Uniform Algebras As Banach Spaces

The Erwin Schrodinger International Boltzmanngasse

ESI Institute for Mathematical Physics A Wien Austria

Uniform Algebras as Banach Spaces

TW Gamelin

SV Kislyakov

Vienna Preprint ESI June

Supp orted by Federal Ministry of Science and Transp ort Austria

Available via httpwwwesiacat

UNIFORM ALGEBRAS AS BANACH SPACES

TW Gamelin SV Kislyakov

June

Abstract

Any Banach space can b e realized as a direct summand of a uniform algebra and

one do es not exp ect an arbitrary uniform algebra to have an abundance of prop erties

not common to all Banach spaces One general result concerning arbitrary uniform

algebras is that no prop er uniform algebra is linearly homeomorphic to a C K space

Nevertheless many sp ecic uniform algebras arising in complex analysis share or are

susp ected to share certain Banach space prop erties of C K We discuss the family

of tight algebras which includes algebras of analytic functions on strictly pseudo con

vex domains and algebras asso ciated with rational approximation theory in the plane

Tight algebras are in some sense close to C K spaces and along with C K spaces

they have the Pelczynski and the DunfordPettis prop erties We also fo cus on certain

prop erties of C K spaces that are inherited by the disk algebra This includes a dis

cussion of interp olation b etween H spaces and Bourgains extension of Grothendiecks

theorem to the disk algebra We conclude with a brief description of linear deformations

of uniform algebras and a brief survey of the known classication results

Supp orted in part by the Russian Foundation for Basic Research grant

Contents

Uniform Algebras

Analytic Functions on Banach Spaces

Characterization of Prop er Subalgebras

Tight Subspaces and Subalgebras of C K

The Pelczynski and DunfordPettis Prop erties

Absolutely Summing and Related Op erators on the Disk Algebra

Interp olation of HardyTyp e Subspaces

Bourgain Pro jections

Perturbation of Uniform Algebras

The Dimension Conjecture

Uniform Algebras

A uniform algebra is a closed subalgebra A of the complex algebra C K that contains the

constants and separates p oints Here K is a compact Hausdor space and A is endowed with

the supremum norm inherited from C K The algebra A is said to b e proper if A C K

Uniform algebras arise naturally in connection with problems in approximation theory The

main examples of prop er uniform algebras come from complex analysis The prototypical

prop er uniform algebra is the disk algebra which we denote simply by C consisting of the

analytic functions on the op en unit disk in the complex plane that extend continuously to

the b oundary More generally if K is a compact subset of C we denote by AK or by

C K the algebra of functions continuous on K and analytic on the interior of K Also we

may consider the uniform closure of the restriction to K of some algebras of elementary

holomorphic functions such as analytic p olynomials or rational functions with singularities

o K The uniform closure of the rational functions with singularities o K is denoted by

RK If K is a compact subset of the complex plane Runges approximation theorem asserts

that RK includes the functions that are analytic in a neighb orho o d of K It may o ccur

that RK is a prop er subalgebra of C K even when K has empty interior Other uniform

algebras asso ciated with a domain D in C are the algebra AD of analytic functions on

D that extend continuously to the closure of D and the algebra H D consisting of all

b ounded analytic functions on D Endowed with the supremum norm on D the algebra

H D b ecomes a uniform algebra on the smallest compactication of D to which the

functions extend continuously In the case of the op en unit disk fjz j g we may

identify H with a closed subalgebra H d of L d via nontangential b oundary

values where d is the arclength measure on the unit circle

One of the goals in studying uniform algebras is to use the to ols of functional analysis

and the Gelfand theory in order to prove approximation theorems or to understand why ap

proximation fails Mergelyans theorem that RK AK whenever K is a compact subset

of the complex plane whose complement has a nite numb er of comp onents was eventually

given a pro of by Glicksb erg and Wermer that dep ends on uniformalgebra techniques and a

less dicult theorem of Walsh on approximation by harmonic functions We quote two other

approximation theorems whose pro ofs dep end on the algebra structure and on techniques

of functional analysis

Theorem Wermer We Let A be a not necessarily closed algebra of analytic

functions on the unit circle in the complex plane Suppose that A separates the points of

and that each function in A extends to be analytic in a neighborhood of Then either A

is dense in C or else there is a nite bordered Riemann surface with border such that

the functions in A extend to be analytic on the surface

Theorem Davie Da Let be the area measure on a compact subset K of the

complex plane and let H be the weakstar closure of RK in L Then each

K K

f H is approximable pointwise ae on K by a sequence of functions f RK

K n

such that jjf jj jjf jj

In some sense uniform algebra theory can b e regarded as an abstract study of the

maximum principle for algebras For an incisive account of this asp ect of the theory see

