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

• Autorzy: Krzysztof Zajkowski
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 44A15: Special transforms (Legendre, Hilbert, etc.)
• 47A10: Spectrum, resolvent

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2005
• Tom: 67
• Numer: 1
• Strony: 397-403

Daty

wydano

2005

Twórcy

autor

Krzysztof Zajkowski

• Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, Poland

Identyfikatory

DOI

10.4064/bc67-0-33