Compact Operators - Core Definitions

Compact operators generalize finite-rank operators and matrices to infinite dimensions, providing a class of operators with particularly nice spectral properties.

DefinitionCompact Operator

Let $X$ and $Y$ be Banach spaces. A bounded linear operator $T : X \to Y$ is compact if for every bounded sequence $(x_n)$ in $X$ , the sequence $(T(x_n))$ has a convergent subsequence in $Y$ .

Equivalently, $T$ is compact if it maps the closed unit ball of $X$ to a relatively compact subset of $Y$ .

Compact operators can be thought of as "almost finite-dimensional" operators. They map bounded sets to precompact sets, exhibiting behavior closer to finite-dimensional linear algebra than general bounded operators.

ExampleExamples of Compact Operators

Finite Rank: Any operator $T : X \to Y$ with finite-dimensional range is compact
Integral Operators: The operator $(Kf)(x) = \int_a^b k(x,y) f(y) \, dy$ on $L^2[a,b]$ with continuous kernel $k$ is compact
Diagonal Operators: On $\ell^2$ , the operator $T((x_n)) = (\lambda_n x_n)$ is compact if and only if $\lambda_n \to 0$
Inclusion Maps: The inclusion $H^1(\Omega) \hookrightarrow L^2(\Omega)$ is compact (Rellich-Kondrachov theorem)
Hilbert-Schmidt Operators: On $L^2$ , operators with kernels $k \in L^2([a,b] \times [a,b])$ are compact

TheoremProperties of Compact Operators

Let $\mathcal{K}(X,Y)$ denote the space of compact operators from $X$ to $Y$ . Then:

$\mathcal{K}(X,Y)$ is a closed subspace of $\mathcal{B}(X,Y)$
If $T$ is compact and $S$ is bounded, then $ST$ and $TS$ are compact
$T$ is compact if and only if $T^*$ is compact
Limits of finite-rank operators are compact
On infinite-dimensional spaces, the identity operator is never compact

Proof

Property 5: If $I : X \to X$ were compact on an infinite-dimensional space, the unit ball would be precompact, hence compact. But by the Riesz Lemma, the unit ball in an infinite-dimensional normed space is never compact.

■

Remark

Compact operators form an ideal in the algebra of bounded operators: products of compact with bounded operators are compact. This algebraic structure is crucial for perturbation theory and operator equations.

The study of compact operators provides a bridge between finite-dimensional linear algebra and infinite-dimensional functional analysis, with applications to integral equations, spectral theory, and PDEs.