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

Examples

Consider the space ${\displaystyle C^{0}([0,1];\mathbb {R} )}$  of all real-valued, continuous functions defined on the unit interval; let ${\displaystyle C^{1}([0,1];\mathbb {R} )}$  denote the subspace consisting of all continuously differentiable functions. Equip ${\displaystyle C^{0}([0,1];\mathbb {R} )}$  with the supremum norm ${\displaystyle \|\,\cdot \,\|_{\infty }}$ ; this makes ${\displaystyle C^{0}([0,1];\mathbb {R} )}$  into a real Banach space. The differentiation operator ${\displaystyle D}$  given by

$${\displaystyle (\mathrm {D} u)(x)=u'(x)}$$

 

is a densely defined operator from ${\displaystyle C^{0}([0,1];\mathbb {R} )}$  to itself, defined on the dense subspace ${\displaystyle C^{1}([0,1];\mathbb {R} ).}$  The operator ${\displaystyle \mathrm {D} }$  is an example of an unbounded linear operator, since

$${\displaystyle u_{n}(x)=e^{-nx}\quad {\text{ has }}\quad {\frac {\left\|\mathrm {D} u_{n}\right\|_{\infty }}{\left\|u_{n}\right\|_{\infty }}}=n.}$$

 

This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator ${\displaystyle D}$  to the whole of ${\displaystyle C^{0}([0,1];\mathbb {R} ).}$ 

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 ${\displaystyle i:H\to E}$  with adjoint ${\displaystyle j:=i^{*}:E^{*}\to H,}$  there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from ${\displaystyle j\left(E^{*}\right)}$  to ${\displaystyle L^{2}(E,\gamma ;\mathbb {R} ),}$  under which ${\displaystyle j(f)\in j\left(E^{*}\right)\subseteq H}$  goes to the equivalence class ${\displaystyle [f]}$  of ${\displaystyle f}$  in ${\displaystyle L^{2}(E,\gamma ;\mathbb {R} ).}$  It can be shown that ${\displaystyle j\left(E^{*}\right)}$  is dense in ${\displaystyle H.}$  Since the above inclusion is continuous, there is a unique continuous linear extension ${\displaystyle I:H\to L^{2}(E,\gamma ;\mathbb {R} )}$  of the inclusion ${\displaystyle j\left(E^{*}\right)\to L^{2}(E,\gamma ;\mathbb {R} )}$  to the whole of ${\displaystyle H.}$  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
• Linear extension (linear algebra) – Mathematical function, in linear algebraPages displaying short descriptions of redirect targets
• 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. MR 2028503.