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

Krzysztof Zajkowski

Banach Center Publications, Tome 68 (2005), p. 397-403 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider operators acting in the space C(X) (X is a compact topological space) of the form $A u (x) = (\sum_{k = 1}^{N} e^{φ_{k}} T_{α_{k}}) u (x) = \sum_{k = 1}^{N} e^{φ_{k} (x)} u (α_{k} (x))$ , u ∈ C(X), where $φ_{k} \in C (X)$ and $α_{k} : X \to X$ 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 $l n (r (A)) = λ (φ) = m a x_{ν \in M e s} \sum_{k = 1}^{N} \int_{X} φ_{k} d ν_{k} - λ * (ν)$ , 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.

Publié le : 2005-01-01

Zbl 1075.47002

EUDML-ID : urn:eudml:doc:282106

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     author = {Krzysztof Zajkowski},
     title = {Spectral radius of operators associated with dynamical systems in the spaces C(X)},
     journal = {Banach Center Publications},
     volume = {68},
     year = {2005},
     pages = {397-403},
     zbl = {1075.47002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc67-0-33}
}

Krzysztof Zajkowski. Spectral radius of operators associated with dynamical systems in the spaces C(X). Banach Center Publications, Tome 68 (2005) pp. 397-403. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc67-0-33/