  The $3x+1$-problem (or Collatz problem) is a notorious conjecture in
arithmetic. It can be viewed as iterating a map and, therefore, it is a
dynamical system on the discrete space $\mathbb{N}$ of natural numbers. The
emerging dynamical system is studied in the present work with methods from the
theory of Koopman operators and $C^*$-algebras. This approach enables us to
"lift" the $3x+1$-dynamical system from the state space (i.e the set
$\mathbb{N}$) to spaces of functions defined on the state space, i.e. to
sequence spaces. The advantage of this lifting is that the Collatz problem can
be described via bounded linear operators, which consist an extensively studied
area of Analysis. We study the properties of these operators and their
relationship to the $3x+1$-problem. Furthermore, we use Fourier transform
techniques to investigate the frequency content of the sequences of signs
emerging from the trajectories of the Collatz map. This enables us to define an
isometry on a Hilbert space. Finally, we utilize the $C^*$-algebra generated by
this isometry in order to study how the sequences of signs correlate with each
other.
