Densely-defined operator

Densely-defined operator

In mathematics — specifically, in operator theory — a densely-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 linear operator "T" from one topological vector space, "X", to another one, "Y", is said to be densely defined if the domain of "T" is a dense subset of "X" and the range of "T" is contained within "Y".

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 ||·||∞; this makes "C"0( [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 "C"0( [0, 1] ; R) to itself, defined on the dense subspace "C"1( [0, 1] ; R). Note also that the operator D is an example of an unbounded linear operator, since

::u_{n} (x) = e^{- n x}

: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 "C"0( [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 "L"2("E", "γ"; R), under which "j"("f") ∈ "j"("E"∗) ⊆ "H" goes to the equivalence class ["f"] of "f" in "L"2("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" → "L"2("E", "γ"; R) of the inclusion "j"("E"∗) → "L"2("E", "γ"; R) to the whole of "H". This extension is the Paley-Wiener map.

References

* cite book
last = Renardy
first = Michael
coauthors = Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 2004
pages = pp. xiv+434
isbn = 0-387-00444-0 MathSciNet|id=2028503


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