AWe

Uniform algebra theory has also served as a source of interesting problems One such

problem originally raised by S Kakutani asked whether the op en unit disk is dense in

the sp ectrum of the algebra H This problem b ecame known as the corona problem

It was answered armatively by L Carleson see Gar While the corona theorem p er se

has not played a really signicant role in analysis the techniques that were devised to solve

the problem have played an imp ortant role in function theory

The question arises as to the extent that prop erties of various uniform algebras dep end

only on their linear structure From the p oint of view of Banach spaces how sp ecial are

uniform algebras We will see in Section that generic uniform algebras are as bad as

generic Banach spaces in the sense that any Banach space is isometric to a complemented

subspace of a uniform algebra On the other hand we will show in Section that a prop er

uniform algebra is distinct as a Banach space from C K In fact no prop er uniform algebra

is linearly isomorphic to a complemented subspace of a C K space or even to a quotient

of a C K space

It is of interest to know which prop erties of C K are inherited by uniform algebras

Towards answering this question a great deal of eort has gone into determining which

prop erties of C are passed down to the disk algebra C Some of the known results are

summarized in the table b elow

At present there is no unied theory but rather only fragmented results on uniform

algebras as Banach spaces Our aim in this article is to present a selection of results in order

to give an idea of what has b een studied and what problems are currently op en Sometimes

pro ofs or indications of pro ofs are also given to convey the avor of the techniques that are

employed

Section includes a brief intro duction to algebras of analytic functions on domains in

Banach spaces In Section we show how some basic facts ab out psumming and pintegral

op erators lead to several characterizations of prop er uniform algebras Sections and are

devoted to certain prop erties that involve weak compactness The prop erties are wellknown

for C K spaces and they are inherited by algebras generated by analytic functions on

planar sets and on strictly pseudo convex domains where the key ingredient is the solvability

of a problem

Prop erty of X C K or X C C K C Reference

A A

X is a linear quotient of C S Yes No x

X is complemented in a Banach lattice Yes No x

X has the DunfordPettis prop erty Yes Yes x

X has the Pelczynski prop erty Yes Yes x

X is weakly sequentially complete Yes Yes x

X has a basis Yes if K is metric Yes Wo j

X veries Grothendiecks theorem Yes Yes x

X has a complemented copy of C Yes if K is metric Yes Wo j

and uncountable

In Sections through we talk of prop erties that are more sp ecic to the disk algebra

C We fo cus on Bourgains extension of Grothendiecks theorem to the disk algebra obtained

in B B This sub ject was treated in detail in the survey K The exp osition here will

follow the ideas of K with slight simplications at some p oints and with emphasis on the

interp olatory nature of the pro ofs The extension of Grothendiecks theorem is discussed in

Section and the results on interp olation used in the pro ofs are dealt with in Section

Section includes a brief account of Bourgain pro jections the main technical to ol used by

Bourgain to transfer results from continuous to analytic functions

A go o d deal of the material presented here has already b een discussed in various mono

graphs and exp ository pap ers We mention particularly the early lecture notes of Pelczynski

Pe and the research monographs of Wo jtaszczyk Wo j and of Diestel Jarchow and Tonge

DJT For background on uniform algebra theory see Gam Sto and AWe For Davies

theorem see also Gam The basic Hardy space theory that we refer to is covered in Du

Gar and Hof One reference for several complex variables and pseudo convexity is Ra

The exp ository pap ers K and K also cover in part the material of the present article

We collect here some standard notation and conventions We denote by the unit

circle fjz j g in the complex plane and by dm d the normalized arclength measure

on The Hilb ert transform on L dm is denoted by H We denote by C the disk algebra

which is the uniform closure of the analytic p olynomials in C The algebra of b ounded

analytic functions on a domain D is denoted by H D If A is a linear space of functions

on K and is a measure on K then H A will denote the weakstar closure of A in

L The generic Banach space or quasiBanach space is denoted by X The closed unit

ball of a Banach space X is denoted by B and the op en unit ball by B The image of a

Banach space X in its bidual X under the canonical emb edding is denoted by X

Analytic Functions on Banach Spaces

We wish to develop some examples of algebras of analytic functions dened on domains in a

