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**Sample text**

In order to estimate this quantity we use an auxiliary result: the dual space to C(1r)/ A(D) is isometric to the space z Hi(D). Namely, (C(1l')/ A(D»' can be identified to (JL E C' (1l'): en (JL) = IT e- in8 dJL(e i9 ) = 0 for n ~ O}. By the theorem ofF. and M. Riesz, JL = f*da, with / E zHi(D). By the density of Jff(D) in A(D) we have J = inf{lIwo + IPIILoo(1l"): IP E A(D)} = d(wo, A(D» = IIwoli = sup {2~ I J~ wordel: / E zHi(D), 11/111 ~ I}, where Wo is the class of wol1l' in C(1r)/ A(D). The last identity is just the HahnBanach theorem.

39. 2. Show that the function I/(l - z) E HP(D) for any p < 1, and that it does not belong to HI (D). Conclude that HP(D) =1= Hq(D) if q =1= p. 3. Show that if I E Jff(D) and I(D) ~ {w E IC: al < arg z < a2}, ~ 0< a2 - al I 4. Let 211", then E HP(D), rp I E HP(D) for p < 1I"/(a2 - al)' E Aut(D). Is I orp E HP(D)? *5. Assume I E H I (D) is such that the set of all products of the form pi, p a polynomial, is dense in HI(D). Show that I is an outer function. 6. Let I E HI(D), 7. (a) Let IIFrllp, 0 < (b) Let I, F, W = Ln~oanZn.

The set M is weakly closed: let 1/1 E M,w (weak-closure of M), then for any f, g E HOO(D) and any e > 0 there is rp E M such that 11/I(g) - rp(g)1 < 11/I(f) -1{J(f)1 < e, e, and 11/I(fg) - rp(fg) I < e, by the definition of the weak topology in (W'O(B»'. Since rp(fg) = rp(f)rp(g), we obtain 11/I(fg) -1/I(f)1/I(g)1 :::: e(l + 11111 + IIgll), which shows that 1/1 is multiplicative. As a consequence, we see that M is a weakly compact subset of (Hoo(D»,. In fact, it is well known that the unit sphere is already weakly compact.