WEIGHTED COMPOSITION OPERATORS BETWEEN n-TH α-WEIGHTED SPACES

Let V-n(alpha) be the Banach space of holomorphic functions on the open unit disk D in the complex plane consisting of those f such that parallel to f parallel to(V)(n alpha):= Sigma(n-1)(i=0) vertical bar f ((i)) (0)vertical bar + sup(z is an element of D) (1 - vertical bar z vertical bar(2))(alpha) vertical bar f((n)) (z)vertical bar < infinity and V-n,0(alpha) be the closed subspace of V-n(alpha) consisting of those f for which lim(vertical bar z vertical bar -> 1) (1 - vertical bar z vertical bar(2))(alpha) vertical bar f((n )(z)vertical bar = 0, where n is any nonnegative integer and alpha > 0. We give boundedness characterizations, norm estimates and essential norm estimates of weighted composition operators W-psi,W-phi :V-n(alpha) -> V-m(beta) and W-psi,W-phi, : V-n,0(alpha) -> V-m,0(beta), respectively, where W-psi,W-phi f (z) = psi (z) f (phi (z)). As a corollary, we characterize the compactness of W-psi,W-phi. Specifically, our characterizations involve not only the classical Julia-Caratheodory type condition, but also the powers phi(k) . In addition, our results extend several well-known results in the literature.

Automated summary

The text discusses the Banach space of holomorphic functions on the open unit disk in the complex plane, as well as the boundedness characterizations, norm estimates, and essential norm estimates of weighted composition operators. It also covers the compactness characterization of the operators, extending various known results in the literature.