Banach Spaces of Analytic Functions and Absolutely Summing - download pdf or read online

By A. Pelczynski

This booklet surveys effects pertaining to bases and diverse approximation homes within the classical areas of analytical services. It includes huge bibliographical reviews.

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Extra resources for Banach Spaces of Analytic Functions and Absolutely Summing Operators (Regional Conference Series in Mathematics ; No. 30)

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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. [H, p. 138]). , there is no IE B H~ and x* E (H~)* with x*(f) = 11/11 = IIx*1I = 1 and such that for every sequence (fn) in B H"" if x*(fn) --+ x*(f) then II/n - III --+ 0. 1(F) = 0. , {s E Ll: iF(s) = I} = {s Ell: liF(s) I = I} = F. E BH~ put Given I for n = 1, 2, In = 1(1 - f~·)/ill - I~II for n = 1,2, .

1, denote by PI and P2 the natural projections from A* = qaD)*/H~ = Ll/H~ Ea1 Vsing onto LI/H~ and Vsing respectively, and let T: qaD)*/H~ ~ qaD)* and TL : L I /H~ ~ L I to be the nearest point cross-sections. Now if W C A * satisfies (4a), then both of the sets PI (W) and P2 (W) have the same property. For subsets of Ll/H~, (4a) is equivalent to (4). Hence the weak closure of T(Pl(W» = TL(Pt(W» is weakly compact. s. sequence (I{Jn) in C(aD). Thus, by a result of [P9], the weak closure of the set T(P2(W» = P2(W) is weakly compact.

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. [H, p. 138]). , there is no IE B H~ and x* E (H~)* with x*(f) = 11/11 = IIx*1I = 1 and such that for every sequence (fn) in B H"" if x*(fn) --+ x*(f) then II/n - III --+ 0. 1(F) = 0. , {s E Ll: iF(s) = I} = {s Ell: liF(s) I = I} = F.

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