Compact Linear Operator

Overview

Compact operators recover a useful part of finite-dimensional behavior inside infinite-dimensional spaces. Their inputs may range over a bounded family with no convergent subsequence, yet their outputs always admit one. This extra control underlies existence arguments, integral equations, and the discrete spectral description of compact self-adjoint operators.

Compactness concerns the operator together with its domain and codomain norms. The same formula can behave differently when the function spaces change, so a statement that an operator is compact must name those spaces.

Prerequisites

Know Banach spaces, bounded linear maps, and norm convergence. A set is relatively compact when its closure is compact. In metric spaces this can be checked using sequences. In a complete metric space, total boundedness is equivalent to relative compactness; total boundedness means that finitely many balls of any prescribed positive radius cover the set.

Definition and conventions

For Banach spaces $X,Y$, a linear map $T:X\to Y$ is compact if $T(B)$ has compact closure in $Y$ for every bounded $B\subseteq X$. It suffices to test the closed unit ball $B_X$.

Equivalently, every bounded sequence $(x_m)$ in $X$ has a subsequence $(x_{m_k})$ for which $Tx_{m_k}$ converges in the norm of $Y$. The inputs themselves need not converge. Since relatively compact sets are bounded, this definition implies that $T$ is a bounded operator. The image $T(B_X)$ need not be closed for general Banach domain spaces.

Intuition

A compact operator makes its unit-ball image approximable, at each resolution, by finitely many output points. This does not say that its range is finite-dimensional. Instead, infinitely many directions may survive while their effects become progressively smaller. The shrinking diagonal operator below makes that distinction concrete.

Examples and non-examples

On $\ell^2$, indexed by $n\ge1$, set $(Tx)_n=x_n/(n+1)$. Let $T_N$ retain only the first $N$ output coordinates. Then $T_N$ has finite rank and

The upper bound follows by summing squared coordinates; equality follows by using $x=e_{N+1}$. Therefore $T$ is compact as an operator-norm limit of compact operators, although its range contains every finitely supported sequence and has infinite dimension.

The identity on an infinite-dimensional Hilbert space is bounded but not compact: an orthonormal sequence is bounded and its distinct terms stay distance $\sqrt2$ apart. It has no norm-convergent subsequence.

For a continuous kernel $k$ on $[0,1]^2$, the operator $Kf(x)=\int_0^1k(x,y)f(y)\,dy$ is compact from $C[0,1]$ to itself. Bounded inputs produce uniformly bounded, equicontinuous outputs, so Arzelà–Ascoli applies. Continuity of the kernel is a sufficient hypothesis in this example, not part of the definition of compactness.

Important properties

Every bounded finite-rank operator is compact. Operator-norm limits of compact maps into a Banach space are compact as well. For a proof sketch, approximate the image of the unit ball uniformly by a compact operator's image, cover that image by finitely many small balls, and obtain total boundedness. Completeness of the codomain then gives compact closure.

Composing a compact map on either side with compatible bounded linear maps preserves compactness. Bounded maps preserve bounded sets, and continuous maps preserve compact sets, which explains the two compositions.

If $x_m$ converges weakly to $x$, a compact operator satisfies $Tx_m\to Tx$ in norm. Weakly convergent sequences are bounded. Any norm-convergent subsequence of their images must have limit $Tx$, since bounded linear functionals distinguish points. A subsequence staying a fixed distance from $Tx$ would contradict compactness.

Connections and further reading

For the zero-Dirichlet Laplacian on a bounded smooth domain, the solution operator maps $L^2$ boundedly into $H^1_0$; the compact inclusion back into $L^2$ makes it compact. This connects weak PDE solutions with spectral theory. Hunter, §4.8 gives this mechanism for elliptic resolvents; Erdman, §7.1 develops the operator framework.