Frame properties induced by iteration of a multiplication operator on Hardy spaces

Motivated by the study of frame properties arising from iterates of linear operators, it was previously established that the multiplication operator T_ϕx(t) = ϕ(t)x(t) cannot generate a frame in L²(a,b) (Results Math, 2019). In this note, we examine the behavior of such operators on the Hardy space H²(D), the Hilbert space of holomorphic functions on the open unit disk D with square integrable boundary values. We show that, in contrast to the L²-setting, the iterates of T_ϕ, for ϕ∈ H∞(D), exhibit fundamentally different frame properties in H²(D), leading to new structural insights and results.