Compact Difference of Fejér-Riesz Composition Operators

We introduce the Fejér-Riesz composition operatorFφ induced by a holomorphic self-map φ of the unit disk D, and defined on the space of holomorphic functions on D by Fφf=χ(-1,1)f∘φ. Denote by Aαp the standard weighted Bergman space of holomorphic functions on D and denote the Hardy space Hp by A-1p. We study the operators Fφ:Aαp→Lp(mα), where mα is the weighted Lebesgue measure dmα(x)=(1-x2)α+1dx, x∈(-1,1). For 0<p<∞ and α≥-1, every such Fφ is bounded, which in the case α=-1 is related to the classical Fejér-Riesz Inequality. We provide characterizations for when Fφ is compact and for when Fφ-Fψ is compact.