@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 -
