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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, aA, we give dynamical conditions implying that the spectrum σ(aT) is invariant under rotation around zero, σ(aT) coincides with the essential spectrum σess (aT) or that σ(aT) is the disc {z ∈ ℂ: ∣z∣ ≤ r(aT)}.

We get the best results when the underlying mapping φ: XX 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, hφ(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.

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Correspondence to Bartosz Kosma Kwaśniewski.

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We thank Andrei Lebedev for numerous discussions on variational formulas appearing in the text. This work was supported by the National Science Centre, Poland, grant number 2019/35/B/ST1/02684.

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Bardadyn, K., Kwaśniewski, B.K. Spectrum of weighted isometries: C*-algebras, transfer operators and topological pressure. Isr. J. Math. 246, 149–210 (2021). https://doi.org/10.1007/s11856-021-2246-6

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  • DOI: https://doi.org/10.1007/s11856-021-2246-6