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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 $C^*$-subalgebra $C(X)\congA\subseteq B(H)$ such that the formula $L(a)=T^*aT$ defines a faithful transferoperator on $A$. Based on the analysis of the $C^*$-algebra $C^*(A,T)$generated by the operators $aT$, $a\in A$, we give dynamical conditionsimplying that the spectrum $\sigma(aT)$ is invariant under rotation aroundzero, $\sigma(aT)$ coincides with essential spectrum $\sigma_{ess}(aT)$ or that$\sigma(aT)=\{z\in \mathbb{C}: |z|\leq r(aT)\}$ is a disk.  We extend classical Ruelle's result and prove that for a general expandingmap $\varphi:X\to X$ and continuous $c:X\to [0,\infty)$ the spectral logarythmof a Ruelle-Perron-Frobenious operator $\mathcal{L}_cf(y)=\sum_{x\in\varphi^{-1}(y)} c(x)f(x)$ is equal to the topological pressure $P(\lnc,\varphi)$. As a consequence we get the variational principle for the spectralradius: $$ r(aT)=\max_{\mu\in \text{Erg}(X,\varphi)}\exp\left(\int_{{X}}\ln(\vert a\vert\sqrt{\varrho })\,d\mu+\frac{h_{\varphi}(\mu)}{2} \right), $$ where $\text{Erg}(X,\varphi)$ standsfor ergodic Borel probability measures, $h_{\varphi}(\mu)$ is theKolmogorov-Sinai entropy, and $\varrho:X\to [0,1]$ is the cocycle associated to$L$. In particular, we clarify the relationship between the Kolmogorov-Sinaientropy and $t$-entropy introduced by Antonevich, Bakhtin and Lebedev.

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