[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 1932.
[6] (Here, the graph Γ(T) is a linear subspace of the direct sum X ⊕ Y, defined as the set of all pairs (x, Tx), where x runs over the domain of T .)
[6] 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:[7] An operator T is said to be densely defined if its domain is dense in X.
The denseness of the domain is necessary and sufficient for the existence of the adjoint (if X and Y are Hilbert spaces) and the transpose; see the sections below.
[nb 1] A densely defined symmetric[clarification needed] operator T on a Hilbert space H is called bounded from below if T + a is a positive operator for some real number a.
[8] Let C([0, 1]) denote the space of continuous functions on the unit interval, and let C1([0, 1]) denote the space of continuously differentiable functions.
[clarification needed] As an example let I ⊂ R be an open interval and consider where: The adjoint of an unbounded operator can be defined in two equivalent ways.
and after extending the linear functional to the whole space via the Hahn–Banach theorem, it is possible to find some
since Riesz representation theorem allows the continuous dual of the Hilbert space
[nb 3] The other equivalent definition of the adjoint can be obtained by noticing a general fact.
is densely defined (for essentially the same reason as to adjoints, as discussed above.)
(For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.)
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.
A core (or essential domain) of a closable operator is a subset C of D(A) such that the closure of the restriction of A to C is A.
Consider the derivative operator A = d/dx where X = Y = C([a, b]) is the Banach space of all continuous functions on an interval [a, b].
[20] On the other hand if D(A) = C∞([a, b]), then A will no longer be closed, but it will be closable, with the closure being its extension defined on C1([a, b]).
[nb 6] 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.
[12] This approach does not cover non-densely defined closed operators.
An operator T on a complex Hilbert space is symmetric if and only if the number
[32][33] A symmetric operator defined everywhere is closed, therefore bounded,[6] which is the Hellinger–Toeplitz theorem.
[5][35] 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.
[43] 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.
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.