“…By Theorem 1.6 there exist discrete dichotomies on Z þ and Z À : By Lemma 1.5, there exist dichotomies fP þ t g tX0 and fP À t g tp0 : This proves ði 0 Þ in Theorem 1.2 and, therefore, (i) in Theorem 1.1 for a ¼ 0 ¼ b: By Proposition 5.2 we also have that Nð0; 0Þ is Fredholm, and, using formulas for the defect numbers and index from Theorem 1.4, we derive ðii 0 Þ in Theorem 1.2 and, by Lemma 5.1, (ii) in Theorem 1.1 for a ¼ 0 ¼ b; and the required formulas for the defect numbers and the index. It remains to prove that (i) and (ii) in Theorem 1.1 imply (1.3), see [9,Theorem 4], and also [7,Theorem 8] for the proof in the case when a ¼ À1 and b ¼ 0: We will present a proof, different form [7], as well as from the corresponding proofs in [3,12,28,36,38,46] given in particular cases. Our proof is based on the following abstract fact from [29, p. 23].…”