Movatterモバイル変換

[0]ホーム

Jump to content

Densely defined operator

Edit links

From Wikipedia, the free encyclopedia

Linear operator on dense subset of its apparent domain

This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(October 2025) (Learn how and when to remove this message)

Inmathematics – specifically, inoperator theory – adensely defined operator orpartially defined operator is a type of partially definedfunction. In atopological sense, it is alinear operator that is defined "almost everywhere". Densely defined operators often arise infunctional analysis as operations that one would like to apply to a larger class of objects than those for which theya priori "make sense".^{[clarification needed]}

Aclosed operator that is used in practice is often densely defined.

Definition

[edit]

Let $X, Y {\displaystyle X,Y}$ betopological vector spaces.

Adensely defined linear operator $T {\displaystyle T}$ from $X {\displaystyle X}$ to $Y {\displaystyle Y}$ is a linear operator of type $T:D(T)\to Y$ , such that $D(T)$ is adense subset of $X {\displaystyle X}$ . In other words, $T {\displaystyle T}$ is apartial function whosedomain is dense in $X {\displaystyle X}$ .

Sometimes this is abbreviated as $T:X\to Y$ when the context makes it clear that $T {\displaystyle T}$ might not be defined for all of $X {\displaystyle X}$ .

Properties

[edit]

Closed Graph Theorem—If $X, Y {\displaystyle X,Y}$ are Hausdorff and metrizable, $T:D(T)\to Y$ is densely defined, with continuous inverse $S:Y\to D(T)$ , then $T {\displaystyle T}$ is closed. That is, the set $\{(x,T(x)):x\in D(T)\}$ is closed in theproduct topology of $X\times Y$ .

Proof

Take anynet $(x_{\alpha })$ in $D(T)$ with $Tx_{\alpha }\to y$ in $Y {\displaystyle Y}$ . By continuity of $S {\displaystyle S}$ , $x_{\alpha }=S(Tx_{\alpha })\to \ S(y)$ . Hence there exists some $x\in D(T)$ such that $x_{\alpha }\to x$ , and $Tx=T(S(y))=y$ .

TheHausdorff property ensures sequential convergence is unique. Themetrizability property ensures thatsequentially closed sets are closed. In functional analysis, these conditions typically hold, as most spaces under consideration areFréchet space, or stronger than Fréchet. In particular,Banach spaces are Fréchet.

Examples

[edit]

Sequence

[edit]

Let $X=\ell ^{2}(\mathbb {N} )$ be theHilbert space ofsquare-summable sequences, with orthonormal basis $(e_{n})_{n\geq 1}$ . Define thediagonal operator $A:D(A)\to \ell ^{2},\qquad (Ax)_{n}:=n\,x_{n},$ with domain $D(A):=\left\{x=(x_{n})_{n\geq 1}\in \ell ^{2}:\sum _{n=1}^{\infty }n^{2}|x_{n}|^{2}<\infty \right\}.$ Then $D(A)$ is dense in $\ell ^{2}$ because thefinitely supported sequences $c_{00}\subset D(A)$ , and $c_{00}$ is dense in $\ell ^{2}$ . The operator $A {\displaystyle A}$ is closed andunbounded, since $\|Ae_{n}\|_{2}=n$ .

There exists a bounded inverse: $A^{-1}:\ell ^{2}\to D(A),\qquad (A^{-1}y)_{n}:={\frac {y_{n}}{n}},\qquad \|A^{-1}\|=\sup _{n\geq 1}{\frac {1}{n}}=1.$ Hence $A:D(A)\to \ell ^{2}$ is bijective with bounded inverse, so $0\in \rho (A)$ and, by theNeumann series argument, theresolvent set of $A {\displaystyle A}$ contains the open unit disk $\{\,\lambda \in \mathbb {C} :\ |\lambda |<1\,\}$ .

In fact, the spectrum of $A {\displaystyle A}$ (that is, the complement of its resolvent set) is precisely the set of positive integers, since for any $\lambda \not \in \{1,2,\dots \}$ , the diagonal formula $(A-\lambda I)^{-1}y={\bigl (}{\tfrac {y_{n}}{n-\lambda }}{\bigr )}_{n\geq 1}$ defines a bounded operator $\ell ^{2}\to D(A)$ .

Thus, $A {\displaystyle A}$ is a densely defined, closed, unbounded operator with bounded inverse and nontrivial, unbounded spectrum.

Differentiation

[edit]

Consider the space $C^{0}([0,1];\mathbb {R} )$ of allreal-valued,continuous functions defined on the unit interval; let $C^{1}([0,1];\mathbb {R} )$ denote the subspace consisting of allcontinuously differentiable functions. Equip $C^{0}([0,1];\mathbb {R} )$ with thesupremum norm $\|\,\cdot \,\|_{\infty }$ ; this makes $C^{0}([0,1];\mathbb {R} )$ into a realBanach space. Thedifferentiation operator $D {\displaystyle D}$ given by $(\mathrm {D} u)(x)=u'(x)$ is a linear operator defined on the dense linear subspace $C^{1}([0,1];\mathbb {R} )\subset C^{0}([0,1];\mathbb {R} )$ , therefore it is a operator densely defined on $C^{0}([0,1];\mathbb {R} )$ .

The operator $\mathrm {D}$ is an example of anunbounded 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 $D {\displaystyle D}$ to the whole of $C^{0}([0,1];\mathbb {R} ).$

Paley–Wiener

[edit]

ThePaley–Wiener integral is a standard example of acontinuous extension of a densely defined operator.

In anyabstract Wiener space $i:H\to E$ withadjoint $j:=i^{*}:E^{*}\to H,$ there is a naturalcontinuous linear operator (in fact it is the inclusion, and is anisometry) 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 theequivalence class $[f]$ of $f {\displaystyle f}$ in $L^{2}(E,\gamma ;\mathbb {R} ).$ It can be shown that $j\left(E^{*}\right)$ is dense in $H . {\displaystyle H.}$ 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 $H . {\displaystyle H.}$ This extension is the Paley–Wiener map.

References

[edit]

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 andL-semi-inner product Hilbert space andPrehilbert 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) withK compact &n<∞ Segal–BargmannF

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 Sobczyk's theorem
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 continuityAC $ba(\Sigma )$ c space Banach coordinateBK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variationBV Bs space ContinuousC(K) withK compact Hausdorff Hardy H^p HilbertH Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–BargmannF 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

Functional analysis (topics –glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic /Topological) Locally convex Reflexive Separable