By P. F. X. Müller, W. Schachermayer

This quantity displays the growth made in lots of branches of contemporary examine in Banach house conception, an analytic method of geometry. together with papers by means of many of the top figures within the zone, it truly is meant to demonstrate the interaction of Banach house idea with harmonic research, chance, complicated functionality concept, and finite dimensional convexity conception. The papers include a range of surveys and unique study.

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UHL. "Vector Measures " Math, surveys, no 15, Amer. Math. Soc. I. 1977. [D] P. L. DUREN, 1970. "Theory of HP spaces". L. L. SHIELDS. Coefficient multipliers on HP and BP spaces. Pacific J. Math. 32 (1970), 69-78. L. W. L. SHIELDS. Linear functionals on HP spaces 0 < p < 1 J. Reine Angew. Math. 238 (1969), 32-60. M. FLETT. On the rate of growth of mean values of holomorphic and harmonic functions. Proc. London Math. Soc. 20 (1970),749-768. M. FLETT. Lipschizt spaces of functions on the circle and the disc; J.

T: L1 - • X be any non-representable operator with T(F)cX. Then there is a Dunford - Pet t is operator D : L1 - • L1 such that the operator T°D is not representab/e. Proof : Suppose T: L1 ~» X is a non-representable operator with T(F)cX. There is a set Ac[0,1] of positive measure and a 6>0 such that for every subset B of A with m(B)>0, diam(T(FB)) > 6. It is well known (see [B2D that every slice S of the set FB = T(FB) contains a set of the form TB- = T(FB0 for some set B'cB of positive measure.

In the last section we obtain most of the results on multipliers in [D-S] as consequence of representability of operators and results of duality for certain Besov-Lipschizt classes -which were proved in [F2] - are obtained from previous vector-valued results. Throughout the paper ( X,ll II ) n IIF(re it )U T~ M p (F,r) means denotes a complex Banach space, , and C will be a constant not -71 necessarily the same at each ocurrence. SPACES OF VECTOR VALUED ANALYTIC FUNCTIONS. 1. Let a < 2 and F: D -> X analytic .

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