Densely defined operator explained

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".

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 \| ⋅ \| _{infty} ; this makes C ^{0([0,} 1 ]; \R) into a real Banach space. The differentiation operator D given by $$(\backslash mathrm\; u)(x)\; =\; u\text{'}(x)$$ 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 D is an example of an unbounded linear operator, since $$u\_n\; (x)\; =\; e^\; \backslash quad\; \backslash text\; \backslash quad\; \backslash frac\; =\; n.$$This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of C ^{0([0,} 1 ]; \R). 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.

References

• Book: Renardy , Michael . Rogers, Robert C.. An introduction to partial differential equations. Texts in Applied Mathematics 13. Second. Springer-Verlag. New York. 2004. xiv+434. 0-387-00444-0. 2028503.