Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of (subsets with compact closure in ). Such an operator is necessarily a bounded operator, and so continuous.[1] Some authors require that are Banach, but the definition can be extended to more general spaces.

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,[2] 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.[3]

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.

Equivalent formulationsEdit

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  .[4]

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[5]

  •   is a compact operator;
  • the image of the unit ball of   under   is relatively compact in  ;
  • the image of any bounded subset of   under   is relatively compact in  ;
  • there exists a neighbourhood   of the origin in   and a compact subset   such that  ;
  • for any bounded sequence   in  , the sequence   contains a converging subsequence.

If in addition   is Banach, these statements are also equivalent to:

  • the image of any bounded subset of   under   is totally bounded in  .

If a linear operator is compact, then it is continuous.

Important propertiesEdit

In the following,   are Banach spaces,   is the space of bounded operators   under the operator norm, and   denotes the space of compact operators  .   denotes the identity operator on  ,  , and  .

  •   is a closed subspace of   (in the norm topology). Equivalently,[6]
    • given a sequence of compact operators   mapping   (where  are Banach) and given that   converges to   with respect to the operator norm,   is then compact.
  • Conversely, if   are Hilbert spaces, then every compact operator from   is the limit of finite rank operators. Notably, this "approximation property" is false for general Banach spaces X and Y.[5]
  •    In particular,   forms a two-sided ideal in  .
  • Any compact operator is strictly singular, but not vice versa.[7]
  • A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (Schauder's theorem).
    • If   is bounded and compact, then:
      • the closure of the range of   is separable.[6][8]
      • if the range of   is closed in Y, then the range of   is finite-dimensional.[6][8]
  • If   is a Banach space and there exists an invertible bounded compact operator   then   is necessarily finite-dimensional.[8]

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.[6]
  • If   then the range of   is closed and the kernel of   is finite-dimensional.[6]
  • If   then the following are finite and equal:  [6]
  • The spectrum   of  , is compact, countable, and has at most one limit point, which would necessarily be the origin.[6]
  • If   is infinite-dimensional then  .[6]
  • If   and   then   is an eigenvalue of both   and  .[6]
  • For every   the set   is finite, and for every non-zero   the range of   is a proper subset of X.[6]

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.[9] 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.

An operator   on an infinite-dimensional Hilbert space  

 

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.

An important subclass of compact operators is the trace-class or nuclear operators.

Completely continuous operatorsEdit

Let X and Y be Banach spaces. A bounded linear operator T : XY is called completely continuous if, for every weakly convergent sequence   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 : XY 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.

ExamplesEdit

  • 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.[10]

See alsoEdit

NotesEdit

  1. ^ Conway 1985, Section 2.4
  2. ^ Conway 1985, Section 2.4
  3. ^ Enflo 1973
  4. ^ Schaefer & Wolff 1999, p. 98.
  5. ^ a b Brézis, H. (2011). Functional analysis, Sobolev spaces and partial differential equations. H.. Brézis. New York: Springer. ISBN 978-0-387-70914-7. OCLC 695395895.
  6. ^ a b c d e f g h i j Rudin 1991, pp. 103-115.
  7. ^ N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cambridge University Press.
  8. ^ a b c Conway 1990, pp. 173-177.
  9. ^ William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000
  10. ^ Kreyszig 1978, Theorems 2.5-3, 2.5-5.

ReferencesEdit