Stability theorems for some combinations of two idempotents

Let P, Q be two bounded linear idempotent operators on a complex Banach space and lambda, mu be two complex numbers with lambda + mu not equal 0. Du [H.K. Du, C. Deng, M. Mbekhta, and V. Muller, On spectral properties of linear combinations of idempotents, Studia Math. 180 (2007), pp. 211-217], Koliha and Rakocevic [J.J. Koliha and V. Rakocevic, Stability theorems for linear combinations of idempotents, Integr. Equ. Oper. Theory 58 (2007), pp. 597-601] pointed out that for lambda P + mu Q the property of being upper semi-Fredholm, lower semi-Fredholm and Fredholm, respectively, is independent of the choice of lambda, mu. In the present note, these results are strengthened by showing that the nullity of lambda P + mu Q - omega PQ are constant for complex numbers lambda, mu, omega. Furthermore, for lambda P + mu Q - omega PQ the property of being upper semi-Fredholm, lower semi-Fredholm and Fredholm, respectively, is independent of the choice of lambda, mu, omega.