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.

**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)**

**Example text**

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"". *