Riccati inequality and functional properties of differential operators on the half line

Given a piecewise continuous function A : R ¯ + → L ( C N ) and a projection P 1 onto a subspace X 1 of C N , we investigate the injectivity, surjectivity and, more generally, the Fredholm properties of the ordinary differential operator with boundary condition ( u ˙ + A u , P 1 u ( 0 ) ) . This operator acts from the “natural” space W A 1 , 2 = { u : u ˙ ∈ L 2 , A u ∈ L 2 } into L 2 × X 1 . A main novelty is that it is not assumed that A is bounded or that u ˙ + A u = 0 has any dichotomy, except to discuss the impact of the results on this special case. We show that all the functional properties of interest, including the characterization of the Fredholm index, can be related to the existence of a selfadjoint solution H of the Riccati differential inequality H A + A ∗ H − H ˙ ⩾ ν ( A ∗ A + H 2 ) . Special attention is given to the simple case when H = A + A ∗ satisfies this inequality. When H is known, all the other hypotheses and criteria are easily verifiable in most concrete problems.