Any bounded operator that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting. When is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the norm topology. Whether this was true in general for Banach spaces (the approximation property) was an unsolved question for many years; in 1973 Per Enflo gave a counter-example, building on work by Grothendieck and Banach.
The origin of the theory of compact operators is in the theory of integral equations, where integral operators supply concrete examples of such operators. A typical Fredholm integral equation gives rise to a compact operator K on function spaces; the compactness property is shown by equicontinuity. The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator is derived from this connection.
A linear map between two topological vector spaces is said to be compact if there exists a neighborhood of the origin in such that is a relatively compact subset of .
Let be normed spaces and a linear operator. Then the following statements are equivalent, and some of them are used as the principal definition by different authors
if the range of is closed in Y, then the range of is finite-dimensional.
If is a Banach space and there exists an invertible bounded compact operator then is necessarily finite-dimensional.
Now suppose that is a Banach space and is a compact linear operator, and is the adjoint or transpose of T.
For any , then is a Fredholm operator of 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, if and are subspaces of where is closed and is finite-dimensional, then is also closed.
If is any bounded linear operator then both and are compact operators.
If then the range of is closed and the kernel of is finite-dimensional.
For every the set is finite, and for every non-zero the range of is a proper subset of X.
Origins in integral equation theoryEdit
A crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution of linear equations of the form
(where K is a compact operator, f is a given function, and u is the unknown function to be solved for) behaves much like as in finite dimensions. The spectral theory of compact operators then follows, and it is due to Frigyes Riesz (1918). It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either a finite subset of C which includes 0, or the spectrum is a countably infinite subset of C which has 0 as its only limit point. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of K with finite multiplicities (so that K − λI has a finite-dimensional kernel for all complex λ ≠ 0).
An important example of a compact operator is compact embedding of Sobolev spaces, which, along with the Gårding inequality and the Lax–Milgram theorem, can be used to convert an elliptic boundary value problem into a Fredholm integral equation. 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-sided ideal in the algebra of all bounded operators on the space. Indeed, the compact operators on an infinite-dimensional separable Hilbert space form a maximal ideal, so the quotient algebra, known as the Calkin algebra, is simple. More generally, the compact operators form an operator ideal.
Compact operator on Hilbert spacesEdit
For Hilbert spaces, another equivalent definition of compact operators is given as follows.
is said to be compact if it can be written in the form
where and are orthonormal sets (not necessarily complete), and is a sequence of positive numbers with limit zero, called the singular values of the operator. The singular values can accumulate only at zero. If the sequence becomes stationary at zero, that is for some and every , then the operator has finite rank, i.e, a finite-dimensional range and can be written as
The bracket is the scalar product on the Hilbert space; the sum on the right hand side converges in the operator norm.
Let X and Y be Banach spaces. A bounded linear operator T : X → Y is called completely continuous if, for every weakly convergentsequence from X, the sequence is norm-convergent in Y (Conway 1985, §VI.3). Compact operators on a Banach space are always completely continuous. If X is a reflexive Banach space, then every completely continuous operator T : X → Y is 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.
Every finite rank operator is compact.
For and a sequence (tn) converging to zero, the multiplication operator (Tx)n = tn xn is compact.
For some fixed g ∈ C([0, 1]; R), define the linear operator T from C([0, 1]; R) to C([0, 1]; R) by
That the operator T is indeed compact follows from the Ascoli theorem.
More generally, if Ω is any domain in Rn and the integral kernel k : Ω × Ω → R is a Hilbert–Schmidt kernel, then the operator T on L2(Ω; R) defined by
is a compact operator.
By Riesz's lemma, the identity operator is a compact operator if and only if the space is finite-dimensional.
Enflo, P. (1973). "A counterexample to the approximation problem in Banach spaces". Acta Mathematica. 130 (1): 309–317. doi:10.1007/BF02392270. ISSN 0001-5962. MR 0402468.
Kreyszig, Erwin (1978). Introductory functional analysis with applications. John Wiley & Sons. ISBN 978-0-471-50731-4.
Kutateladze, S.S. (1996). Fundamentals of Functional Analysis. Texts in Mathematical Sciences. Vol. 12 (2nd ed.). New York: Springer-Verlag. p. 292. ISBN 978-0-7923-3898-7.
Lax, Peter (2002). Functional Analysis. New York: Wiley-Interscience. ISBN 978-0-471-55604-6. OCLC 47767143.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
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. ISBN 978-0-387-00444-0. (Section 7.5)