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.