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

Examples

Consider the space C⁰([0, 1]; R) of all real-valued, continuous functions defined on the unit interval; let C¹([0, 1]; R) denote the subspace consisting of all continuously differentiable functions. Equip C⁰([0, 1]; R) with the supremum norm ||·||_∞; this makes C⁰([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 C⁰([0, 1]; R) to itself, defined on the dense subspace C¹([0, 1]; R). Note also that the operator D is an example of an unbounded linear operator, since

u_{n}(x)=e^{{-nx}}\,

has

{\frac {\|{\mathrm {D}}u_{n}\|_{{\infty }}}{\|u_{n}\|_{\infty }}}=n.

This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of C⁰([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 → E with adjoint j = i^∗ : E^∗ → H, there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from j(E^∗) to L²(E, γ; R), under which j(f) ∈ j(E^∗) ⊆ H goes to the equivalence class [f] of f in L²(E, γ; R). It is not hard to show that j(E^∗) is dense in H. Since the above inclusion is continuous, there is a unique continuous linear extension I : H → L²(E, γ; R) of the inclusion j(E^∗) → L²(E, γ; R) to the whole of H. This extension is the Paley–Wiener map.

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. MR 2028503.

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