On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator

Förster, K.-H.; Nagy, B.

doi:10.1016/0024-3795(89)90378-9

Cited by 20 publications

(12 citation statements)

References 7 publications

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“…Bohl assumes that the cone is solid, some power of B is compact and that B is order preserving in some strict sense (actually it is enough that B commutes with such a map). Bohl's proof is constructive as it provides the convergence of the (ratio) power method (von Mises procedure [41]) to the eigenvector x and the eigenvalue λ, and it also establishes convergence from below and above to the spectral radius by what are sometimes called Collatz-Wielandt numbers [10,15,42].…”

Section: Cone and Orbital Spectral Radius For Bounded Homogeneous Mapsmentioning

confidence: 99%

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Eigenvectors of homogeneous order-bounded order-preserving maps

Thieme¹

2017

Discrete &Amp; Continuous Dynamical Systems - B

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This paper considers homogeneous order preserving continuous maps on the normal cone of an ordered normed vector space. It is shown that certain operators of that kind which are not necessarily compact themselves but have a compact power have a positive eigenvector that is associated with the cone spectral radius. We also derive conditions for the existence of homogeneous order preserving eigenfunctionals. Our results are illustrated in a model for spatially distributed two-sex populations.In the following, X, Y and Z are ordered normed vector spaces with cones X + , Y + and Z + respectively,Since we do not consider maps that are homogeneous in other ways, we will simply call them homogeneous maps. If follows from the definition that B0 = 0.Homogeneous maps are not Frechet differentiable at 0 unless B(x + y) = Bx + By for all x, y ∈ X + . For the following holds.

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Section: Cone and Orbital Spectral Radius For Bounded Homogeneous Mapsmentioning

confidence: 99%

“…Mallet-Paret and Nussbaum [25,26] call r + (B) the Bonsall cone spectral radius and r o (B) the cone spectral radius. For x ∈ X + , the number γ B (x) has been called local spectral radius of B at x by Förster and Nagy [15].…”

Section: Cone and Orbital Spectral Radius For Bounded Homogeneous Mapsmentioning

confidence: 99%

Eigenvectors of homogeneous order-bounded order-preserving maps

Thieme¹

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…Mallet-Paret and Nussbaum [26,27] call r + (B) the Bonsall cone spectral radius and r o (B) the cone spectral radius. For x ∈ X + , the number γ B (x) has been called local spectral radius of B at x by Förster and Nagy [16].…”

Section: Cone and Orbital Spectral Radiusmentioning

confidence: 99%

Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces

Thieme

2014

Preprint

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“…Our goal is to give other characterizations of λ w (x). Theorem 4.2 shows that, for every n ≥ 1 [2], [3] and [8]). We note that equation (4.11) is particularly useful to compute the value of λ w (indeed solving the linear inequality therein is easier than solving the nonlinear inequality in (4.10)).…”

Section: Infinite-type Branching Processesmentioning

confidence: 99%

“…Indeed in [1] we proved that in the irreducible case, λ w ≥ 1/M w and we gave sufficient conditions for equality (for instance all site-breeding BRWs satisfy these conditions). In this paper we use a different approach which allows us to characterize λ w (x) in terms of the existence of solutions of certain infinite-dimensional linear systems (Theorem 4.2); in particular we show that λ w (x) is related to the so-called Collatz-Wielandt numbers of some linear operator (see [2], [3] and [8] for the definition).…”

Section: Introductionmentioning

confidence: 99%

Characterization of Critical Values of Branching Random Walks on Weighted Graphs through Infinite-Type Branching Processes

Bertacchi

Zucca

2008

J Stat Phys

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We study the branching random walk on weighted graphs; site-breeding and edgebreeding branching random walks on graphs are seen as particular cases. Two kinds of survival can be identified: a weak survival (with positive probability there is at least one particle alive somewhere at any time) and a strong survival (with positive probability the colony survives by returning infinitely often to a fixed site). The behavior of the process depends on the value of a certain parameter which controls the birth rates; the threshold between survival and (almost sure) extinction is called critical value. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.

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On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator

Cited by 20 publications

References 7 publications

Eigenvectors of homogeneous order-bounded order-preserving maps

Eigenvectors of homogeneous order-bounded order-preserving maps

Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces

Characterization of Critical Values of Branching Random Walks on Weighted Graphs through Infinite-Type Branching Processes

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