Compact operator

Equivalent formulations

A linear operator T : X → Y is compact if and only if any of the following is true

Image of the closed unit ball in X under T is relatively compact in Y.

Image of any bounded set under T is relatively compact in Y.

Image of any bounded set under T is totally bounded in Y.

there exists a neighbourhood of 0, U ⊂ X , and compact set V ⊂ Y such that T ( U ) ⊂ V .

For any sequence ( x n ) n ∈ N from the unit ball in X, the sequence ( T x n ) n ∈ N contains a Cauchy subsequence.

If a linear operator is compact, then it is easy to see that it is bounded, and hence continuous.

Important properties

In the following, X, Y, Z, W are Banach spaces, B(X, Y) is the space of bounded operators from X to Y with the operator norm, K(X, Y) is the space of compact operators from X to Y, B(X) = B(X, X), K(X) = K(X, X), i d X is the identity operator on X.

K(X, Y) is a closed subspace of B(X, Y): Let T_n, n ∈ N, be a sequence of compact operators from one Banach space to the other, and suppose that T_n converges to T with respect to the operator norm. Then T is also compact.

Conversely, if X, Y are Hilbert spaces, then every compact operator from X to Y is the limit of finite rank operators. Notably, this is false for general Banach spaces X and Y.

B ( Y , Z ) ∘ K ( X , Y ) ∘ B ( W , X ) ⊆ K ( W , Z ) .   In particular, K(X) forms a two-sided ideal in B(X).

i d X is compact if and only if X has finite dimension.

For any T ∈ K(X),  i d X − T   is a Fredholm operator of index 0. In particular,  im ( i d X − T )   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 M and N are subspaces of a Banach space where M is closed and N is finite-dimensional, then M + N is also closed.

Any compact operator is strictly singular, but not vice versa.

An operator is compact if and only if its adjoint is compact (Schauder's theorem).

Origins in integral equation theory

A crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution of linear equations of the form

( λ K + I ) u = f

(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 spaces

For Hilbert spaces, another equivalent definition of compact operators is given as follows.

An operator T on an infinite-dimensional Hilbert space H

T : H → H

is said to be compact if it can be written in the form

T = ∑ n = 1 ∞ λ n ⟨ f n , ⋅ ⟩ g n ,

where f 1 , f 2 , … and g 1 , g 2 , … are orthonormal sets (not necessarily complete), and λ 1 , λ 2 , … 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 λ N + k = 0 for some N ∈ N ,  and every  k = 1 , 2 , … , then the operator has finite rank, i.e, a finite-dimensional range and can be written as

T = ∑ n = 1 N λ n ⟨ f n , ⋅ ⟩ g n .

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 operators

Let X and Y be Banach spaces. A bounded linear operator T : X → Y is called completely continuous if, for every weakly convergent sequence ( x n ) from X, the sequence ( T x n ) 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 lights of today's terminology.

Examples

Every finite rank operator is compact.

For ℓ p and a sequence (t_n) converging to zero, the multiplication operator (Tx)_n = t_n x_n 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.