Compact operator
Infunctional analysis,a branch ofmathematics,acompact operatoris alinear operator,wherearenormed vector spaces,with the property thatmapsbounded subsetsoftorelatively compactsubsets of(subsets with compactclosurein). Such an operator is necessarily abounded operator,and so continuous.[1]Some authors require thatare Banach, but the definition can be extended to more general spaces.
Any bounded operatorthat has finiterankis a compact operator; indeed, the class of compact operators is a natural generalization of the class offinite-rank operatorsin an infinite-dimensional setting. Whenis aHilbert space,it is true that any compact operator is a limit of finite-rank operators,[1]so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in thenorm topology.Whether this was true in general for Banach spaces (theapproximation property) was an unsolved question for many years; in 1973Per Enflogave a counter-example, building on work byGrothendieckandBanach.[2]
The origin of the theory of compact operators is in the theory ofintegral equations,where integral operators supply concrete examples of such operators. A typicalFredholm integral equationgives rise to a compact operatorKonfunction spaces;the compactness property is shown byequicontinuity.The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea ofFredholm operatoris derived from this connection.
Equivalent formulations
[edit]A linear mapbetween twotopological vector spacesis said to becompactif there exists a neighborhoodof the origin insuch thatis a relatively compact subset of.[3]
Letbe normed spaces anda linear operator. Then the following statements are equivalent, and some of them are used as the principal definition by different authors[4]
- is a compact operator;
- the image of the unit ball ofunderisrelatively compactin;
- the image of any bounded subset ofunderisrelatively compactin;
- there exists aneighbourhoodof the origin inand a compact subsetsuch that;
- for any bounded sequencein,the sequencecontains a converging subsequence.
If in additionis Banach, these statements are also equivalent to:
- the image of any bounded subset ofunderistotally boundedin.
If a linear operator is compact, then it is continuous.
Important properties
[edit]In the following,are Banach spaces,is the space of bounded operatorsunder theoperator norm,anddenotes the space of compact operators.denotes theidentity operatoron,,and.
- is a closed subspace of(in the norm topology). Equivalently,[5]
- given a sequence of compact operatorsmapping(whereare Banach) and given thatconverges towith respect to theoperator norm,is then compact.
- Conversely, ifare Hilbert spaces, then every compact operator fromis the limit of finite rank operators. Notably, this "approximation property"is false for general Banach spacesXandY.[4]
- In particular,forms a two-sidedidealin.
- Any compact operator isstrictly singular,but not vice versa.[6]
- A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (Schauder's theorem).
- Ifis a Banach space and there exists aninvertiblebounded compact operatorthenis necessarily finite-dimensional.[7]
Now suppose thatis a Banach space andis a compact linear operator, andis theadjointortransposeofT.
- For any, is aFredholm operatorof index 0. In particular,is closed. This is essential in developing the spectral properties of compact operators. One can notice the similarity between this property and the fact that, ifandare subspaces ofwhereis closed andis finite-dimensional, thenis also closed.
- Ifis any bounded linear operator then bothandare compact operators.[5]
- Ifthen the range ofis closed and the kernel ofis finite-dimensional.[5]
- Ifthen the following are finite and equal:[5]
- Thespectrumofis compact,countable,and has at most onelimit point,which would necessarily be the origin.[5]
- Ifis infinite-dimensional then.[5]
- Ifandthenis an eigenvalue of bothand.[5]
- For everythe setis finite, and for every non-zerothe range ofis aproper subsetofX.[5]
Origins in integral equation theory
[edit]A crucial property of compact operators is theFredholm alternative,which asserts that the existence of solution of linear equations of the form
(whereKis a compact operator,fis a given function, anduis the unknown function to be solved for) behaves much like as in finite dimensions. Thespectral theory of compact operatorsthen follows, and it is due toFrigyes Riesz(1918). It shows that a compact operatorKon an infinite-dimensional Banach space has spectrum that is either a finite subset ofCwhich includes 0, or the spectrum is acountably infinitesubset ofCwhich has 0 as its onlylimit point.Moreover, in either case the non-zero elements of the spectrum areeigenvaluesofKwith finite multiplicities (so thatK− λIhas a finite-dimensionalkernelfor all complex λ ≠ 0).
An important example of a compact operator iscompact embeddingofSobolev spaces,which, along with theGårding inequalityand theLax–Milgram theorem,can be used to convert anelliptic boundary value probleminto a Fredholm integral equation.[8]Existence of the solution and spectral properties then follow from the theory of compact operators; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues. One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarily high vibration frequencies always exist.
The compact operators from a Banach space to itself form a two-sidedidealin thealgebraof all bounded operators on the space. Indeed, the compact operators on an infinite-dimensional separable Hilbert space form a maximal ideal, so thequotient algebra,known as theCalkin algebra,issimple.More generally, the compact operators form anoperator ideal.
