# Abstract Analytic Function Theory And Hardy Algebras by K. Barbey, H. König

By K. Barbey, H. König

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And vice relation into equivalence and part our different after iii) occur. is << some for A. 3 R E F O R M U L A T I O N : ii) M(~)VcM(~)Vor part of all G l e a - for A. for any ~6P. 2 w e h a v e or M ( 9 ) ~ c M ( ~ ) ^. T h e QED. 2 with M(~)~cM(~) ~ and M ( ~ ) ~ c M ( ~ ) ^ c a n n o t or M ( ~ ) ~ c M ( ~ ) ^. This in c o n n e c t i o n likewise the b a n d b(P),b(Q) b(P) b(P) :=M(~) v as follows. is r e d u c i n g . are m u t u a l l y singular to b ( P ) n b ( Q ) = { O } ) . is a r e d u c i n g band then for e a c h P6F(A) either b(P)c_B or b(P)c-B ^.

That the relation relation H ( u , v ) < ~ is s y m m e t r i c . two or H ( u , v ) < ~ set The , relation that the they on B(X). Vu,v6X. 1~H(u,v)~ ~ that fix a n o n v o i d with llfll<=1} , supnorm of X. We h a v e O ~ G ( u , v ) ~ 2 H(u,w)~H(u,v)H(v,w) rem shows the the e s s e n t i a l s to be rE(v) If-If d e n o t e s We contains H:H(u,v) _ contains It is a s u r p r i s e structure: = Sup{Lf(u)-f(v)I:fCA is sym/netric, Thus section theorem. 1 The present representation relations is an e q u i v a l e n c e are G(u,v)<2 The in f a c t relation next is theo- identical.

4 to the function F+s for 40 e>O and thus obtain f(log(F+e))+dme appropriate m6M(~) well-defined. 3 is me<O. F6LI(o) and hence Inf : g(x) := FO 6 P o s ( X , Z ) . Choose a fixed function = Then such flog G dm e x i s t s I f(x) if f ( x ) ~ O 0 if f ( x ) = O 0 ~ G:=g i) to T a n d G. T h u s there in the modT exist 6 LI(T) extended sense, whenever f(x)~O. GF=I log G + log F = 0 in L(Fo) so t h a t L e t us r e f o r m u l a t e : flog Fdm exists and There in the these exist extended Inf{exp(-flog with We m< 0 and GT=gT=gfo~o.