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".

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 $\mathrm{dom}(T)$ of $X$ and takes values in $Y,$ written $T:\mathrm{dom}(T)\subseteq X\to Y.$ 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 $C^0([0,1];\mathbb{ℝ})$ of all real-valued, continuous functions defined on the unit interval; let $C^1([0,1];\mathbb{ℝ})$ denote the subspace consisting of all continuously differentiable functions. Equip $C^0([0,1];\mathbb{ℝ})$ with the supremum norm $\Vert \cdot {\Vert }_{\mathrm{\infty }}$; this makes $C^0([0,1];\mathbb{ℝ})$ into a real Banach space. The differentiation operator $D$ given by $$(\mathrm{D}u)(x)=u^{\prime }(x)$$ is a densely defined operator from $C^0([0,1];\mathbb{ℝ})$ to itself, defined on the dense subspace $C^1([0,1];\mathbb{ℝ}).$ The operator $\mathrm{D}$ is an example of an unbounded linear operator, since $$u_n(x)=e^{-nx}\text{ has }\frac{{\Vert \mathrm{D}{u}_n\Vert }_{\mathrm{\infty }}}{{\Vert u_n\Vert }_{\mathrm{\infty }}}=n.$$ This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator $D$ to the whole of $C^0([0,1];\mathbb{ℝ}).$

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 $L^2(E,\gamma ;\mathbb{ℝ}),$ under which $j(f)\in j\left(E^{*}\right)\subseteq H$ goes to the equivalence class $[f]$ of $f$ in $L^2(E,\gamma ;\mathbb{ℝ}).$ 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 $I:H\to L^2(E,\gamma ;\mathbb{ℝ})$ of the inclusion $j\left(E^{*}\right)\to L^2(E,\gamma ;\mathbb{ℝ})$ to the whole of $H.$ This extension is the Paley–Wiener map.

See also

• Blumberg theorem – Any real function on R admits a continuous restriction on a dense subset of R
• Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
• Partial function – Function whose actual domain of definition may be smaller than its apparent domain

References

• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. xiv+434. ISBN 0-387-00444-0.