On Generalized Toeplitzness of Weighted Composition Operators on H2.

In this paper, we study the weighted composition operators W f , φ on the Hardy space H 2 that are (T u , λ) -Toeplitz, where u is a nonconstant inner function and λ is a complex number with | λ | ≤ 1 ; that is, T u ∗ W f , φ T u = λ W f , φ . In particular, we prove that for an analytic self-map φ of D , not identically zero on D , and f ∈ H ∞ \ { 0 } , if λ ∈ D ¯ \ { 0 } such that 1 λ (u ∘ φ) is a self-map of D , then W f , φ is (T u , λ) -Toeplitz precisely when C φ u = λ u . We also give several applications of this result. [ABSTRACT FROM AUTHOR]