  We investigate the problems related to the Collatz map $T$ from the point of
view of functional analysis. We associate with $T$ certain linear operator
$\mathcal{T}$ and show that cycles and (hypothetical) diverging trajectory
(generated by $T$) correspond to certain classes of fixed points of operator
$\mathcal{T}$. Furthermore, we demonstrate connection between dynamical
properties of operator $\mathcal{T}$ and map $T$. We prove that absence of
nontrivial cycles of $T$ leads to hypercyclicity of operator $\mathcal{T}$. In
the second part we show that the index of operator
$Id-\mathcal{T}\in\mathcal{L}(H^2(D))$ gives upper estimate on the number of
cycles of $T$. For the proof we consider the adjoint operator
$\mathcal{F}=\mathcal{T}^*$ \[ \mathcal{F}: g\to
g(z^2)+\frac{z^{-\frac{1}{3}}}{3}\left(g(z^{\frac{2}{3}})+e^{\frac{2\pi
i}{3}}g(z^{\frac{2}{3}}e^{\frac{2\pi i}{3}})+e^{\frac{4\pi
i}{3}}g(z^{\frac{2}{3}}e^{\frac{4\pi i}{3}})\right), \] first introduced by
Berg, Meinardus in \cite{BM1994}, and show it does not have non-trivial fixed
points in $H^2(D)$. Moreover, we calculate resolvent of operator $\mathcal{F}$
and as an application deduce equation for the characteristic function of total
stopping time $\sigma_{\infty}$.
  Furthermore, we construct an invariant measure for $\mathcal{T}$ in a
slightly different setup, and investigate how the operator $\mathcal{T}$ acts
on generalized arithmetic progressions.
