Compact Operators - Applied Functional Analysis

In finite dimensions, every linear operator maps bounded sets to bounded sets, and since closed bounded sets are compact (Heine-Borel), bounded sequences always have convergent subsequences. This is the engine behind most of finite-dimensional linear algebra: eigenvalue decompositions, the SVD, and the Fredholm alternative all rely on extracting convergent subsequences.

In infinite dimensions, closed bounded sets are no longer compact (Example 2), and this machinery breaks down for general bounded operators. Compact operators are precisely the class of operators for which it does not break down—they are the infinite-dimensional operators that still behave like matrices.

The eigenvalue decomposition and the SVD both reduce to finding a vector that achieves an extremum. The method is always:

Take a sequence approaching the supremum/infimum.
Extract a convergent subsequence (compactness of the unit ball).
The limit achieves the optimum (continuity).

Eigenvalues of a symmetric matrix are found by maximizing the Rayleigh quotient $R(x) = \langle Ax, x \rangle / \|x\|^2$ over the unit sphere. In $\mathbb{R}^n$ the unit sphere is compact, so a maximizing sequence has a convergent subsequence whose limit is an eigenvector. Restrict to its orthogonal complement and repeat—the full eigenvalue decomposition follows by induction. Each eigenvalue $\lambda_i$ yields a projection $P_i = \langle \cdot, \psi_i \rangle \psi_i$ onto the corresponding eigenspace, and the spectral decomposition $A = \sum \lambda_i P_i$ says the operator is a sum of scaled orthogonal projections.

The SVD of a general (non-symmetric) matrix follows the same pattern by reducing to the symmetric case: form $K^*K$ (which is self-adjoint and positive), apply the Rayleigh quotient argument to find its eigenvalues $\sigma_i^2$ and eigenvectors $v_i$ , then set $u_i = Kv_i / \sigma_i$ . The result is $K = \sum \sigma_i \langle \cdot, v_i \rangle \, u_i$ . Each step requires extracting a convergent subsequence from a maximizing sequence on the unit sphere.

A compact operator maps bounded sequences to sequences with convergent subsequences—by definition. This is step 2, transplanted from a property of the space to a property of the operator. For the Hilbert-Schmidt spectral theorem, you maximize the Rayleigh quotient on the unit sphere. You cannot extract a convergent subsequence from the maximizing sequence $(x_n)$ directly, but since $A$ is compact, $(Ax_n)$ has a convergent subsequence, which suffices to show $(x_n)$ converges to an eigenvector. Compactness of the operator substitutes for compactness of the ball.

Definition and Basic Properties¶

The sequential characterization is the one we use most often in practice: bounded sequence in, convergent subsequence out. Compare this with a general bounded operator, which only guarantees bounded sequence in, bounded sequence out.

The converse is false in infinite dimensions: the identity operator $I : \ell^2 \to \ell^2$ is bounded but not compact, since the orthonormal sequence $(e_n)$ is bounded but has no convergent subsequence.

The Space of Compact Operators¶

In algebraic language, $\mathcal{K}(X, Y)$ is a closed two-sided ideal in $\mathcal{L}(X)$ when $X = Y$ . Composing a compact operator with any bounded operator (on either side) produces another compact operator.

The closedness statement is the most important: it says that the operator-norm limit of compact operators is again compact. This is the key tool for proving compactness of specific operators.

Finite-Rank Operators: Compact Operators as “Infinite-Dimensional Matrices”¶

The connection between compact operators and matrices runs through finite-rank operators.

This is the precise sense in which compact operators generalize matrices. A matrix $A \in \mathbb{R}^{m \times n}$ defines an operator of rank at most $\min(m, n)$ —always finite. The key theorem (Theorem 1) now tells us:

Compact operators are precisely the operators that can be approximated by “matrices” (finite-rank operators) in the operator norm.

In Hilbert spaces this is exact: every compact operator is the operator-norm limit of finite-rank operators. In general Banach spaces, this is the approximation property (which most natural spaces satisfy, though Enflo showed it can fail).

