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

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 -