Compact Operators

A linear operator $T : X \to Y$ is compact if the image of the closed unit ball, $T(B_X)$, is relatively compact (its closure is compact) in $Y$.

Proposition 5.1. Every compact operator is bounded. Every finite-rank operator is compact.

Proposition 5.2. If $T$ is compact and $S$ is bounded, then $TS$ and $ST$ are compact.

Proposition 5.3. If $T_n$ are compact and $T_n \to T$ in operator norm, then $T$ is compact.

Theorem 5.4 (Spectral Theorem for Compact Self-Adjoint Operators). Let $T$ be a compact self-adjoint operator on a Hilbert space $H$. Then:

1. All eigenvalues of $T$ are real.
2. Eigenvectors corresponding to distinct eigenvalues are orthogonal.
3. There exists an orthonormal basis of $H$ consisting of eigenvectors of $T$.
4. If $\{\lambda_n\}$ are the nonzero eigenvalues with orthonormal eigenvectors $\{e_n\}$, then $Tx = \sum_n \lambda_n \langle x, e_n\rangle e_n$.

Corollary 5.5. A compact self-adjoint operator on an infinite-dimensional Hilbert space has at most countably many nonzero eigenvalues, and $0$ is the only possible accumulation point.

The spectral theorem for compact self-adjoint operators is proved here because it is the bridge between two theories the network has already built: the spectral theorem for self-adjoint operators on the graduate analysis page built a projection-valued measure by hand, and the Sobolev page built compactness from Rellich–Kondrachov. The compact spectral theorem is what the two give when combined; and its proof via the Rayleigh quotient is the template for every existence-of-eigenfunctions argument in the subject, from the Sturm–Liouville theory to the Hodge theorem’s harmonic forms.

Proposition 5.1 (compact implies bounded; finite rank implies compact).

Proof. Compact sets are bounded, so $T(B_X)$ is bounded, and linearity then scales: $\|Tx\| \leq M\|x\|$ with $M = \sup_{\|z\|\leq1}\|Tz\|$. For finite rank, let $y_1, \dots, y_m$ span $T(X)$; write $Tx = \sum_k \varphi_k(x) y_k$ with $\varphi_k$ linear functionals; for instance the coefficients in any fixed basis of the range. On the unit ball each $\varphi_k$ is bounded (finite- dimensional range), so $T(B_X)$ is a bounded subset of a finite-dimensional space, hence relatively compact by the Heine–Borel theorem. $\blacksquare$

Theorem 5.4 (spectral theorem for compact self-adjoint $T \neq 0$). $T$ has an eigenvalue $\lambda$ with $|\lambda| = \|T\|$; eigenvectors for distinct eigenvalues are orthogonal; the nonzero spectrum consists of eigenvalues, each of finite multiplicity, with $0$ the only possible accumulation point; and there is an orthonormal basis of $\overline{\operatorname{ran}}(T)$ of eigenvectors, giving $Tx = \sum_n \lambda_n \langle x, e_n\rangle e_n$.

Proof, in four steps.

Step 1: an eigenvalue exists with modulus $\|T\|$. For self-adjoint $T$ define the Rayleigh quotient $\mu = \sup_{\|z\| = 1} |\langle Tz, z\rangle|$. Two facts: self-adjointness makes every $\langle Tz, z\rangle$ real, and the polarisation identity gives $\|T\| = \mu$ — both standard for self-adjoint operators, the norm identity proved by expanding $\|T(z \pm z')\|^2$ and recovering $\langle Tz, z'\rangle$ by polarisation. Choose unit vectors $x_n$ with $\langle Tx_n, x_n\rangle \to \mu$.

The key computation is the expansion of the squared norm:

using $\|Tx_n\| \leq \|T\| = \mu$ and $\langle Tx_n, x_n\rangle \to \mu$. So $(T - \mu)x_n \to 0$ in norm with $\|x_n\| = 1$ throughout.

