densely defined operator

{{short description|Function that is defined almost everywhere (mathematics)}}

In mathematics – specifically, in operator theory – a densely defined operator or partially 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".{{clarify|what’s a priori make sense?|date=August 2024}}

A closed operator that is used in practice is often densely defined.

Definition

A densely defined linear operator $T$ from one topological vector space, $X,$ to another one, $Y,$ is a linear operator that is defined on a dense linear subspace $\operatorname{dom}(T)$ of $X$ and takes values in $Y,$ written $T : \operatorname{dom}(T) \subseteq X \to Y.$ Sometimes this is abbreviated as $T : X \to Y$ when the context makes it clear that $X$ might not be the set-theoretic domain of $T.$

Examples

Consider the space $C^0([0, 1]; \R)$ of all real-valued, continuous functions defined on the unit interval; let $C^1([0, 1]; \R)$ denote the subspace consisting of all continuously differentiable functions. Equip $C^0([0, 1]; \R)$ with the supremum norm $\|\,\cdot\,\|_\infty$ ; this makes $C^0([0, 1]; \R)$ into a real Banach space. The differentiation operator $D$ given by $(\mathrm{D} u)(x) = u'(x)$ is a densely defined operator from $C^0([0, 1]; \R)$ to itself, defined on the dense subspace $C^1([0, 1]; \R).$ The operator $\mathrm{D}$ is an example of an unbounded linear operator, since

$u_n (x) = e^{- n x} \quad \text{ has } \quad \frac{\left\|\mathrm{D} u_n\right\|_{\infty}}{\left\|u_n\right\|_\infty} = n.$

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

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 $i : H \to E$ with adjoint $j := i^* : E^* \to H,$ there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from $j\left(E^*\right)$ to $L^2(E, \gamma; \R),$ under which $j(f) \in j\left(E^*\right) \subseteq H$ goes to the equivalence class $[f]$ of $f$ in $L^2(E, \gamma; \R).$ It can be shown that $j\left(E^*\right)$ is dense in $H.$ Since the above inclusion is continuous, there is a unique continuous linear extension $I : H \to L^2(E, \gamma; \R)$ of the inclusion $j\left(E^*\right) \to L^2(E, \gamma; \R)$ to the whole of $H.$ This extension is the Paley–Wiener map.