On Generalized Toeplitzness of Weighted Composition Operators on H2

In this paper, we study the weighted composition operators Wf,? on the Hardy space H-2 that are (T-u, lambda)-Toeplitz, where u is a nonconstant inner function and lambda is a complex number with |lambda| <= 1; that is, (TuW)-W-*; T-f,phi(u) = lambda W (f,phi). In particular, we prove that for an analytic self map ? of D, not identically zero on D, and f is an element of H-infinity \ {0}, if lambda is an element of (D) over bar \ {0} such that 1/lambda (u ? ?) is a self-map of D, then W (f,phi) is (T-u, lambda)-Toeplitz precisely when C(phi)u = lambda u. We also give several applications of this result.