# Spectral radius of operators associated with dynamical systems in the spaces C(X)

• Volume: 67, Issue: 1, page 397-403
• ISSN: 0137-6934

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## Abstract

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We consider operators acting in the space C(X) (X is a compact topological space) of the form $Au\left(x\right)=\left({\sum }_{k=1}^{N}{e}^{{\phi }_{k}}{T}_{{\alpha }_{k}}\right)u\left(x\right)={\sum }_{k=1}^{N}{e}^{{\phi }_{k}\left(x\right)}u\left({\alpha }_{k}\left(x\right)\right)$, u ∈ C(X), where ${\phi }_{k}\in C\left(X\right)$ and ${\alpha }_{k}:X\to X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $\phi ={\left({\phi }_{k}\right)}_{k=1}^{N}$. We prove that $ln\left(r\left(A\right)\right)=\lambda \left(\phi \right)=ma{x}_{\nu \in Mes}{\sum }_{k=1}^{N}{\int }_{X}{\phi }_{k}d{\nu }_{k}-\lambda *\left(\nu \right)$, where Mes is the set of all probability vectors of measures $\nu ={\left({\nu }_{k}\right)}_{k=1}^{N}$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on $\left({C}^{N}\left(X\right)\right)*$. In other words λ* is the Legendre conjugate to λ.

## How to cite

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Krzysztof Zajkowski. "Spectral radius of operators associated with dynamical systems in the spaces C(X)." Banach Center Publications 67.1 (2005): 397-403. <http://eudml.org/doc/282106>.

@article{KrzysztofZajkowski2005,
abstract = {We consider operators acting in the space C(X) (X is a compact topological space) of the form $Au(x) = (∑_\{k=1\}^\{N\} e^\{φ_k\}T_\{α_k\})u(x) = ∑_\{k=1\}^\{N\} e^\{φ_k(x)\}u(α_k(x))$, u ∈ C(X), where $φ_k ∈ C(X)$ and $α_k: X → X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $φ = (φ_k)_\{k=1\}^\{N\}$. We prove that $ln(r(A)) = λ(φ) = max_\{ν∈Mes\} \{∑_\{k=1\}^\{N\} ∫_\{X\} φ_\{k\}dν_\{k\} - λ*(ν)\}$, where Mes is the set of all probability vectors of measures $ν = (ν_k)_\{k=1\}^\{N\}$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on $(C^N(X))*$. In other words λ* is the Legendre conjugate to λ.},
author = {Krzysztof Zajkowski},
journal = {Banach Center Publications},
keywords = {spectral radius; Legendre transform; lower semicontinuous convex functional},
language = {eng},
number = {1},
pages = {397-403},
title = {Spectral radius of operators associated with dynamical systems in the spaces C(X)},
url = {http://eudml.org/doc/282106},
volume = {67},
year = {2005},
}

TY - JOUR
AU - Krzysztof Zajkowski
TI - Spectral radius of operators associated with dynamical systems in the spaces C(X)
JO - Banach Center Publications
PY - 2005
VL - 67
IS - 1
SP - 397
EP - 403
AB - We consider operators acting in the space C(X) (X is a compact topological space) of the form $Au(x) = (∑_{k=1}^{N} e^{φ_k}T_{α_k})u(x) = ∑_{k=1}^{N} e^{φ_k(x)}u(α_k(x))$, u ∈ C(X), where $φ_k ∈ C(X)$ and $α_k: X → X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $φ = (φ_k)_{k=1}^{N}$. We prove that $ln(r(A)) = λ(φ) = max_{ν∈Mes} {∑_{k=1}^{N} ∫_{X} φ_{k}dν_{k} - λ*(ν)}$, where Mes is the set of all probability vectors of measures $ν = (ν_k)_{k=1}^{N}$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on $(C^N(X))*$. In other words λ* is the Legendre conjugate to λ.
LA - eng
KW - spectral radius; Legendre transform; lower semicontinuous convex functional
UR - http://eudml.org/doc/282106
ER -

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