Spectrum of weighted isometries: C*-algebras, transfer operators and topological pressure

Abstract

We study the spectrum of operators $aT\in B(H)$ on a Hilbert space $H$ where $T$ is an isometry and $a$ belongs to a commutative $C^{\ast }$-subalgebra $C(X)\cong A\subseteq B(H)$ such that the formula $L(a)=T^{\ast }aT$ defines a faithful transfer operator on $A$. Based on the analysis of the $C^{\ast }$-algebra $C^{\ast }(A,T)$ generated by the operators $aT$, $a\in A$, we give dynamical conditions implying that the spectrum $\sigma (aT)$ is invariant under rotation around zero, $\sigma (aT)$ coincides with essential spectrum ${\sigma }_{ess}(aT)$ or that $\sigma (aT)=\{z\in \mathbb{C}:|z|\le r(aT)\}$ is a disk. We extend classical Ruelle's result and prove that for a general expanding map $\phi :X\to X$ and continuous $c:X\to [0,\infty )$ the spectral logarythm of a Ruelle-Perron-Frobenious operator ${\mathcal{L}}_{c}f(y)={\sum }_{x\in {\phi }^{-1}(y)}c(x)f(x)$ is equal to the topological pressure $P(\ln c,\phi )$. As a consequence we get the variational principle for the spectral radius: $$r(aT)=\underset{\mu \in \text{Erg}(X,\phi )}{\max }\exp \left({\int }_X\ln (|a|\sqrt{\varrho })d\mu +\frac{h_{\phi }(\mu )}{2}\right),$$ where $\text{Erg}(X,\phi )$ stands for ergodic Borel probability measures, $h_{\phi }(\mu )$ is the Kolmogorov-Sinai entropy, and $\varrho :X\to [0,1]$ is the cocycle associated to $L$. In particular, we clarify the relationship between the Kolmogorov-Sinai entropy and $t$-entropy introduced by Antonevich, Bakhtin and Lebedev.

37 pages, some typos fixed, accepted for publication in the Israel Journal of Mathematics