Orthogonal decomposition of composition operators on the $H^2$ space of Dirichlet series

Let $mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $mathscr{C}_varphi$ on $mathscr{H}^2$ which are generated by symbols of the form $varphi(s) = c_0s + sum_{ngeq1} c_n n^{-s}$, in the case that $c_0 geq 1$. If only a subset $mathbb{P}$ of prime numbers features in the Dirichlet series of $varphi$, then the operator $mathscr{C}_varphi$ admits an associated orthogonal decomposition. Under sparseness assumptions on $mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $mathscr{C}_varphi$. Furthermore, in the case that $varphi$ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of $mathscr{C}_varphi$ in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.