On the shift semigroup on the Hardy space of Dirichlet series

Abstract

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