Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator that behaves, in several important respects, like a finite-dimensional operator such as a matrix. In infinite-dimensional spaces, bounded sets are usually not compact, and bounded sequences need not have convergent subsequences. Compact operators partly restore this finite-dimensional behavior by sending bounded sets to sets whose closures are compact, or equivalently, in normed spaces, by sending bounded sequences to sequences with convergent subsequences.

Compact operators first arose in the theory of integral equations, where many integral operators have compactness properties. They play a central role in the Fredholm alternative, in the spectral theory of linear operators, and in applications to differential equations and Sobolev spaces. For example, compactness often implies that the nonzero spectrum of an operator consists of isolated eigenvalues of finite multiplicity, with possible accumulation only at zero.

Compact operators were first studied in connection with integral equations. A typical example is an operator of the form

\((Kf)(x)=\int_a^b k(x,y)f(y)\,dy,\)

where the function \(k\) is called the integral kernel. Under suitable regularity assumptions on \(k\), the operator \(K\) sends bounded families of functions to families that are uniformly bounded and equicontinuous. By the Arzelà-Ascoli theorem, such families have compact closure in spaces such as \(C([a,b])\). Thus many integral operators are compact.

This compactness is useful because it makes some infinite-dimensional linear equations resemble finite-dimensional systems of linear equations. For example, the Fredholm equation

\(u-\lambda Ku=f\)

can often be studied by finite-dimensional approximation and by the Fredholm alternative. In this way, compact operators provide a bridge between concrete integral equations and the abstract spectral theory of operators on function spaces.

The same idea appears in other parts of analysis. Compact embeddings of function spaces, such as certain embeddings of Sobolev spaces into Lp spaces, allow differential and boundary value problems to be treated by methods similar to those used for compact integral operators.

Let \(X\) and \(Y\) be normed vector spaces, and let \(T:X\to Y\) be a linear operator. The operator \(T\) is called compact if it maps bounded subsets of \(X\) to relatively compact subsets of \(Y\); that is, if for every bounded set \(B\subseteq X\), the closure of \(T(B)\) in \(Y\) is compact.

Equivalently, \(T\) is compact if the image of the closed unit ball of \(X\) is relatively compact in \(Y\). In metric terms, this means that every sequence \((x_n)\) bounded in \(X\) has a subsequence \((x_{n_k})\) such that \((Tx_{n_k})\) converges in \(Y\).

Every compact linear operator between normed spaces is bounded, and hence continuous.

Many standard results on compact operators are stated for Banach spaces. If \(Y\) is Banach, a subset of \(Y\) is relatively compact if and only if it is totally bounded. Thus, for operators into Banach spaces, compactness may also be expressed by saying that the image of every bounded subset of \(X\) is totally bounded in \(Y\).

Let \(X,Y\) be topological vector spaces and \(T: X \to Y\) a linear operator.

The following statements are equivalent, and different authors may pick any one of these as the principal definition for "\(T\) is a compact operator":

• there exists a neighborhood \(U\) of the origin in \(X\) and \(T(U)\) is a relatively compact subset of \(Y\);
• there exists a neighborhood \(U\) of the origin in \(X\) and a compact subset \(V\subseteq Y\) such that \(T(U)\subseteq V\);
• there exists a nonempty open set \(U\) in \(X\) and \(T(U)\) is a relatively compact subset of \(Y\).

Throughout this section, \(X,Y,Z\) denote normed or Banach spaces, as specified, and \(B(X,Y)\) denotes the space of bounded linear operators from \(X\) to \(Y\). The compact operators from \(X\) to \(Y\) are denoted by \(K(X,Y)\).

Every finite-rank operator is compact. Indeed, if the range of \(T:X\to Y\) is finite-dimensional, then the image of a bounded subset of \(X\) is a bounded subset of a finite-dimensional normed space, and hence has compact closure.

Compactness is therefore automatic in finite-dimensional linear algebra. Conversely, the identity operator on an infinite-dimensional Banach space is not compact: the closed unit ball of an infinite-dimensional Banach space is not compact. Thus a Banach space \(X\) is finite-dimensional if and only if its identity operator \(\operatorname{Id}_X\) is compact.

More generally, if a Banach space \(X\) admits an invertible compact operator \(T:X\to X\), then \(X\) is finite-dimensional.

If \(X\) and \(Y\) are Banach spaces, then \(K(X,Y)\) is a closed linear subspace of \(B(X,Y)\) in the operator norm. Equivalently, if a sequence of compact operators \(T_n:X\to Y\) converges in operator norm to an operator \(T\), then \(T\) is compact.

Compact operators are stable under composition with bounded operators. If \(T:X\to Y\) is compact and \(A:W\to X\), \(B:Y\to Z\) are bounded linear operators, then

\(BTA:W\to Z\)

is compact. In particular, \(K(X)=K(X,X)\) is a two-sided ideal in the Banach algebra \(B(X)=B(X,X)\).