Compact operator on Hilbert spaces
[edit]For Hilbert spaces, another equivalent definition of compact operators is given as follows.
An operatoron an infinite-dimensionalHilbert space,
- ,
is said to becompactif it can be written in the form
- ,
whereandare orthonormal sets (not necessarily complete), andis a sequence of positive numbers with limit zero, called thesingular valuesof the operator, and the series on the right hand side converges in the operator norm. The singular values canaccumulateonly at zero. If the sequence becomes stationary at zero, that isfor someand every,then the operator has finite rank,i.e.,a finite-dimensional range, and can be written as
- .
An important subclass of compact operators is thetrace-classornuclear operators,i.e., such that.While all trace-class operators are compact operators, the converse is not necessarily true. For exampletends to zero forwhile.
Completely continuous operators
[edit]LetXandYbe Banach spaces. A bounded linear operatorT:X→Yis calledcompletely continuousif, for everyweakly convergentsequencefromX,the sequenceis norm-convergent inY(Conway 1985,§VI.3). Compact operators on a Banach space are always completely continuous. IfXis areflexive Banach space,then every completely continuous operatorT:X→Yis compact.
Somewhat confusingly, compact operators are sometimes referred to as "completely continuous" in older literature, even though they are not necessarily completely continuous by the definition of that phrase in modern terminology.
Examples
[edit]- Every finite rank operator is compact.
- Forand a sequence(tn)converging to zero, the multiplication operator (Tx)n= tnxnis compact.
- For some fixedg∈C([0, 1];R), define the linear operatorTfromC([0, 1];R) toC([0, 1];R) byThat the operatorTis indeed compact follows from theAscoli theorem.
- More generally, if Ω is any domain inRnand the integral kernelk:Ω × Ω →Ris aHilbert–Schmidt kernel,then the operatorTonL2(Ω;R) defined byis a compact operator.
- ByRiesz's lemma,the identity operator is a compact operator if and only if the space is finite-dimensional.[9]
See also
[edit]- Compact embedding
- Compact operator on Hilbert space
- Fredholm alternative– mathematical theorem
- Fredholm integral equation
- Fredholm operator– bounded operator with kernel and cokernel both having finite dimension
- Strictly singular operator
- Spectral theory of compact operators
Notes
[edit]- ^abConway 1985,Section 2.4
- ^Enflo 1973
- ^Schaefer & Wolff 1999,p. 98.
- ^abBrézis, H. (2011).Functional analysis, Sobolev spaces and partial differential equations.H.. Brézis. New York: Springer.ISBN978-0-387-70914-7.OCLC695395895.
- ^abcdefghijRudin 1991,pp. 103–115.
- ^N.L. Carothers,A Short Course on Banach Space Theory,(2005) London Mathematical Society Student Texts64,Cambridge University Press.
- ^abcConway 1990,pp. 173–177.
- ^William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000
- ^Kreyszig 1978,Theorems 2.5-3, 2.5-5.
References
[edit]- Conway, John B.(1985).A course in functional analysis.Springer-Verlag. Section 2.4.ISBN978-3-540-96042-3.
- Conway, John B.(1990).A Course in Functional Analysis.Graduate Texts in Mathematics.Vol. 96 (2nd ed.). New York:Springer-Verlag.ISBN978-0-387-97245-9.OCLC21195908.
- Enflo, P.(1973)."A counterexample to the approximation problem in Banach spaces".Acta Mathematica.130(1): 309–317.doi:10.1007/BF02392270.ISSN0001-5962.MR0402468.
- Kreyszig, Erwin (1978).Introductory functional analysis with applications.John Wiley & Sons.ISBN978-0-471-50731-4.
- Kutateladze, S.S. (1996).Fundamentals of Functional Analysis.Texts in Mathematical Sciences. Vol. 12 (2nd ed.). New York: Springer-Verlag. p. 292.ISBN978-0-7923-3898-7.
- Lax, Peter(2002).Functional Analysis.New York: Wiley-Interscience.ISBN978-0-471-55604-6.OCLC47767143.
- Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces.Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN978-1584888666.OCLC144216834.
- Renardy, M.; Rogers, R. C. (2004).An introduction to partial differential equations.Texts in Applied Mathematics. Vol. 13 (2nd ed.). New York:Springer-Verlag.p. 356.ISBN978-0-387-00444-0.(Section 7.5)
- Rudin, Walter(1991).Functional Analysis.International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:McGraw-Hill Science/Engineering/Math.ISBN978-0-07-054236-5.OCLC21163277.
- Schaefer, Helmut H.;Wolff, Manfred P. (1999).Topological Vector Spaces.GTM.Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN978-1-4612-7155-0.OCLC840278135.
- Trèves, François(2006) [1967].Topological Vector Spaces, Distributions and Kernels.Mineola, N.Y.: Dover Publications.ISBN978-0-486-45352-1.OCLC853623322.
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