Densely defined operator
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 [math]\displaystyle{ T }[/math] from one topological vector space, [math]\displaystyle{ X, }[/math] to another one, [math]\displaystyle{ Y, }[/math] is a linear operator that is defined on a dense linear subspace [math]\displaystyle{ \operatorname{dom}(T) }[/math] of [math]\displaystyle{ X }[/math] and takes values in [math]\displaystyle{ Y, }[/math] written [math]\displaystyle{ T : \operatorname{dom}(T) \subseteq X \to Y. }[/math] Sometimes this is abbreviated as [math]\displaystyle{ T : X \to Y }[/math] when the context makes it clear that [math]\displaystyle{ X }[/math] might not be the set-theoretic domain of [math]\displaystyle{ T. }[/math]
Examples
Consider the space [math]\displaystyle{ C^0([0, 1]; \R) }[/math] of all real-valued, continuous functions defined on the unit interval; let [math]\displaystyle{ C^1([0, 1]; \R) }[/math] denote the subspace consisting of all continuously differentiable functions. Equip [math]\displaystyle{ C^0([0, 1]; \R) }[/math] with the supremum norm [math]\displaystyle{ \|\,\cdot\,\|_\infty }[/math]; this makes [math]\displaystyle{ C^0([0, 1]; \R) }[/math] into a real Banach space. The differentiation operator [math]\displaystyle{ D }[/math] given by [math]\displaystyle{ (\mathrm{D} u)(x) = u'(x) }[/math] is a densely defined operator from [math]\displaystyle{ C^0([0, 1]; \R) }[/math] to itself, defined on the dense subspace [math]\displaystyle{ C^1([0, 1]; \R). }[/math] The operator [math]\displaystyle{ \mathrm{D} }[/math] is an example of an unbounded linear operator, since [math]\displaystyle{ u_n (x) = e^{- n x} \quad \text{ has } \quad \frac{\left\|\mathrm{D} u_n\right\|_{\infty}}{\left\|u_n\right\|_\infty} = n. }[/math] This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator [math]\displaystyle{ D }[/math] to the whole of [math]\displaystyle{ C^0([0, 1]; \R). }[/math]
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 [math]\displaystyle{ i : H \to E }[/math] with adjoint [math]\displaystyle{ j := i^* : E^* \to H, }[/math] there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from [math]\displaystyle{ j\left(E^*\right) }[/math] to [math]\displaystyle{ L^2(E, \gamma; \R), }[/math] under which [math]\displaystyle{ j(f) \in j\left(E^*\right) \subseteq H }[/math] goes to the equivalence class [math]\displaystyle{ [f] }[/math] of [math]\displaystyle{ f }[/math] in [math]\displaystyle{ L^2(E, \gamma; \R). }[/math] It can be shown that [math]\displaystyle{ j\left(E^*\right) }[/math] is dense in [math]\displaystyle{ H. }[/math] Since the above inclusion is continuous, there is a unique continuous linear extension [math]\displaystyle{ I : H \to L^2(E, \gamma; \R) }[/math] of the inclusion [math]\displaystyle{ j\left(E^*\right) \to L^2(E, \gamma; \R) }[/math] to the whole of [math]\displaystyle{ H. }[/math] 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.
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