  Over the last few years Pohl (partly jointly with coauthors) developed dual
`slow/fast' transfer operator approaches to automorphic functions, resonances,
and Selberg zeta functions for a certain class of hyperbolic surfaces
$\Gamma\backslash\mathbb{H}$ with cusps and all finite-dimensional unitary
representations $\chi$ of $\Gamma$.
  The eigenfunctions with eigenvalue $1$ of the fast transfer operators
determine the zeros of the Selberg zeta function for $(\Gamma,\chi)$. Further,
if $\Gamma$ is cofinite and $\chi$ is the trivial one-dimensional
representation then highly regular eigenfunctions with eigenvalue $1$ of the
slow transfer operators characterize Maass cusp forms for $\Gamma$.
Conjecturally, this characterization extends to more general automorphic
functions as well as to residues at resonances.
  In this article we study, without relying on Selberg theory, the relation
between the eigenspaces of these two types of transfer operators for any Hecke
triangle surface $\Gamma\backslash\mathbb{H}$ of finite or infinite area and
any finite-dimensional unitary representation $\chi$ of the Hecke triangle
group $\Gamma$. In particular we provide explicit isomorphisms between relevant
subspaces. This solves a conjecture by M\"oller and Pohl, characterizes some of
the zeros of the Selberg zeta functions independently of the Selberg trace
formula, and supports the previously mentioned conjectures.
