Generalizations of Furuta's inequality

Abstract

We obtain amongst others the following result for any x, y ∈ H, where is a function defined by power series with real coefficients and convergent on the open disk D(0, R) ⊂ ℂ, R > 0, , T ∈ ℬ(H), α, β ≥ 0 with α + β ≥ 1 and ‖T ‖2α, ‖T‖2β < R. For constant functions this produces Furuta's inequality which in its turn is a generalization of Kato's inequality that is obtained for α ∈ [0, 1] and β = 1 − α.