Preface
This is a revised versio n o f m y ten-lectur e maratho n a t th e America n Mathematica l So -
ciety Regiona l Conferenc e "O n Banac h space s of analyti c functions " hel d a t Ken t Stat e Uni -
versity, July 11— 16, 1976. I t i s based o n th e cours e whic h I taught a t th e Ohi o Stat e Uni -
versity durin g th e fal l an d winte r quarter s of th e academi c yea r 1975 1976.
In th e past , the "classical " spaces of analyti c functions , th e dis c algebr a A an d th e
Hardy space s
Hp,
hav e bee n studie d mainl y fro m th e poin t o f vie w of th e theor y o f Banac h
algebras an d harmoni c analysis . However , several result s obtaine d mor e recentl y ca n b e nat -
urally formulate d i n th e languag e o f genera l Banac h space s and operato r theory , althoug h
they ar e stil l closel y relate d t o som e question s i n hard analysi s an d ofte n us e analyti c tools .
An example i s the Fefferman-Stei n dualit y betwee n BMO and H
l
(cf . [F-S]) . Anothe r on e
is Henkin's theor y o f analyti c measure s fo r space s o f analyti c function s o f severa l comple x
variables i n strictly pseudoconve x domain s (cf . [He3]) . Als o th e technique s o f p-absolutel y
summing operator s an d relate d idea s whic h hav e bee n recentl y th e objec t o f intens e stud y i n
the theor y o f Banac h space s seem t o offe r a new approac h t o space s o f analyti c functions .
To explai n this , note tha t th e natura l injectio n A Hp i s an exampl e o f a p-absolutely
summing operator , an d th e fac t tha t thi s operato r i s p-integral fo r p 1 is equivalent t o th e
M. Riesz theore m o n th e boundednes s i n Lp-norm o f th e orthogona l projectio n fro m LP on -
to Hp. Mor e generally considerin g th e completio n i n Lp(fi)-norm o f th e uniforml y contin -
uous holomorphic function s i n a fixed bounded domai n o f holomorphy , wher e j u is a mea-
sure o n th e boundar y o f th e domain , is nothing mor e tha n studyin g certai n propertie s o f a
certain p-absolutely summin g operator : "natura l injection" . A more sophisticate d exampl e
is a map/—• (/(2 n ))~
= 1
fro m A int o I 2. A n inequalit y discovere d b y Pale y aroun d 1932
says tha t
( Z !/(2")l
2
Y ^ 2 fdD |/1 dm fo r ft A.
It mean s exactl y tha t th e ma p i n question i s absolutely summing . Paley' s inequalit y depend s
heavily o n th e analyticit y of/ . Ther e i s no counterpar t fo r continuou s function s o n th e cir -
cle an d thi s reflects th e fac t tha t ever y absolutel y summin g operato r fro m a C(S)-space int o
a Hilber t spac e i s nuclear, an d therefor e compact . Thi s produces a linear topologica l invari -
ant whic h enable s u s t o sho w tha t th e dis c algebr a (an d mor e generall y a uniform algebr a
with a nontrivial Gleaso n part ) i s not isomorphic , as a Banach space , to an y quotien t o f a
C(5)-space (cf . §§ 4 an d 5) . O n th e othe r hand , Paley' s inequalit y ca n b e applie d t o prov e
the basi c fac t i n th e theor y o f absolutel y summin g operators , discovered b y Grothendieck ,
that ever y operato r fro m a n L J(/i)-space t o a Hilbert spac e i s absolutely summin g (cf . § 3 fo r
details).
Some word s abou t th e organizatio n o f th e paper :
1
http://dx.doi.org/10.1090/cbms/030/01
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