On the multiplication operators on spaces of analytic functions
We consider Hilbert spaces of analytic functions on a plane domain Ω and multiplication operators on such spaces induced by functions from H∞(Ω). Recently, K. Zhu has given conditions under which the adjoints of multiplication operators on Hilbert spaces of analytic functions belong to the Cowen–Douglas classes. In this paper, we provide some sufficient conditions which give the converse of the main result obtained by K. Zhu. We also characterize the commutant of certain multiplication operators. Introduction. LetH be a Hilbert space of functions analytic on a plane domain Ω such that: (1) Every point w ∈ Ω is a nonzero bounded linear functional on H, so that H has a reproducing kernel kw ∈ H such that f(w) = 〈f, kw〉 for all f ∈ H. (2) If f is a function in H, then so is zf . (3) If f ∈ H and f(λ) = 0, then there is a function g ∈ H such that (z − λ)g = f . A space H satisfying the above conditions is called a Hilbert space of analytic functions on Ω. The Hardy and Bergman spaces are examples of Hilbert spaces of analytic functions on the open unit disk. A complex-valued function φ on Ω for which φf ∈H for every f ∈H is called a multiplier of H, and every multiplier φ on H determines a multiplication operator Mφ on H by Mφf = φf , f ∈ H. The set of all multipliers of H is denoted by M(H). Clearly M(H) ⊂ H∞(Ω), where H∞(Ω) is the space of all bounded analytic functions on Ω. In fact ‖φ‖∞ ≤ ‖Mφ‖. A good source on this topic is [4]. 2000 Mathematics Subject Classification: 47B38, 47A10.