On Transfer Operators for Continued Fractions with Restricted Digits

For any non‐empty subset I of the natural numbers, let Λ I denote those numbers in the unit interval whose continued fraction digits all lie in I. Define the corresponding transfer operatorLI , β f ( z ) = ∑ n ∈ I( 1 n + z ) 2 β f ( 1 n + z )for Re ( β ) > max ( 0 , θ I) , where Re (rβ) = θ I is the abscissa of convergence of the series ∑ n ∈ In − 2 β . When acting on a certain Hilbert space H I , rβ, we show that the operator L I , rβ is conjugate to an integral operator K I , rβ. If furthermore rβ is real, then K I , rβ is selfadjoint, so that L I , rβ : H I , rβ → H I , rβ has purely real spectrum. It is proved that L I , rβ also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space C ω [0, 1], and on the Fréchet space C ∞ [0, 1]. The analytic properties of the map rβ ↦ L I , rβ are investigated. For certain alphabets I of an arithmetic nature (for example, I = primes, I = squares, I an arithmetic progression, I the set of sums of two squares it is shown that rβ ↦ L I , rβ admits an analytic continuation beyond the half‐plane Re rβ > θ I . 2000 Mathematics Subject Classification 37D35, 37D20, 30B70.