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

Banach Center Publications (2005)

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

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topKrzysztof 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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