Densely-defined operator

In mathematics — specifically, in operator theory — a densely-defined operator is a type of partially-defined function; in a topological sense, it is a linear operator that is defined "almost everywhere". Densely-defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they a priori "make sense".

Definition

A linear operator T from one topological vector space, X, to another one, Y, is said to be densely defined if the domain of T is a dense subset of X and the range of T is contained within Y.

Examples

• Consider the space C⁰([0, 1]; R) of all real-valued, continuous functions defined on the unit interval; let C¹([0, 1]; R) denote the subspace consisting of all continuously-differentiable functions. Equip C⁰([0, 1]; R) with the supremum norm ||·||_∞; this makes C⁰([0, 1]; R) into a real Banach space. The differentiation operator D given by

$(mathrm\{D\}\; u)(x)\; =\; u\text{'}(x)$

is a densely-defined operator from C⁰([0, 1]; R) to itself, defined on the dense subspace C¹([0, 1]; R). Note also that the operator D is an example of an unbounded linear operator, since

$u\_\{n\}\; (x)\; =\; e^\{-\; n\; x\}$

has

$frac\{|\; mathrm\{D\}\; u\_\{n\}\; |\_\{infty\}\}\{|\; u\_\{n\}\; |\_\{infty\}\}\; =\; n.$

This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of C⁰([0, 1]; R).

• The Paley-Wiener integral, on the other hand, is an example of a continuous extension of a densely-defined operator. In any abstract Wiener space i : H → E with adjoint j = i^∗ : E^∗ → H, there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from j(E^∗) to L²(E, γ; R), under which j(f) ∈ j(E^∗) ⊆ H goes to the equivalence class [f] of f in L²(E, γ; R). It is not hard to show that j(E^∗) is dense in H. Since the above inclusion is continuous, there is a unique continuous linear extension I : H → L²(E, γ; R) of the inclusion j(E^∗) → L²(E, γ; R) to the whole of H. This extension is the Paley-Wiener map.

References

• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Second edition, New York: Springer-Verlag.