Densely defined

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".

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 $\operatorname(T)$ of $X$ and takes values in $Y,$ written $T : \operatorname(T) \subseteq X \to Y.$ 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 $C^0($, 1 \R) of all real-valued, continuous functions defined on the unit interval; let $C^1($, 1 \R) denote the subspace consisting of all continuously differentiable functions. Equip $C^0($, 1 \R) with the supremum norm $\, \,\cdot\,\, _\infty$; this makes $C^0($, 1 \R) into a real Banach space. The differentiation operator $D$ given by $(\mathrm u)(x) = u'(x)$ is a densely defined operator from $C^0($, 1 \R) to itself, defined on the dense subspace $C^1($, 1 \R). The operator $\mathrm$ is an example of an unbounded linear operator, since
$u_n (x) = e^ \quad \text \quad \frac = n.$
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator $D$ to the whole of $C^0($, 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 /math> 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.

See also

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References

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{{DEFAULTSORT:Densely-Defined Operator

Functional analysis
Hilbert space
Linear operators
Operator theory