scholarly journals Generalized spectral radius and its max algebra version

2013 ◽  
Vol 439 (4) ◽  
pp. 1006-1016 ◽  
Author(s):  
Vladimir Müller ◽  
Aljoša Peperko
Keyword(s):  
Spectral Radius ◽  
Max Algebra ◽  
Generalized Spectral Radius

Generalized spectral radius and norm inequalities for Hilbert space operators

2015 ◽  
Vol 26 (12) ◽  
pp. 1550097
Author(s):  
Amer Abu-Omar ◽  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Spectral Radius ◽  
Operator Matrices ◽  
Norm Inequalities ◽  
Hilbert Space Operators ◽  
Generalized Spectral Radius

We apply spectral radius and norm inequalities to certain [Formula: see text] operator matrices to give simple proofs, refinements and generalizations of known norm inequalities. New norm inequalities are also given. Our analysis uncovers the interplay between different spectral radius and norm inequalities.

Approximating the spectral radius of sets of matrices in the max-algebra is NP-hard

2000 ◽  
Vol 45 (9) ◽  
pp. 1762-1765 ◽  
Author(s):  
V.D. Blondel ◽  
S. Gaubert ◽  
J.N. Tsitsiklis
Keyword(s):  
Spectral Radius ◽  
Np Hard ◽  
Max Algebra

scholarly journals Joint spectral radius, Sturmian measures and the finiteness conjecture

2017 ◽  
Vol 38 (8) ◽  
pp. 3062-3100 ◽  
Author(s):  
OLIVER JENKINSON ◽  
MARK POLLICOTT
Keyword(s):  
Spectral Radius ◽  
Short Proof ◽  
Optimal Growth ◽  
Joint Spectral Radius ◽  
New Approach ◽  
Matrix Anal ◽  
Generalized Spectral Radius ◽  
Optimal Growth Rate ◽  
And Control ◽  
The Relationship

The joint spectral radius of a pair of $2\times 2$ real matrices $(A_{0},A_{1})\in M_{2}(\mathbb{R})^{2}$ is defined to be $r(A_{0},A_{1})=\limsup _{n\rightarrow \infty }\max \{\Vert A_{i_{1}}\cdots A_{i_{n}}\Vert ^{1/n}:i_{j}\in \{0,1\}\}$, the optimal growth rate of the norm of products of these matrices. The Lagarias–Wang finiteness conjecture [Lagarias and Wang. The finiteness conjecture for the generalized spectral radius of a set of matrices. Linear Algebra Appl.214 (1995), 17–42], asserting that $r(A_{0},A_{1})$ is always the $n$th root of the spectral radius of some length-$n$ product $A_{i_{1}}\cdots A_{i_{n}}$, has been refuted by Bousch and Mairesse [Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture. J. Amer. Math. Soc.15 (2002), 77–111], with subsequent counterexamples presented by Blondel et al [An elementary counterexample to the finiteness conjecture. SIAM J. Matrix Anal.24 (2003), 963–970], Kozyakin [A dynamical systems construction of a counterexample to the finiteness conjecture. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (Seville, Spain, December 2005). IEEE, Piscataway, NJ, pp. 2338–2343] and Hare et al [An explicit counterexample to the Lagarias–Wang finiteness conjecture. Adv. Math.226 (2011), 4667–4701]. In this article, we introduce a new approach to generating finiteness counterexamples, and use this to exhibit an open subset of $M_{2}(\mathbb{R})^{2}$ with the property that each member $(A_{0},A_{1})$ of the subset generates uncountably many counterexamples of the form $(A_{0},tA_{1})$. Our methods employ ergodic theory; in particular, the analysis of Sturmian invariant measures. This approach allows a short proof that the relationship between the parameter $t$ and the Sturmian parameter ${\mathcal{P}}(t)$ is a devil’s staircase.

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