On the shift semigroup on the Hardy space of Dirichlet series

We develop a Wold decomposition for the shift semigroup on the Hardy space $ mathcal{H}^2 $ mathcal{H}^2 of square summable Dirichlet series convergent in the half-plane $ Re (s) > 1/2 $ Re (s) > 1/2 . As an application we have that a shift invariant subspace of $ mathcal{H}^2 $ mathcal{H}^2 is unitarily equivalent to $ mathcal{H}^2 $ mathcal{H}^2 if and only if it has the form $ phi mathcal{H}^2 $ phi mathcal{H}^2 for some $ mathcal{H}^2 $ mathcal{H}^2 -inner function φ.