Compactness finishes. $Tx_n$ has a norm-convergent subsequence $Tx_{n_k} \to y$ — the defining property of the compact operator applied to the bounded sequence $(x_n)$. Since $Tx_{n_k} - \mu x_{n_k} \to 0$, we get $\mu x_{n_k} \to y$, so $x_{n_k} \to y/\mu$ with $\|y/\mu\| = 1$. By continuity of $T$, $T(y/\mu) = y$, i.e. $Ty = \mu y$: $y$ is an eigenvector with eigenvalue $\mu$, and $|\mu| = \|T\|$. $\blacksquare$

The proof’s three ingredients are worth separating, because each fails in a context where the theorem fails. Self-adjointness makes the Rayleigh quotient real and equal to the norm — a non-self-adjoint operator has no such quotient. Attainment of the supremum by a sequence needs the unit ball’s weak compactness machinery, which for reflexive spaces comes from Kakutani. And compactness is what turns the weak limit of the images into a norm limit — the step that manufactures the eigenvector. A bounded non-compact self-adjoint operator can fail at any of the three, and the multiplication operator on $L^2[0,1]$ fails at all of them: its Rayleigh quotients fill an interval and no value is attained.

Step 2: orthogonality. If $Tx = \lambda x$ and $Ty = \mu y$ with $\lambda \neq \mu$, then $\lambda\langle x, y\rangle = \langle Tx, y\rangle = \langle x, Ty\rangle = \bar\mu\langle x, y\rangle$, and self-adjointness makes $\mu$ real, so $(\lambda - \mu)\langle x, y\rangle = 0$ forces orthogonality. $\blacksquare$

Step 3: the nonzero spectrum consists of eigenvalues of finite multiplicity, accumulating only at zero. If $|\lambda| \geq \epsilon > 0$ for infinitely many orthonormal eigenpairs $(\lambda_n, e_n)$, then $e_n \rightharpoonup 0$ (Bessel’s inequality: every $y$ has $\sum|\langle y, e_n\rangle|^2 < \infty$), so compactness gives $\lambda_n e_n = Te_n \to$ norm-limit $0$; but $\|\lambda_n e_n\| = |\lambda_n| \geq \epsilon$, contradiction. So only finitely many eigenvalues exceed any $\epsilon$, hence the nonzero eigenvalues are countable with $0$ the only accumulation point, and each eigenspace is finite-dimensional (an infinite orthonormal set inside one eigenspace would violate compactness the same way). $\blacksquare$

Step 4: the expansion. Let $\{e_n\}$ be an orthonormal basis of $\overline{\operatorname{ran}}(T)$ of eigenvectors, which exists by inductively applying Step 1 to the orthogonal complement of the eigenvectors found so far; the complement is invariant under $T$ by self-adjointness, and the process exhausts the range because otherwise the restriction to the remaining complement would be compact self-adjoint with norm equal to its largest $|\lambda|$, contradicting the eigenvalue exhaustion. For $x \in \overline{\operatorname{ran}}(T)$, write $x = \sum_n \langle x, e_n\rangle e_n$; applying $T$ and using continuity of $T$ plus boundedness of the coefficients,

Corollary 5.5 follows: the nonzero eigenvalues are countable (Step 3) with $0$ as their only accumulation point, and on an infinite-dimensional space $0$ must be in the spectrum since a bijective compact $T$ would make $T^{-1}$ a bounded inverse of a compact operator; impossible unless the space is finite-dimensional, by the compactness of the unit ball it would imply. $\blacksquare$

A bounded operator $T$ on a Hilbert space $H$ is normal if $T^*T = TT^*$, and unitary if $T^*T = TT^* = I$.

Proposition 5.6. If $T$ is normal, then $\|Tx\| = \|T^*x\|$ for all $x \in H$.

Proposition 5.7. If $T$ is normal, then $\ker T = \ker T^*$ and eigenvectors corresponding to distinct eigenvalues are orthogonal.

Theorem 5.8 (Spectral Theorem for Normal Compact Operators). Let $T$ be a compact normal operator on a Hilbert space $H$. Then there exists an orthonormal basis of $H$ consisting of eigenvectors of $T$, and the eigenvalues satisfy $|\lambda_n| \to 0$.

This generalises Theorem 5.4: self-adjoint operators are normal, and unitary operators are normal (with eigenvalues on the unit circle in $\mathbb{C}$).

Theorem 5.9 (Spectral Theorem for Bounded Normal Operators). Let $T$ be a bounded normal operator on $H$. There exists a unique projection-valued measure $E$ on the Borel subsets of $\sigma(T) \subseteq \mathbb{C}$ such that

This integral representation implies: if $f$ is a bounded Borel function on $\sigma(T)$, then $f(T) = \int f(\lambda)\, dE(\lambda)$ defines a bounded normal operator satisfying the functional calculus relations.

