Boundedness From Below of Multiplication Operators Between α-Bloch Spaces

In this paper, the boundedness from below of multiplication operators between α-Bloch spaces , α > 0, on the unit disk D is studied completely. For a bounded multiplication operator , defined by Muƒ = u ƒ for f ∈ , we prove the following result:(i) If 0 < β < α, or 0 < α ≤ 1 and α < β, Mu is not bounded below;(ii) if 0 < α = β ≤ 1, Mu is bounded below if and only if lim infz→∂D |u(z)| > 0;(iii) if 1 < α ≤ β,Mu is bounded below if and only if there exist a δ > 0 and a positive r < 1 such that for every point z ∈ D there is a point z′ ∈ D with the property d(z′, z) < r and (1–|z′|2)β–α|u(z′)| ≥ δ, where d( · , · ) denotes the pseudo-distance on D.