Generalizations of Furuta's inequality

摘要

We obtain amongst others the following result, for any x, y ∈ H, where f(z) = ∑_(n=0) ~∞a_nz~n is a function defined by power serients and convergent on the open disk D(0, R) ? C, R > 0, f_A(z) = ∑_(n=0) ~∞ {pipe}a_n{pipe}z~n, T ∈ ?(H), α, β ≥ 0 with α + β ≥ 1 and {norm of matrix}T{norm of matrix}2α, {norm of matrix}T{norm of matrix}~(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.