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

Abstract

We study the spectrum of operators $aT \in {\cal B}(H)$ on a Hilbert space H where T is an isometry and a belongs to a commutative C*-subalgebra $C(X) \cong A \subseteq {\cal B}(H)$ such that the formula L(a) = T*aT defines a faithful transfer operator on A. Based on the analysis of the C*-algebra C* (A, T) generated by the operators aT, a ∈ A, we give dynamical conditions implying that the spectrum σ(aT) is invariant under rotation around zero, σ(aT) coincides with the essential spectrum σ_ess (aT) or that σ(aT) is the disc {z ∈ ℂ: ∣z∣ ≤ r(aT)}.

We get the best results when the underlying mapping φ: X → X is expanding and open. We prove for any such map and a continuous map c: X → [0, ∞) that the spectral logarithm of a Ruelle—Perron—Frobenius operator ${{\cal L}_c}f(y) = \sum\nolimits_{x \in {\varphi ^{ - 1}}(y)} {c(x)f(x)}$ is equal to the topological pressure P(ln c, φ). This extends Ruelle’s classical result and implies the variational principle for the spectral radius:

$$r(aT) = \mathop {\max }\limits_{\mu \in {\rm{Erg}}(X,\varphi )} {\rm{exp}}\left( {\int_X {\ln (\left| a \right|\sqrt \varrho )} \,d\mu + {{{h_\varphi }(\mu )} \over 2}} \right),$$

where Erg(X, φ) is the set of ergodic Borel probability measures, h_φ(μ) is the Kolmogorov—Sinai entropy, and ϱ: X → [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.