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".

A closed operator that is used in practice is often densely defined.

Definition

A densely defined linear operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} from one topological vector space, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X,} to another one, $Y,$ is a linear operator that is defined on a dense linear subspace $\operatorname {dom} (T)$ of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 $C^{0}([0,1];\mathbb {R} )$ of all real-valued, continuous functions defined on the unit interval; let $C^{1}([0,1];\mathbb {R} )$ denote the subspace consisting of all continuously differentiable functions. Equip $C^{0}([0,1];\mathbb {R} )$ with the supremum norm $\|\,\cdot \,\|_{\infty }$; this makes $C^{0}([0,1];\mathbb {R} )$ into a real Banach space. The differentiation operator $D$ given by

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

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

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

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

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 $i:H\to E$ 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\left(E^*\right)} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2(E, \gamma; \R),} under which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j(f) \in j\left(E^*\right) \subseteq H} goes to the equivalence class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [f]} of $f$ in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2(E, \gamma; \R).} It can be shown that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\left(E^*\right)} is dense in $H.$ Since the above inclusion is continuous, there is a unique continuous linear extension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I : H \to L^2(E, \gamma; \R)} of the inclusion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\left(E^*\right) \to L^2(E, \gamma; \R)} to the whole of $H.$ 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)
• 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.