In a Hilbert space \(X\), the ring \(B(X)\) of bounded operators modulo the two-sided ideal \(K(X)\) is the Calkin algebra. This quotient is used to define one common form of the essential spectrum: for a bounded operator \(T\) on a Hilbert space, the essential spectrum is the spectrum of the coset \(T+K(H)\) in the Calkin algebra, so compact perturbations do not change it.

Since every finite-rank operator is compact, every operator-norm limit of finite-rank operators is compact. On Hilbert spaces the converse also holds: every compact operator between Hilbert spaces is the operator-norm limit of finite-rank operators.

For general Banach spaces this converse need not hold. The question of whether compact operators can always be approximated by finite-rank operators is related to the approximation property, which fails for some Banach spaces. 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.

A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact; this result is known as Schauder's theorem.

If \(T:X\to Y\) is compact, then the closure of the range of \(T\) is separable. If the range of \(T\) is closed, then the range is finite-dimensional.

Every compact operator between Banach spaces is strictly singular, but the converse is false: there are strictly singular operators that are not compact.

Compact operators are closely connected with Fredholm operators and the Fredholm alternative. This connection is one reason compact operators behave, in many respects, like finite-dimensional linear maps.

Let \(X\) be a Banach space and let \(K:X\to X\) be compact. Then

\(I-K\)

is a Fredholm operator of index zero. Equivalently, the kernel of \(I-K\) is finite-dimensional, the range of \(I-K\) is closed, and

\(\dim \ker(I-K)=\dim\bigl(X/\operatorname{Im}(I-K)\bigr).\)

More generally, for every scalar \(\lambda\neq 0\), the operator

\(\lambda I-K\)

is Fredholm of index zero.

A standard consequence is the Fredholm alternative. For a compact operator \(K:X\to X\), the equation

\(u-Ku=f\)

has a unique solution \(u\in X\) for every \(f\in X\) if and only if the homogeneous equation

\(u-Ku=0\)

has only the zero solution. If the homogeneous equation has a nonzero solution, then its solution space is finite-dimensional, and the inhomogeneous equation is solvable only for those \(f\) satisfying finitely many compatibility conditions.

In terms of the adjoint operator \(K^*:X^*\to X^*\), these compatibility conditions can be written as

\(\varphi(f)=0 \quad\text{for every } \varphi\in\ker(I-K^*).\)

Thus the obstruction to solving \(u-Ku=f\) is finite-dimensional. This is analogous to the finite-dimensional situation in which a system of linear equations is solvable precisely when the right-hand side is orthogonal to the nullspace of the transpose.

For the parameter-dependent equation

\(u-\lambda Ku=f,\)

with \(\lambda\neq 0\), the same alternative applies to the operator \(I-\lambda K\). Failure of invertibility occurs precisely when \(1/\lambda\) is a nonzero eigenvalue of \(K\). Since the nonzero spectrum of a compact operator consists only of isolated eigenvalues of finite multiplicity, these exceptional parameters form a discrete set, with possible accumulation only at infinity.

The spectral theory of compact operators is closer to finite-dimensional linear algebra than the spectral theory of general bounded operators.

Let \(X\) be an infinite-dimensional complex Banach space, and let \(T:X\to X\) be a compact operator. Then \(0\) belongs to the spectrum \(\sigma(T)\). Apart from \(0\), every spectral value of \(T\) is an eigenvalue of finite multiplicity. More precisely, if \(\lambda\neq 0\) and \(\lambda\in\sigma(T)\), then

\(\ker(T-\lambda I)\)

is finite-dimensional, the range of \(T-\lambda I\) is closed, and \(\lambda\) is an isolated point of the spectrum.

The nonzero spectrum is therefore either finite or countably infinite. If it is infinite, its only possible accumulation point is \(0\). Equivalently, for every \(r>0\), the set

\(\{\lambda\in\sigma(T):|\lambda|>r\}\)

is finite. Thus compact operators cannot have a continuous band of nonzero spectral values.

The finite-dimensional character of the nonzero spectrum can also be expressed in Fredholm-theoretic terms. For \(\lambda\neq 0\), the operator

\(T-\lambda I\)

has finite-dimensional kernel and cokernel, and these dimensions are equal:

\(\dim \ker(T-\lambda I)=\dim\bigl(X/\operatorname{Im}(T-\lambda I)\bigr).\)

\(\dim \ker(T-\lambda I)=\dim \ker(T^*-\lambda I)\)

Condensed: the full section is in Wikipedia.

Compact operators on Hilbert spaces have a particularly close relationship with finite-dimensional linear algebra. In addition to the general Banach-space spectral properties of compact operators, Hilbert-space compact operators admit orthogonal decompositions, singular-value expansions, and, in the self-adjoint case, a spectral theorem resembling the spectral theorem for finite-dimensional Hermitian matrices.