Banach Spaces of Analytic Functions and Absolutely Summing by A. Pelczynski

By A. Pelczynski

This e-book surveys effects relating bases and diverse approximation houses within the classical areas of analytical capabilities. It comprises huge bibliographical reviews.

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Wojtaszczyk [W2] observed recently that Havin's lemma can be generalized to uniform algebras with unique representing measures for linear multiplicative functionals. 2 is taken from [A-L]. The construction is classical (cf. [Z, Vol. I, p. 105]). 3 shows that the "Remarque" in [A-L] is false. 1 is due to Amar and Lederer [A-L] and Fisher [Fi]. An analogous result for the disc algebra is due to Phelps [Ph2]. Let us recall that an IE BH~ is an extreme point of B H~ iff I aDlog(l - III) dm = --00 (cf.

Hence m(e",,) > o. 10). Let <1>* E (L"")* be defined by *(g) = m(E)-l fEg(z)signf(z)m(dz) for gEL"". ALEKSANDER PElCZYNSKI 42 Let x* be the restriction of <1>* onto H~. Clearly IIx*1I = Ix*(f)1 = 11<1>*11 = 1. Hence Re x*(g) :(; 1 for g E B H~' Finally if, for some g E B~, *(g) = x*(g) = I, then Ig(z)1 = 1 and g(z)sign/(z) = 1 for z E E almost everywhere. Hence g(z) = I(z) for z E E almost everywhere. Since m(£) > 0, we infer that I = g. 0 REMARK. Note that if I is an exposed point for B H~ then a linear functional which strictly supports B H~ at I can be chosen from L 1/H ~, a predual of H~.

S. 1), s s+ 1 Ix,iz) I ~ (lfkJz)1 + IgkJz)1) = 1 n L j== 1 j== 1 J for z E aD and for s = 1,2, . . 13) we get IfaD GSUk s+1 dm I;;;. IJ-aD Gsu dm 1- 2-s- 2a1/ ; ;. 15) j==1 s ;;;. a - L If aD Gjudm - ( JaD T~-2a1/ - 2-s- 2a1/ > a(1 Gj_1udm 1- 2-S- 2a1/ - 1/). 1», SUPzEeks+llfks+l(z) - 11- 0 as s sUPkllu';1I1 ~ sUPnllunli =M < lim 8=="" 00. 8) and the relation get f aD\ek 11 - fk IIGslu ks+ s+ ~ lim sup f Uk dm 8==00 aD\ek s+ ~ a - lim inf f. uk dm s==oo e ks + 1 8+1 = lim sup 8==00 -0. 16) we get = 1.

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