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".^{[clarification needed]}

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 $C^{0}([0,1];\mathbb {R} )$ 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 $j\left(E^{*}\right)$ to $L^{2}(E,\gamma ;\mathbb {R} ),$ under which $j(f)\in j\left(E^{*}\right)\subseteq H$ goes to the equivalence class $[f]$ of $f$ in $L^{2}(E,\gamma ;\mathbb {R} ).$ 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 $I:H\to L^{2}(E,\gamma ;\mathbb {R} )$ of the inclusion $j\left(E^{*}\right)\to L^{2}(E,\gamma ;\mathbb {R} )$ 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.

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Definition

Examples

See also

References