By Earl R. Berkson, N. Tenney Peck, J. Jerry Uhl

During the educational yr 1986-7, the collage of Illinois used to be host to a symposium on mathematical research which was once attended via a few of the major figures within the box. This e-book arises out of this targeted yr and lays emphasis at the synthesis of recent and classical research. The contributed articles through the members disguise the gamut of mainstream subject matters. This e-book could be necessary to researchers in mathematical research.

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A complex Banach space B is said to have the analyticRadon-Nikodym properiy(kMP) if every B-valued bounded holomorphic function from the disc has boundary limits almost everywhere, in other words if the Poisson integral is an isometry between { f e L - B d ) : f(n) - 0 for n < 0 ) and ITBCD). There they showed that is equivalent to consider the same property for any value of p, 1 < p < <» and also that RNP implies ARNP, proving that LL(T) which fails to have RNP does have this other property. Our objective here will be to look for a substitute for { f e LPB(T) : f(n) - 0 for n < 0 ) which is valid independently of the Banach space considered, and here Operator Theory plays a role.

And in our case Fr(9) goes to (ei^®)n£z as r goes to 1, which does not belong to co. From these two facts we have that the Poisson integral embeds the space { f € L P B ( T ) : f ( n ) - 0 for n < 0 ) into HPB(D) but it is not surjective unless the Banach space has some property. In [31 Bukhvalov and Danilevich considered the following definition. - [31. A complex Banach space B is said to have the analyticRadon-Nikodym properiy(kMP) if every B-valued bounded holomorphic function from the disc has boundary limits almost everywhere, in other words if the Poisson integral is an isometry between { f e L - B d ) : f(n) - 0 for n < 0 ) and ITBCD).

2), once the following lemma has been established. 2) LEMMA. where p Given k € L^(G) is an arbitrary and measure X a closed and subspace 1 < p < <» , we of L^{M,fj) , have Np j^(k) < Np(k) . Proof. At the outset, we can assume without loss of generality that 00 the set of points where each r^ k is non-zero has the form is a compact subset of function in G . Let U r m=l L^(G,X) of the form j=l • °2 ' '' ' °n where f: G —^ X be a simple n where , ^ compact subsets of G , and 30 Berkson & Gillespie: Spectral decompositions , belong to X .