Compact Operators - Examples and Constructions

Approximation by finite-rank operators is a fundamental technique for studying compact operators, connecting abstract theory with concrete computations.

TheoremApproximation by Finite Rank

An operator $T \in \mathcal{B}(X,Y)$ is compact if and only if it can be approximated in operator norm by finite-rank operators. That is, there exist finite-rank operators $T_n$ such that $\|T - T_n\| \to 0$ .

This characterization is extremely useful: to prove an operator is compact, it suffices to approximate it by finite-rank operators.

DefinitionHilbert-Schmidt Operators

Let $H$ be a Hilbert space with orthonormal basis $\{e_n\}$ . An operator $T \in \mathcal{B}(H)$ is Hilbert-Schmidt if $\|T\|_{HS}^2 = \sum_{n=1}^\infty \|T(e_n)\|^2 < \infty$

The Hilbert-Schmidt norm $\|\cdot\|_{HS}$ is independent of the choice of orthonormal basis.

TheoremHilbert-Schmidt Operators are Compact

Every Hilbert-Schmidt operator is compact. Moreover, the space of Hilbert-Schmidt operators forms a Hilbert space with inner product $\langle S, T \rangle_{HS} = \sum_{n=1}^\infty \langle S(e_n), T(e_n) \rangle$

Proof

For finite-rank projection $P_N = \sum_{n=1}^N \langle \cdot, e_n \rangle e_n$ , we have $\|T - P_N T\|^2 \leq \|T - P_N T\|_{HS}^2 = \sum_{n=N+1}^\infty \|T(e_n)\|^2 \to 0$

Thus $T$ is approximated by finite-rank operators $P_N T$ , hence compact.

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ExampleIntegral Operators

Consider $(Kf)(x) = \int_\Omega k(x,y) f(y) \, dy$ on $L^2(\Omega)$ .

Continuous kernel: If $k \in C(\Omega \times \Omega)$ and $\Omega$ is compact, then $K$ is compact
$L^2$ kernel: If $k \in L^2(\Omega \times \Omega)$ , then $K$ is Hilbert-Schmidt, hence compact, with $\|K\|_{HS}^2 = \int_\Omega \int_\Omega |k(x,y)|^2 \, dx \, dy$
Degenerate kernel: If $k(x,y) = \sum_{i=1}^n f_i(x) g_i(y)$ , then $K$ has finite rank

DefinitionTrace Class Operators

An operator $T$ on a Hilbert space is trace class if for some (equivalently all) orthonormal basis $\{e_n\}$ , $\|T\|_1 = \sum_{n=1}^\infty \langle |T|(e_n), e_n \rangle < \infty$ where $|T| = \sqrt{T^*T}$ . Trace class operators are compact.

Remark

The hierarchy is: trace class $\subset$ Hilbert-Schmidt $\subset$ compact $\subset$ bounded. Each inclusion is strict, and each class has progressively weaker properties but broader applicability.

Understanding these operator classes is essential for quantum mechanics, where observables are represented by operators and trace class operators correspond to density matrices.