Weak-star generators of<i>H</i>^∞

Abstract

Let H" denote the algebra of bounded analytic functions in the unit disk D = {z: \ z \ < 1}.A function ψ in H °° is called a generator if the polynomials in ψ are weak-star dense in H°*.The problem to be considered here is that of characterizing the generators of H°°.The weak-star topology of H°° can be thought of as arising in the following way.By Fatou's theorem, each function ψ in H°° has radial limits at almost every point of the unit circle C = {z: \ z \ = 1} and thus gives rise to a bounded measurable function ψ a on C. The map ψ -• ψ σ sends H°° isomorphically and isometrically onto a certain subspace of L°°(C); we denote this subspace by H°°(G).(We regard C as endowed with normalized Lebesgue measure.)The space H°°(C) is the dual of a quotient space of L\C) and as such has a weak-star topology (which is simply the topology induced on H°°(C) by the weak-star topology of L°°(C)).Because of the natural correspondence between H°° and H°°(C) y the weak-star topology on the latter induces a topology on the former, and this is what we mean by the weak-star topology of H°°.The convergent sequences of this topology are easily characterized.LEMMA 1.A sequence {ψ n }T in H°* converges weak-star to the function ψ if and only if it is uniformly bounded and converges to ψ at every point of D. 11az IWe then have 2;for all φ in if 00 and all a in Zλ Now suppose the sequence {ψ n } in H°° is uniformly bounded and converges to the function ψ at each point of D. Then it follows from

Locations

• Pacific Journal of Mathematics - View - PDF