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

By A. Pelczynski

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

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Read or Download Banach Spaces of Analytic Functions and Absolutely Summing Operators (Regional Conference Series in Mathematics ; No. 30) PDF

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

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Clearly Q n : Xl ~ H ci is a linear operator with IIQ n II ,,;; I, for n = 1, 2, .... Let us put Q(v) = LimQn(v) for v E Xl. n Here Limnh n denotes a fixed Banach limit of a bounded sequence (h n ) with respect to the a(H~, C(aD)/A)-topology. Since H~ is the dual of C(aD)/A, the Banach limit exists and limnh n is a cluster point of the set U;=I {h n } in the a(H~, C(aD)/A)-topology. Clearly Q: Xl ~ H~ is a linear operator with IIQII ,,;; 1. 9) and if g = dv/dp, f aD Q(v)· f dm = then, for every f E C(aD), fs f(F(s))g(s)p(ds).

F F ) which contradicts the definition of the exposed point, becausefF being nonconstant implies thatf· fF EBH",,\{f}. 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.

2, there is a nonconstant fF E H"" with IlfF11 = 1 such that iFI (I) :J F. For n = 1, 2, ... , we have 1*(n - *(fFnl == <1>; n «(1 - fF)n <: Ilill sup 11 - fF(z)l. zEe n n; Taking into account thatel:J e2 :J ... :J F = = I en and iFI (I) :J F, we infer that limnsuPzEen 11 - fF(z) I = O. Hence *(n = *(f. f F ) which contradicts the definition of the exposed point, becausefF being nonconstant implies thatf· fF EBH",,\{f}. Hence m(e",,) > o. 10). Let <1>* E (L"")* be defined by *(g) = m(E)-l fEg(z)signf(z)m(dz) for gEL"".

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