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

Tom 198 / 2010

Krzysztof Zajkowski Studia Mathematica 198 (2010), 301-307 MSC: Primary 47B37; Secondary 47A10, 44A15. DOI: 10.4064/sm198-3-8


We prove that for the spectral radius of a weighted composition operator $aT_\alpha$, acting in the space $L^p(X,\mathcal{B},\mu)$, the following variational principle holds: $$ \ln r(aT_\alpha)=\max_{\nu\in M^1_{\alpha,{\rm e}}}\int_X\ln|a|\,d\nu, $$ where $X$ is a Hausdorff compact space, $\alpha:X\to X$ is a continuous mapping preserving a Borel measure $\mu$ with $\mathop{\rm supp}\mu=X$, $M^1_{\alpha,{\rm e}}$ is the set of all $\alpha$-invariant ergodic probability measures on~$X$, and $a:X\to \mathbb{R}$ is a continuous and $\mathcal{B}_\infty$-measurable function, where $\mathcal{B}_\infty=\bigcap_{n=0}^\infty\alpha^{-n}(\mathcal{B})$. This considerably extends the range of validity of the above formula, which was previously known in the case when $\alpha$ is a homeomorphism.


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

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