domain of T. Thus, S is the adjoint of T.
It follows immediately from the above definition that the adjoint T∗ is closed. In particular, a self-adjoint operator (i.e., T = T∗) is closed. An operator T is closed and densely defined if and only if T∗∗ = T.
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 : H1 → H2 coincides with the orthogonal complement of the range of the adjoint. That is,
von Neumann's theorem states that T∗T and TT∗ are self-adjoint, and that I + T∗T and I + TT∗ both have bounded inverses. If has trivial kernel, has dense range (by the above identity.) Moreover, T is surjective if and only if there is a such that
- for every .
(This is essentially a variant of the so-called closed range theorem.) In particular, T has closed range if and only if T∗ has closed range.
In contrast to the bounded case, it is not necessary that we have: (TS)∗ = S∗T∗, since, for example, it is even possible that (TS)∗ doesn't exist. This is, however, the case if, for example, T is bounded.
- T∗T = T T∗;
- the domain of T is equal to the domain of T∗, and for every x in this domain;
- there exist self-adjoint operators A, B such that T = A + iB, T∗ = A – iB, and for every x in the domain of T.
Every self-adjoint operator is normal.
Let T : B1 → B2 be an operator between Banach spaces. Then the transpose (or dual) of T is an operator satisfying:
for all x in B1 and y in B2*. Here, we used the notation: .
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:
given by where . Through this isomorphism, the transpose T' relates to the adjoint T∗ in the following way:
where . (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 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.
Given a linear operator , not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of , and we say that is closable. Denote the closure of by It follows easily that is the restriction of to
A core of a closable operator is a subset of such that the closure of the restriction of to is
Any closed linear operator defined on the whole space is bounded. This is the closed graph theorem. Additionally, the following properties are easily checked:
- If is closed then is closed where is a scalar and is the identity function;
- If is closed, then its kernel (or nullspace) is a closed subspace of ;
- If is closed and injective, then its inverse is also closed;
- An operator admits a closure if and only if for every pair of sequences and in both converging to , such that both and converge, one has .
Consider the derivative operator
If one takes to be instead the set of all infinitely differentiable functions, A will no longer be closed, but it will be closable, with the closure being its extension defined on
Symmetric operators and self-adjoint operators
An operator T on a Hilbert space is symmetric if and only if for each x and y in the domain of T we have . A densely definied operator T is symmetric if and only if it agrees with its adjoint restricted to the domain of T, in other words when T∗ is an extension of T. 
In general, the domain of the adjoint 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. Note that, when T is self-adjoint, the existence of the adjoint implies that T is dense and since is necessarily closed, 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)).
Equivalently, an operator T is self-adjoint if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators T – i, 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 Ty – iy = x and Tz + iz = x.
An operator T is self-adjoint, if the two subspaces Γ ( T ), J ( Γ ( T ) ) are orthogonal and their sum is the whole space 
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 is real for all x in the domain of T.
The spectral theorem applies to self-adjoint operators  and moreover, to normal operators, but not to densely defined, closed operators in general, since in this case the spectrum can be empty.
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.
- 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 converge to 0 and also Txn converge to some y it holds that y = 0.
Not all operators are closable.
If S is densely defined and T is an extension of S then S∗ is an extension of T∗.
Every symmetric operator is closable.
A symmetric operator is called maximal symmetric if it has no symmetric extensions, except for itself.
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.
An 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 T – i, T + i have dense range.
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.
- 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∗.
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 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.
- Reed & Simon 1980, Notes to Chapter VIII, page 305
- von Neumann, J. (1930), "Allgemeine Eigenwerttheorie Hermitescher Functionaloperatoren (General Eigenvalue Theory of Hermitian Functional Operators)", Mathematische Annalen 102 (1): 49–131, doi:10.1007/BF01782338
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- Pedersen 1989, 5.1.1
- Pedersen 1989, 5.1.4
- Berezansky, Sheftel & Us 1996, page 5
- Suppose fj is a sequence in the domain of T that converges to g ∈ B1. Since T is uniformly continuous on its domain, Tfj is Cauchy in B2. Thus, (fj, Tfj) is Cauchy and so converges to some (f, Tf) since the graph of T is closed. Hence, f = g, and the domain of T is closed.
- Pedersen 1989, 5.1.12
- Verifying that T∗ is linear trivial.
- Berezansky, Sheftel & Us 1996, Example 3.2 on page 16
- Reed & Simon 1980, page 252
- Berezansky, Sheftel & Us 1996, Example 3.1 on page 15
- Proof: being closed, the everywhere defined T∗ is bounded, which implies boundedness of T∗∗, the latter being the closure of T. See also (Pedersen 1989, 2.3.11) for the case of everywhere defined T.
- Pedersen 1989, 5.1.5
- Proof: We have: . So, if is bounded, then its adjoint is bounded.
- Berezansky, Sheftel & Us 1996, page 12
- Proof: If T is closed densely defined, then T∗ exists and is densely defined. Thus, T∗∗ exists. The graph of 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 T is densely defined. Since T∗∗ is closed, T is densely defined and closed.
- Brezis, pp. 28.
- Yoshida, pp. 200.
- If T is surjective, then has bounded inverse, which we denote by S. The estimate then follows since
- Yoshida, pp. 195.
- Pedersen 1989, 5.1.11
- Yoshida, pp. 193.
- Yoshida, pp. 196.
- Pedersen 1989, 5.1.3
- Kato 1995, 5.3.3
- Follows from (Pedersen 1989, 5.1.5) and the definition via adjoint operators.
- Pedersen 1989, 5.2.5
- Reed & Simon 1980, page 256
- Pedersen 1989, 5.1.16
- Reed & Simon 1980, Example on pages 257-259
- Berezansky, Sheftel & Us 1996, page 25
- Pedersen 1989, 5.1.9
- Pedersen 1989, 5.3.8
- Berezansky, Sheftel & Us 1996, page 89
- Pedersen 1989, 5.3.19
- Reed & Simon 1980, Example 5 on page 254
- Pedersen 1989, 5.2.12
- Reed & Simon 1980, page 84
- Reed & Simon 1980, page 250
- Berezansky, Sheftel & Us 1996, pages 6,7
- Berezansky, Sheftel & Us 1996, page 7
- Reed & Simon 1980, page 253
- Pedersen 1989, 5.1.2
- Pedersen 1989, 5.1.6
- Pedersen 1989, 5.2.6
- Reed & Simon 1980, page 257
- Reed & Simon 1980, pages 255, 256
- Berezansky, Y.M.; Sheftel, Z.G.; Us, G.F. (1996), Functional analysis II, Birkhäuser (see Chapter 12 "General theory of unbounded operators in Hilbert spaces").
- Brezis, Haïm (1983), Analyse fonctionnelle — Théorie et applications (in French), Paris: Mason
- Hazewinkel, Michiel, ed. (2001), "Unbounded operator", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Hall, B.C. (2013), "Chapter 9. Unbounded Self-adjoint Operators", Quantum Theory for Mathematicians, Graduate Texts in Mathematics, Springer
- Kato, Tosio (1995), "Chapter 5. Operators in Hilbert Space", Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, ISBN 3-540-58661-X
- Pedersen, Gert K. (1989), Analysis now, Springer (see Chapter 5 "Unbounded operators").
- Reed, Michael; Simon, Barry (1980), Methods of Modern Mathematical Physics, 1: Functional Analysis (revised and enlarged ed.), Academic Press (see Chapter 8 "Unbounded operators").
- Yoshida, Kôsaku (1980), Functional Analysis (sixth ed.), Springer