On Generalized Toeplitzness of Weighted Composition Operators on H^2

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,φT_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_φu=λ u . We also give several applications of this result.