Classical Perron-Frobenius Theory Research Articles

A dynamical approach to the Perron-Frobenius theory and generalized Krein-Rutman type theorems

We present a dynamical approach to the classical Perron-Frobenius theory by using some elementary knowledge on linear ODEs. It is completely self-contained and significantly different from those in the literature. As a result, we develop a complex version of the Perron-Frobenius theory and prove some generalized Krein-Rutman type theorems.

Eventual Cone Invariance

Eventually nonnegative matrices are square matrices whose powers become and remain (entrywise) nonnegative. Using classical Perron-Frobenius theory for cone preserving maps, this notion is generalized to matrices whose powers eventually leave a proper cone K ⊂ R^n invariant, that is, A^mK ⊆ K for all sufficiently large m. Also studied are the related notions of eventual cone invariance by the matrix exponential, as well as other generalizations of M-matrix and dynamical system notions.

Irreducible semigroups of positive operators on Banach lattices

The classical Perron–Frobenius theory asserts that an irreducible matrix has cyclic peripheral spectrum and its spectral radius is an eigenvalue corresponding to a positive eigenvector. This was extended by Radjavi and Rosenthal to semigroups of matrices and of compact operators on -spaces. We extend this approach to operators on an arbitrary Banach lattice . We prove, in particular, that if is a commutative irreducible semigroup of positive operators on containing a compact operator then there exist positive disjoint vectors in such that every operator in acts as a positive scalar multiple of a permutation on . Compactness of may be replaced with the assumption that is peripherally Riesz, i.e. the peripheral spectrum of is separated from the rest of the spectrum and the corresponding spectral subspace is finite dimensional. Applying the results to the semigroup generated an irreducible peripherally Riesz operator , we show that is a cyclic permutation on , , and if for some in and then for some and . We also extend results of Drnovšek and Levin about peripheral spectra of irreducible operators.

Extensions of Perron–Frobenius theory

The classical Perron–Frobenius theory asserts that, for two matrices $$A$$ and $$B$$ , if $$0\le B \le A$$ and $$r(A)=r(B)$$ with $$A$$ being irreducible, then $$A=B$$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that $$r(A)$$ is a pole of the resolvent of $$A$$ . In this paper, we prove that the same result holds if $$B$$ is irreducible and $$r(B)$$ is a pole of the resolvent for $$B$$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.

Variational characterizations of the sign-real and the sign-complex spectral radius

The sign-real and the sign-complex spectral radius, also called the generalized spec- tral radius, proved to be an interesting generalization of the classical Perron-Frobenius theory (for nonnegative matrices) to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one correspondences to classical Perron-Frobenius theory. In this paper the corresponding inf-max characterization as well as variational characterizations of the generalized (real and com- plex) spectral radius are presented. Again those are almost identical to the corresponding results in classical Perron-Frobenius theory.

Concave Perron–Frobenius Theory and applications

Many important results of classical Perron–Frobenius Theory can be extended from linear selfmappings of the standard cone in finite dimensional real space to concave selfmappings of this cone. This is in particular true for minima of linear mappings, albeit the spectrum of these special concave mappings is more intricate than that for linear mappings. As classical Perron–Frobenius Theory has numerous applications there are many new applications for its concave extension. 1991 Mathematical Subject Classification: Primary 39A11, 15A48; secondary 47H07, 47H12.

A hyperplane approach to the EMS algorithm

By proving that the image of the EMS (Estimate, Maximize and Smooth) map lies in a certain hyperplane, the existence of EMS solutions is established. Classical Perron-Frobenius theory is then used to define a new class of smoothing matrices which identify implicit conditions where the convergence can be tuned to the underlying maximum likelihood solution.

The Perron-Frobenius theory for almost periodic representations of semigroups in the spaces Lp

The classical Perron-Frobenius theory of nonnegative matrices is generalized to nonnegative almost periodic representations of topological semigroups in the spaces Lp(Ω, σ, Μ), where (Ω, σ, Μ) is a space with a σ-finite measure, 1≤p<∞. With each such representation one connects the associated action of its Sushkevich kernel onto some naturally arising space with measure; this allows that the investigation of the spectral properties of the representation be reduced to the investigation of the ergodic properties of the corresponding action. In particular, it is established, that the boundary spectrum of an indecomposable representation is a subgroup of the dual group of the Sushkevich kernel (coincides with it if the considered semigroup is Abelian). In the general case the boundary spectrum is cyclic (i.e., the union of the subgroups of the dual group of the Sushkevich kernel). The results of the paper are new even at the consideration of the semigroups of degree one of an operator (if other words, the representations of the semigroup Z+); this yields the generalized Perron-Frobenius theory for nonnegative a. p. operators.

The lattice of faces of a finite dimensional cone

In recent years the classical Perron-Frobenius theory has been extended to matrices which leave invariant a cone in the finite dimensional real space V. Here we shall be concerned with the geometry of those cones which are suitable for Perron-Frobenius theory. l Although there is an extensive theory dealing with the lattice of faces of a polyhedral convex set, our work is of a different spirit, especially as parts of it extend to cones which are not polyhedral.

Extensions of a Limit Theorem of Everett, Ulam and Harris on Multitype Branching Processes to a Branching Process with Countably Many Types

Extensions of a Limit Theorem of Everett, Ulam and Harris on Multitype Branching Processes to a Branching Process with Countably Many Types