Dedicated to Professor Yuri I. Karlovich on his 65th anniversarywhere ω ij (t) = log[α i (β −j (t))/t] and α i , β −j are iterations of α, β. This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for γ = 0.Ind A := dim ker A − dim ker A * is referred to as the index of A (see, e.g., [7, Chap. 4]). For A, B ∈ B(X), we will write A ≃ B if A − B ∈ K(X). Recall that an operator B r ∈ B(X) (resp. B l ∈ B(X)) is said to be a right (resp. left) regularizer for A if AB r ≃ I (resp. B l A ≃ I).