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.