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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.
$Au(x) = (∑_{k=1}^{N} e^{φ_k}T_{α_k})u(x) = ∑_{k=1}^{N} e^{φ_k(x)}u(α_k(x))$, 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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
397-403
Opis fizyczny
Daty
wydano
2005
Twórcy
autor
- Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, Poland
Bibliografia
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-bc67-0-33