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