Abstract
Let P, Q be two bounded linear idempotent operators on a complex Banach space and λ, μ be two complex numbers with λ + μ ≠ 0. Du [H.K. Du, C. Deng, M. Mbekhta, and V. Müller, On spectral properties of linear combinations of idempotents, Studia Math. 180 (2007), pp. 211–217], Koliha and Rakočević [J.J. Koliha and V. Rakočević, Stability theorems for linear combinations of idempotents, Integr. Equ. Oper. Theory 58 (2007), pp. 597–601] pointed out that for λP + μQ the property of being upper semi-Fredholm, lower semi-Fredholm and Fredholm, respectively, is independent of the choice of λ, μ. In the present note, these results are strengthened by showing that the nullity of λP + μQ − ωPQ are constant for complex numbers λ, μ, ω. Furthermore, for λP + μQ − ωPQ the property of being upper semi-Fredholm, lower semi-Fredholm and Fredholm, respectively, is independent of the choice of λ, μ, ω.
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