Irreducible Multiplication Operators on Spaces of Analytic Functions

Mφ : H → H for φ ∈ H∞(Ω). We mention some known results in this area that serve as a motivation for the present paper. First, if H is the classical Hardy space of the unit disk D, and if φ is an inner function on D, then Mφ is a pure isometry and a shift operator on H , and so its reducing subspaces are in a one-to-one correspondence with the closed subspaces of H (φH). Therefore, the reducing subspace lattice of such an operator Mφ is isomorphic to the lattice of closed subspaces of H (φH). In particular, if φ is any inner function other than a Möbius map, then Mφ has infinitely many reducing subspaces. See [5] for this result and related references. Second, if H is the Bergman space of D and φ is a Blaschke product of two zeros in D, then Mφ : H → H has exactly two nontrivial reducing subspaces. See [11] for proof and a description of these reducing subspaces. Finally, if H is the Hilbert space on D induced by a positive weight sequence {ωn} (with ωn+1/ωn ≤M for some constant M and all n ≥ 0):