Spectral radius of weighted composition operators in $L^{p}$-spaces

Krzysztof Zajkowski

Spectral radius of weighted composition operators in

L^{p}

-spaces

Krzysztof Zajkowski

Studia Mathematica, Tome 196 (2010), p. 301-307 / Harvested from The Polish Digital Mathematics Library

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Résumé

We prove that for the spectral radius of a weighted composition operator $a T_{α}$ , acting in the space $L^{p} (X,, μ)$ , the following variational principle holds: $l n r (a T_{α}) = m a x_{ν \in M ¹_{α, e}} \int_{X} l n | a | d ν$ , where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, $M ¹_{α, e}$ is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and $_{\infty}$ -measurable function, where $_{\infty} = ⋂_{n = 0}^{\infty} α^{- n} ()$ . This considerably extends the range of validity of the above formula, which was previously known in the case when α is a homeomorphism.

Publié le : 2010-01-01

Zbl 1189.47031

EUDML-ID : urn:eudml:doc:285428

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     title = {Spectral radius of weighted composition operators in $L^{p}$-spaces},
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     volume = {196},
     year = {2010},
     pages = {301-307},
     zbl = {1189.47031},
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Krzysztof Zajkowski. Spectral radius of weighted composition operators in $L^{p}$-spaces. Studia Mathematica, Tome 196 (2010) pp. 301-307. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm198-3-8/