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Spectral radius of operators associated with dynamical systems in the spaces \(C(X)\). (English) Zbl 1075.47002

Jarosz, Krzysztof (ed.) et al., Topological algebras, their applications, and related topics. Proceedings of the conference to celebrate the 70th birthday of Professor Wiesław Żelazko, Bȩdlewo, Poland, May 11–17, 2003. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 67, 397-403 (2005).
Summary: We consider operators acting in the space \(C(X)\) (\(X\) is a compact topological space) of the form \[ Au(x) = \Big(\sum_{k=1}^N e^{\varphi_k} T_{\alpha_k}\Big)u(x) = \sum_{k=1}^N e^{\varphi_k(x)}u(\alpha_k(x)),\quad\;u\in C(X), \] where \(\varphi_k\in C(X)\) and \(\alpha_k:X\to X\) are given continuous mappings (\(1 \leq k \leq N\)). 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 \(\lambda\) depending on the vector of functions \(\varphi=(\varphi_k)_{k=1}^N\). We prove that \[ \ln(r(A)) = \lambda(\varphi) = \max_{\nu\in \text{Mes}} \bigg\{\sum_{k=1}^N \int_X \varphi_k \,d\nu_k - \lambda^*(\nu)\bigg\}, \] where Mes is the set of all probability vectors of measures \(\nu=(\nu_k)_{k=1}^N\) on \(X\times \{1,\ldots ,N\}\) and \(\lambda^*\) is some convex lower-semicontinuous functional on \((C^N(X))^\star\). In other words, \(\lambda^*\) is the Legendre conjugate to \(\lambda\).
For the entire collection see [Zbl 1063.46001].

MSC:

47A10 Spectrum, resolvent
44A15 Special integral transforms (Legendre, Hilbert, etc.)
46E10 Topological linear spaces of continuous, differentiable or analytic functions
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