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. Some authors require that be 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. Every compact operator in a Hilbert space ' is a limit (in operator norm) of finite-rank operators, so 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 Alexander Grothendieck and Stefan 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.
Let be topological vector spaces and a linear operator.
The following statements are equivalent, and different authors may pick any one of these as the principal definition for "' is a compact operator":
If in addition are normed spaces, these statements are also equivalent to:
If in addition ' is Banach, these statements are also equivalent to:
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 .
Now suppose that is a Banach space and is a compact linear operator, and is the adjoint or transpose of T.
A crucial property of compact operators is the Fredholm alternative in the solution of linear equations. Let be a compact operator, a given function, and the unknown function to be solved for. Then the Fredholm alternative states that the equationbehaves 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 on an infinite-dimensional Banach space has spectrum that is either a finite subset of which includes 0, or the spectrum is a countably infinite subset of which has as its only limit point. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of with finite multiplicities (so that has a finite-dimensional kernel for all complex ).
An important example of a compact operator is compact embedding of Sobolev spaces, which, along with the GÃÂ¥rding inequality and the LaxâÂÂMilgram 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.
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, and the series on the right hand side converges in the operator norm. 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
An important subclass of compact operators is the trace-class or nuclear operators, i.e., such that . While all trace-class operators are compact operators, the converse is not necessarily true. For example tends to zero for while .
Let be Banach spaces. A bounded linear operator is called completely continuous if, for every weakly convergent sequence from , the sequence is norm-convergent in .
Compact operators between Banach spaces are always completely continuous, but the converse is false, because there exists a completely continuous operator that is not compact. However, the converse is true if is a reflexive Banach space: then every completely continuous operator is compact.
Somewhat confusingly, compact operators are sometimes referred to as "completely continuous" in older literature, even though the latter is a weaker condition in modern terminology.