NOTE ON THE NORM OF LINEAR COMBINATIONS OF A CLASS OF LINEAR OPERATORS

Abstract

Let H(D) be the space of analytic functions on the open unit disk D in the complex plane C, S(D) = {f ε H(D): f (D) ⊆ D}, K (D) the space of Cauchy transforms on D, W^(m) <inf>w</inf> (D) the mth weighted type space on D with the weight function w and m εN<inf>0</inf>, W^(m) <inf>w,0</inf> (D) the little m th weighted type space on D, and where We find the norm of the operator Ls: K (D) → W^(m) <inf>w</inf> (D) in terms of above-mentioned parameters and symbols, and present a characterization for the boundedness of the operator Ls: K (D) → W^(m) <inf>w,0</inf> (D), considerably extending several recent results in the literature.