APPLICATIONS OF KATO'S INEQUALITY TO OPERATOR-VALUED INTEGRALS ON HILBERT SPACES

Abstract

By the use of the celebrated Kato's inequality we obtain in this paper some inequalities for operator-valued integrals on a complex Hilbert space H. Among others, we show that [Formula: see text] for any x, y ∈ H, provided [Formula: see text] and p : E → [0, ∞) are μ-measurable functions on E and such that [Formula: see text] and [Formula: see text] are Bochner integrable on E for some α ∈ [0, 1]. Natural applications for various norms and numerical radii associated with the Bochner integral of operator-valued functions and some examples for the operator exponential are presented as well.