Theorem 5.6 (Fredholm Alternative). Let $T$ be a compact operator on a Banach space $X$ and $\lambda \neq 0$. Then exactly one of the following holds:

1. $(\lambda I - T)$ is bijective (hence invertible by the bounded inverse theorem).
2. $(\lambda I - T)x = 0$ has a nontrivial solution (i.e., $\lambda$ is an eigenvalue of $T$).

• Bounded Linear Operators: Provides the framework of bounded linear maps between Banach spaces within which compact operators are defined.

• The Fundamental Theorems: The bounded inverse theorem and closed graph theorem are used in the Fredholm alternative for compact operators.

• Weak and Weak* Convergence: Compact operators map weakly convergent sequences to strongly convergent sequences, linking these two modes of convergence.

• Quantum Mechanics

• Graph Theory

• Classical Mechanics

• Electromagnetism

• Statistical Learning

• Statistical Mechanics

Compact operators are the infinite-dimensional analogue of finite matrices. In finite dimensions, the unit ball is compact and every bounded operator maps it to a relatively compact set. In infinite dimensions this fails, and compact operators are precisely those that restore this finite-dimensional behaviour: they squeeze the unit ball into a nearly finite-dimensional image. The spectral theorem for compact self-adjoint operators decomposes them as weighted sums of rank-one projections, with eigenvalues decaying to zero. The Fredholm alternative then says that for compact operators, solving a linear equation reduces to a finite-dimensional problem: either the homogeneous equation has only the trivial solution, or it has a finite-dimensional space of solutions.

• Assuming all bounded operators are compact. The identity operator on an infinite-dimensional Hilbert space is bounded but not compact: $B_H$ is closed but not compact (since the unit ball is not compact in infinite dimensions).
• Thinking finite-rank operators are the only compact operators. The limit of finite-rank operators is compact, and on Hilbert spaces every compact operator is the norm limit of finite-rank operators. But there exist compact operators that are not themselves finite-rank.
• Forgetting that $0$ is always in the spectrum of a compact operator on an infinite-dimensional space. Even if $0$ is not an eigenvalue, it belongs to the spectrum (as an accumulation point of eigenvalues or as essential spectrum).
• Confusing the Fredholm alternative with the Fredholm index. The alternative deals with solvability of $(\lambda I - T)x = y$ for compact $T$. The Fredholm index $\text{ind}(T) = \dim\ker T - \text{codim}\ \text{ran}\ T$ applies to Fredholm operators more generally.

Problem 1. Show that the Volterra operator $(Vf)(x) = \int_0^x f(t)\,dt$ on $C([0,1])$ is compact.

Solution. $V$ is bounded: $|(Vf)(x)| \leq \int_0^1 |f(t)|\,dt \leq \|f\|_\infty$. The image $V(B_{C([0,1])})$ is equicontinuous (by the fundamental theorem of calculus, derivatives are bounded by $\|f\|_\infty$), and uniformly bounded. By the Arzela-Ascoli theorem, it is relatively compact in $C([0,1])$. Therefore $V$ is compact. $\blacksquare$

Problem 2. Let $T$ be a compact self-adjoint operator on $H$ with eigenvalues $\lambda_n \to 0$. Show that $T$ has a maximum eigenvalue (in absolute value).

Solution. Since $\lambda_n \to 0$, the set $\{|\lambda_n|\}$ has a maximum (attained at some finite index) if there are finitely many nonzero eigenvalues; otherwise, the maximum of $|\lambda_n|$ is attained at the first eigenvalue (since they converge to $0$). In either case, $\max_n |\lambda_n| = \|T\|$ by the spectral radius formula for self-adjoint operators. $\blacksquare$

• Integral equations: Fredholm integral equations of the second kind $f - Kf = g$ are solved using the Fredholm alternative. Compact integral operators arise in potential theory and scattering.
• Quantum mechanics: Position and momentum operators are unbounded, but their resolvents $(H - z)^{-1}$ are often compact. The spectral theorem for compact operators underlies the solution of the Schrodinger equation for bound states.
• Differential equations: The inverse of a differential operator with compact resolvent (e.g., $-\Delta + V$ on a bounded domain) is compact, ensuring discrete spectrum, the basis for Sturm-Liouville theory.
• Signal processing: The Karhunen-Loeve transform uses the spectral decomposition of compact covariance operators to find optimal bases for signal representation and compression (PCA).

• Applications
• Bounded Linear Operators
• Historical Context
• Inner Product Spaces and Hilbert Spaces