<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">max</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math> and S properties and asymptotic stability in the max algebra

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We present several results describing the interplay between the max algebraic joint spectral radius (JSR) for compact sets of matrices and suitably defined matrix norms. In particular, we extend a classical result for the conventional algebra, showing that the JSR can be described in terms of induced norms of the matrices in the set. We also show that for a set generating an irreducible semigroup (in a cone-theoretic sense), a monotone Barabanov norm always exists. This fact is then used to show that the max algebraic JSR is locally Lipschitz continuous on the space of compact irreducible sets of matrices with respect to the Hausdorff distance. We then prove that the JSR is Hoelder continuous on the space of compact sets of nonnegative matrices. Finally, we prove a strict monotonicity property for the max algebraic JSR that echoes a fact for the classical JSR.

Section: The Max-algebraic Joint Spectral Radius -Elementary Propertiesmentioning

confidence: 90%

Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius

Guglielmi

Mason

Wirth

2018

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Fiedler–Pták scaling in max algebra

Sergeev

2013

Bounds on the generalized and the joint spectral radius of Hadamard products of bounded sets of positive operators on sequence spaces

Peperko

2012

Let Ψ 1 ,. .. Ψ m be bounded sets of positive kernel operators on a Banach function space L. We prove that for the generalized spectral radius ρ and the joint spectral radiusˆρradiusˆ radiusˆρ the inequalities ρ Ψ (1 m) 1 • · · · • Ψ (1 m) m ≤ ρ(Ψ 1 Ψ 2 · · · Ψ m) 1 m , ˆ ρ Ψ (1 m) 1 • · · · • Ψ (1 m) m ≤ ˆ ρ(Ψ 1 Ψ 2 · · · Ψ m) 1 m hold, where Ψ (1 m) 1 • · · · • Ψ (1 m) m denotes the Hadamard (Schur) geometric mean of the sets Ψ 1 ,. .. , Ψ m .