Integral equations Eigenvalue problems for integral equations Theorem

Jentzsch's theorem on positive kernels

Builds on Continuous kernel Eigenfunction of an integral operator Spectral radius Krein–Rutman theorem Perron–Frobenius theorem Compactness of integral operators

Statement

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \R$ be a continuous kernel with $K(x, y) > 0$ for all $x, y \in [a, b]$ . Let $T_K$ be the integral operator $(T_K \varphi)(x) = \int_a^b K(x, y) \varphi(y) \, dy$ , and let $r$ be its spectral radius. Then:

$r > 0$ , and $r$ is an eigenvalue of $T_K$ with an eigenfunction $u$ that is continuous and satisfies $u(x) > 0$ for all $x \in [a, b]$ ;
the eigenspace of $r$ is spanned by $u$ ;
every other eigenvalue $\mu$ of $T_K$ , real or complex, satisfies $\abs{\mu} < r$ .

Remarks

The operator $T_K$ is compact on $C[a, b]$ by the compactness of integral operators, and it maps the cone of nonnegative functions into itself. Since $K$ has a positive minimum on the square, $T_K$ sends every nonnegative function that is not identically $0$ to a strictly positive function. So it maps the cone minus $0$ into the interior of the cone. The strong form of the Krein–Rutman theorem then gives all three parts.

Example: for $K(x, y) = 1 + xy$ on $[0, 1]$ , $T_K$ has rank $2$ and maps into the span of $1$ and $x$ . Its matrix in this basis is $\begin{pmatrix} 1 & 1/2 \\ 1/2 & 1/3 \end{pmatrix}$ , with eigenvalues $\frac{4 \pm \sqrt{13}}{6}$ . The larger one, $r = \frac{4 + \sqrt{13}}{6}$ , has the eigenfunction $u(x) = 1 + 2(r - 1) x = 1 + \frac{\sqrt{13} - 2}{3} x$ , which is positive on $[0, 1]$ . The other eigenvalues, $\frac{4 - \sqrt{13}}{6}$ and $0$ , have modulus less than $r$ .

Strict positivity matters. Take $K(x, y) = 1$ when $y \le x$ and $0$ otherwise, which is nonnegative but neither continuous nor positive. Its operator is the Volterra operator, whose spectral radius is $0$ and which has no eigenvalue at all.

Going deeper

For a student

The theorem explains the long-time behaviour of models with a positive integral operator, such as population models structured by size. After many steps, the state lines up with the positive eigenfunction $u$ and grows by the factor $r$ per step. It is a special case of the Krein–Rutman theorem for strongly positive compact operators on ordered Banach spaces.

Reading path

To understand this page from the ground up, read these 101 pages in order. The foundations come first.

Banach space concept
Bernstein polynomial definition
Characteristic polynomial concept
Compact operator concept
Compact space concept
Continuous kernel concept
Degenerate kernel concept
Dense set concept
Eigenfunction of an integral operator concept
Eigenvalue concept
Eigenvector concept
Element definition
Extended real number line object
Finite-rank operator concept
Hausdorff space concept
Hilbert–Schmidt integral operator concept
Irreducible matrix concept
Limit point of a set definition
Nonnegative matrix concept
Orthonormal basis concept
Proposition concept
Real numbers object
Sequence definition
Set bounded above definition
Spectral radius concept
Square-integrable kernel concept
Uniform metric on bounded functions example
Krein–Rutman theorem statement
Set definition
Perron–Frobenius theorem statement
Limit of a function definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Real sequence definition
Subsequence definition
Supremum definition
Empty set object
Family of sets definition
Ordered pair definition
Continuity at a point definition
Intersection definition
Union definition
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Cartesian product definition
Continuous function definition
Epsilon-delta definition of continuity statement
Union of a family of sets definition
Subset definition
Uniqueness quantifier definition
Convergent sequence definition
Monotone convergence theorem for sequences statement
Pointwise convergence definition
Every sequence has a monotone subsequence statement
Binary relation definition
Uniform continuity definition
Topological space definition
Bolzano–Weierstrass theorem statement
Uniform convergence definition
Uniform continuity on a set definition
Sequential compactness definition
Function definition
Subspace topology definition
Continuous functions on a closed bounded interval are uniformly continuous statement
Preimage of a set definition
Restriction of a function definition
Metric space definition
Weierstrass approximation theorem statement
Convergence of a sequence in a metric space definition
Open ball definition
Subspace metric definition
Diameter of a set definition
Metric topology definition
Epsilon characterisation of Cauchy sequences statement
Totally bounded set definition
Open set definition
Complete metric space definition
Continuous map of topological spaces definition
Open sets of a metric space statement
Open cover definition
Open sets characterised by neighbourhoods statement
Compact space definition
Continuity at a point of a topological space definition
Completeness of the space of bounded continuous functions statement
Epsilon-delta continuity in metric spaces statement
Compact set definition
Lebesgue number lemma statement
Sup norm on bounded continuous functions example
Compactness and sequential compactness agree in metric spaces statement
Equicontinuity definition
Stone–Weierstrass theorem statement
Compact iff complete and totally bounded statement
Approximation of a kernel by degenerate kernels statement
Arzelà–Ascoli theorem statement
Compactness of integral operators statement
Jentzsch's theorem on positive kernels statement

Tag	`P8CAK0FHGJ`
URI	`m2://P8CAK0FHGJ`
Kind	statement, theorem
Writing	W0 stub
MSC	45C05, 47B65
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About	Continuous kernel
Uses	Continuous kernel, Eigenfunction of an integral operator, Spectral radius, Krein–Rutman theorem, Perron–Frobenius theorem, Compactness of integral operators