Remark 1

The analogy between compact operators and matrices is not just a slogan. Here is a dictionary:

Finite dimensions ( $\mathbb{R}^n \to \mathbb{R}^m$ )	Compact operators ( $X \to Y$ )
Every operator has finite rank	Approximated by finite-rank operators
Spectrum = eigenvalues	Spectrum = eigenvalues $\cup\ \{0\}$
Each eigenvalue has finite multiplicity	Each nonzero eigenvalue has finite multiplicity
SVD: $A = \sum_{i=1}^r \sigma_i u_i v_i^T$	SVD: $K = \sum_{i=1}^\infty \sigma_i \langle \cdot, v_i \rangle \, u_i$
Symmetric: $A = Q\Lambda Q^T$	Self-adjoint: $K = \sum_{i=1}^\infty \lambda_i \langle \cdot, \psi_i \rangle \, \psi_i$
Fredholm alternative holds	Fredholm alternative holds

The SVD holds for every compact operator between Hilbert spaces. The eigenvalue decomposition into an orthonormal eigenbasis requires the additional assumption that the operator is self-adjoint (just as $A = Q\Lambda Q^T$ requires symmetry in finite dimensions). Without self-adjointness, a compact operator’s eigenvectors need not form a basis.

Everything that makes finite-dimensional linear algebra work carries over to compact operators. What fails for general bounded operators in infinite dimensions is exactly what compact operators restore.

The Key Example: Integral Operators¶

Our main source of compact operators is integral operators with square-integrable kernels.

Example 1 (The solution operator for Poisson’s equation is compact)

Consider the Poisson equation on $[0, L]$ with Neumann boundary conditions:

-u'' = f, \quad u'(0) = u'(L) = 0, \quad \int_0^L u(x) \, dx = M.

(7)

Integrating twice and applying Fubini gives the solution

u(x) = \int_0^L g(x, s) f(s) \, ds + c

(8)

where $g$ is the Green’s function (an $L^2$ kernel). The solution operator $A^{-1} : f \mapsto u$ is an integral operator with an $L^2$ kernel, hence compact by Theorem 2.

This is a fundamental pattern: differential operators are unbounded, but their inverses (solution operators) are often compact. The compactness of $A^{-1}$ is what gives the Laplacian a discrete spectrum of eigenvalues.

Spectral Theory of Compact Operators¶

The spectral theory of compact operators is the payoff: it tells us that compact operators have a spectrum that looks just like the spectrum of a matrix, up to a possible accumulation point at zero.

The Fredholm Alternative¶

Theorem 3 (Spectral Fredholm Alternative)

Let $A : H \to H$ be a compact linear operator on a Hilbert space $H$ . Then:

The spectrum $\sigma(A)$ is a compact subset of $\mathbb{C}$ whose only possible accumulation point is $\lambda = 0$ .
For each $\lambda \in \mathbb{C} \setminus \{0\}$ , exactly one of the following holds:
- (Alternative 1): $\lambda \in \rho(A)$ , i.e., $(A - \lambda I)^{-1}$ exists and is bounded. The equation $Ax - \lambda x = y$ has a unique solution for every $y$ .
- (Alternative 2): $\lambda \in \sigma_p(A)$ is an eigenvalue of finite multiplicity. The equation $Ax - \lambda x = 0$ has a nontrivial finite-dimensional solution space.

The Hilbert-Schmidt Theorem¶

When the compact operator is additionally self-adjoint, we get a complete spectral decomposition—the infinite-dimensional analogue of the eigenvalue decomposition of a symmetric matrix.

This is the infinite-dimensional eigendecomposition for self-adjoint operators: the operator $A$ is completely determined by its eigenvalues and eigenfunctions, just as a symmetric matrix is determined by its eigenvalues and eigenvectors. The spectral representation $A = \sum \lambda_i \langle \cdot, \psi_i \rangle \psi_i$ is the direct analogue of the matrix diagonalization $A = Q \Lambda Q^T$ . Note that this is not the SVD—it is the eigenvalue decomposition, which requires self-adjointness. The SVD $K = \sum \sigma_i \langle \cdot, v_i \rangle u_i$ is a separate factorization that works for all compact operators (see Remark 1).

Application: Spectral Theory of the Laplacian¶

Looking Ahead¶

Compact operators are the bridge between the abstract operator theory of this chapter and several later topics in the course:

Weak convergence + compact operator $\Rightarrow$ strong convergence. If $x_n \rightharpoonup x$ weakly and $K$ is compact, then $Kx_n \to Kx$ strongly. This is a key tool in the calculus of variations for passing to the limit in nonlinear problems.
Rellich-Kondrachov compactness. The Sobolev embedding $H^1(\Omega) \hookrightarrow L^2(\Omega)$ is compact for bounded $\Omega$ . This is the source of compactness in elliptic PDE theory and the reason the direct method of the calculus of variations works.
Fixed point theory. Compact operators are the setting for Schauder’s fixed point theorem and the Leray-Schauder degree, which extend Brouwer’s fixed point theorem to infinite dimensions.