  Consider the operator $E$ on arithmetic functions such that $Ef$ is the
multiplicative arithmetic function defined by $(Ef)(p^a) = f(a)$ for every
prime power $p^a$. We investigate the behaviour of $E^m\tau_k$, where $\tau_k$
is a $k$-dimensional divisor function and $E^m$ stands for the $m$-fold iterate
of $E$. We estimate the error terms of $\sum_{n\le x} E^m\tau_k(n)$ for various
combinations of $m$ and $k$. We also study properties of $E^mf$ for arbitrary
$f$ and sufficiently large $m$.
  Our study provides a unified approach to functions with exponential divisors.
We improve special cases of the Dirichlet asymmetric divisor problem and
several results on the exponential divisor and totient functions.