Banach space

A complexvalued function on an op en subset of a Banach space X is analytic if it

is lo cally b ounded and its restriction to every complex onedimensional ane subspace of

X is analytic In other words f is analytic on D if f is lo cally b ounded and if for every

x D and direction x X the function f x x dep ends analytically on Sums

0 0

pro ducts and uniform limits of analytic functions are analytic

A lo cally b ounded function f on D is analytic just as so on as its restriction to D Y

is analytic for every nitedimensional subspace Y of X Thus any statement ab out analytic

functions that involves only a nite numb er of p oints of X will hold in general once it holds

for analytic functions of several complex variables

Let f b e analytic on a domain D in a Banach space X Supp ose D and supp ose

jf xj C for kxk r For xed x X the function f x has a Taylor series

expansion

f x A x j j r kxk

m=0

One checks easily that each A x is mhomogeneous that is A x A x From the

m m m

m m

Cauchy estimates we have jA xj C kxk r and consequently A x is lo cally b ounded

m m

If we restrict the expansion f x A x to a nite dimensional subspace of X we obtain

the usual expansion of an analytic function as a series of mhomogeneous p olynomials In

particular A x dep ends analytically on x in any nitedimensional subspace of X and

since it is lo cally b ounded it is analytic

Let P P X denote the space of analytic functions on X that are mhomogeneous

m m

We endow P with the supremum norm over the unit ball B of X and then P b ecomes a

m X m

Banach space The Cauchy estimates show that the corresp ondence f A is a normone

pro jection from the space H B of b ounded analytic functions on the op en unit ball of

X to P

The rst Taylor co ecient in the expansion of an analytic function f at coincides

with the usual Frechet derivative f of f at whose dening prop erty is that f x

f f x okxk as x The Frechet derivative f is a continuous linear

functional on X and the space P coincides with the dual space X of X

Theorem Milne Mi Any Banach space is isometric to a complemented

subspace of a uniform algebra

b e the closed unit ball of the dual space X To prove the theorem we let K B

of X endowed with the weakstar top ology Recall that X denotes the canonical image of

X in X The restriction of X to K is a closed subspace of C K Let A b e the uniform

algebra on K generated by X The functions in A are analytic on the op en unit ball of X

hence have Taylor expansions f x A x The normone pro jection f A into

m 1

P X is the identity on X and it pro jects any mhomogeneous p olynomial in elements of

X to if m Thus if we pass to uniform limits of sums of p olynomials in elements of X

X we obtain a normone pro jection of A onto X

of weak We could as well obtain the same result by considering the algebra AB

that are analytic on the op en unit ball of X It is not star continuous functions on B

onto X maps AB onto P X dicult to check that the pro jection of AB

X 1 X

thus onto X However it is shown linear functionals that are weakstar continuous on B

need not coincide with the algebra generated by the in ACG that the algebra AB

weakstar continuous linear functionals Along these lines it is not even known whether the

coincides with B maximal ideal space of AB

X X

There is an expanding literature ab out p olynomials on Banach spaces and ab out

uniform algebras asso ciated to Banach spaces see Din GJL The study of p olynomials

fo cuses on the spaces P which can b e viewed as spaces of multilinear functionals on X

Every continuous mhomogeneous p olynomial f on X is the restriction to the diagonal of a

unique continuous symmetric mlinear functional F on X X This F is given by the

p olarization formula

f x x F x x

1 m 1 1 m m 1 m

the summation b eing extended over the indep endent choices of exercise The

same formula shows that the norm in P is equivalent to the multilinear functional norm

though the spaces are not isometric in general The space P can also b e viewed as the dual

space of the mfold symmetric pro jective tensor pro duct of X with itself For background

see Mu Gam

Characterization of Prop er Subalgebras

In this section we shall prove that if a uniform algebra is prop er then it diers as a Banach

space from any space C K The crux of the pro of is that every absolutely summing op erator

from C K to a reexive space is compact while we construct on any prop er uniform algebra

an op erator to that is absolutely summing but not compact

Let X and Y b e Banach spaces and let p q An op erator T X Y is

said to b e q psumming if

X X

1q 1p

q p

kT x k C sup jx x j x X kx k

j j

for every nite collection fx g of elements of X The b est constant C is denoted by T

j q p

The class of such op erators forms an op erator ideal in the sense that precomp ositions and

p ostcomp ositions with b ounded op erators remain within the class and further the usual

estimates for norms hold

If p q coincides with the denition of a psumming op erator see Basic

Concepts in this case the notation T is used By absolutely summing we mean

summing We remind the reader that the psumming op erators are characterized as those

that factor through a part of the inclusion L L in the sense that T can b e

represented as a comp osition

U V

T X M M Y

where is a probability measure M is a subspace of L and M is a subspace of L

containing M or alternatively M is the closure of M in L For p T is said to

b e strictly pintegral if it factors through the entire inclusion L L that is we

can take M L and M L in This notion diers slightly from that of a

pintegral op erator as dened in Basic Concepts However the two notions coincide if say

