Spectral radius of weighted composition operators in $L^p$-spaces
Volume 198 / 2010
Abstract
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.
