Abstract: In this paper, we give some new results of the coefficient multiplier of some analytic function spaces, characterize the coefficient multiplier spaces (H_α,H_β) and (A^p,α), A^q,β) with 0 < p ≤ 1, p ≤ q ≤∞, 0 ≤α < β < ∞ and show (H_α, H_β) = H_β-α with 0 < α < β < ∞.

Key words: analytic function space, Hadamard product, multiplier, closed graph theorem, operator