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

We study the spectrum of operators $$aT \in {\cal B}(H)$$ on a Hilbert space H where T is an isometry and belongs to commutative C*-subalgebra $$C(X) \cong A \subseteq such that formula L(a) = T*aT defines faithful transfer operator A. Based analysis C*-algebra C* (A, T) generated by aT, ∈ A, we give dynamical conditions implying σ(aT) invariant under rotation around zero, coincides with essential σess (aT) or disc {z ℂ: ∣z∣ ≤ r(aT)}. get best results when underlying mapping φ: X → expanding open. prove for any map continuous c: [0, ∞) spectral logarithm Ruelle—Perron—Frobenius $${{\cal L}_c}f(y) \sum\nolimits_{x {\varphi ^{ - 1}}(y)} {c(x)f(x)} $$ equal topological pressure P(ln c, φ). This extends Ruelle’s classical result implies variational principle radius: $$r(aT) \mathop {\max }\limits_{\mu {\rm{Erg}}(X,\varphi )} {\rm{exp}}\left( {\int_X {\ln (\left| \right|\sqrt \varrho \,d\mu + {{{h_\varphi }(\mu \over 2}} \right),$$ Erg(X, φ) set ergodic Borel probability measures, hφ(μ) Kolmogorov—Sinai entropy, ϱ: 1] cocycle associated L. In particular, clarify relationship between entropy t-entropy introduced Antonevich, Bakhtin Lebedev.