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Concept# Unbounded operator

Summary

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 1920s and early 1930s 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 1932.
Let X, Y be Banach spaces. An unbounded operator (or simply operator) T : D(T) → Y is a linear map T from a linear subspace D(T) ⊆ X—the domain of T—to the space Y. Contrary to the usual convention, T may not be defined on the whole space X.
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 X ⊕ Y, defined as the set of all pairs (x, Tx), where x runs over the domain of T .

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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.

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

Operator theory

In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra.

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