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

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 $C(X)\cong A\subseteq B(H)$suchthattheformula$L(a)=T^*aT$ 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 $\sigma (aT)=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le r(aT)\}$ is a disk. We extend classical Ruelle’s result and prove that for a general expanding map $\phi :X\to X$ and continuous $c:X\to [0,\mathrm{\infty })$ the spectral logarythm of a Ruelle-Perron-Frobenious operator $$\mathcal{L}$cf(y)=\sum{x\in \varphi^{-1}(y)} c(x)f(x)$isequaltothetopologicalpressure$P(\ln c,\varphi)$. As a consequence we get the variational principle for the spectral radius: $$r(aT)=\underset{\mu \in \text{Erg}(X,\phi )}{\max }\exp ({\int }_X\ln (\mathrm{\mid }a\mathrm{\mid }\sqrt{\varrho })d\mu +\frac{h_{\phi }(\mu )}{2}),$$ 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.