Y is reexive As explained in Basic Concepts for p every psumming op erator on C K

is strictly pintegral so that the psumming and the strictly pintegral op erators on C K

coincide

We recall that any absolutely summing op erator is weakly compact For integral

op erators we can use the DunfordPettis prop erty of L to say more

Lemma A strictly integral operator from X to a reexive Banach space is

compact In particular an absolutely summing operator from C K to a reexive Banach

space is compact

To see this consider the factorization ab ove with p M L M

L Since Y is reexive V is weakly compact By the DunfordPettis theorem V maps

weakly compact subsets of L to normcompact subsets of Y Since b ounded subsets of

L are weakly precompact in L the comp osed op erator T maps b ounded subsets of

X into normcompact subsets of Y and T is compact

The prototyp e for an absolutely summing op erator that is not integral is the Paley

op erator P on the disk algebra C The Paley op erator assigns to a function f on the unit

k k

Paleys inequality is circle the sequence of th Fourier co ecients ff g

k =1

k 2 1

jf j c kf k f H m

P 1

k =1

for some constant c In other words the restriction of the Paley op erator P to H m

1 2

is a b ounded op erator from H m to For the pro of see Hof Z

2 k

Let M b e the closed linear span in L m of the exp onential functions expi

k It is a classical fact see Z that the L norms on M are equivalent for p

Further for p there is a continuous pro jection Q of L m onto M Thus for

p we can factor the Paley op erator on C through the inclusion L L

p 2

P C L m L m M

and P op erating on C is pintegral for p On the other hand P is not compact

so P is not absolutely summing on C

The story is dierent if we restrict P to C Paleys inequality yields the factorization

1 2

P A H m

where V is the Paley op erator on H m Thus P is absolutely summing on C On the

other hand P maps the exp onential functions expi k to the standard basis vectors

of so that P is not compact and P is not integral This shows incidentally that C is

not complemented in C nor even isomorphic to any quotient space of a C K space or

else the comp osition of the pro jection and P would pro duce an absolutely summing op erator

on C K that is not compact

Our aim is to transfer the Paley op erator and this nal observation to an arbitrary

uniform algebra Paleys inequality transfers directly as follows

Lemma Let A be a uniform algebra on K Let f A satisfy kf k and let

be any measure on K orthogonal to A Then the sequence

P d f

k =1

belongs to and kP k c k k where c is the best constant for Paleys inequality The

2 P P

estimate persists for f H j j the weakstar closure of A in L j j

Proof Dene U C C K by U g s gf s for g C and s K

n n n n

where g is the Poisson integral of g Then kU k U z f and U z f Now

U M K M sends A to C which by the F and M Riesz theorem is identied

1 1

with H m Thus U h dm for some h H m and

Z Z

d f z hz dm k

Paleys inequality for h yields kP k c khk c kU k c k k This proves the

2 P 1 P P

rst statement of Lemma and the second is obtained by applying the rst to the uniform

j j and noting that generates a functional on L j j orthogonal to H j j algebra H

Theorem Kislyakov K If A is a proper uniform subalgebra of C K

then there is an absolutely summing operator from A to that is not compact hence not

integral

The pro of dep ends on the Paley op erator asso ciated with an extremal function F

for a certain dual extremal problem Since A is prop er there is a measure on K such

that A but the complex conjugate of is not orthogonal to A We assume that the

functional f f d on A has unit norm By the HahnBanach and Riesz representation

theorems there is a measure on K such that kk and A Let ff g b e a

sequence of functions in A such that kf k and f d and let F H jj jj b e

n n

a weakstar limit p oint of the sequence ff g Then jF j and F d from which it

follows that jF j ae d Now f A for all f A hence for all f H jj jj

2 2

In particular F A and F A We dene

T g g A g d F

k =1

By Lemma applied to F H jj jj and the orthogonal measure g the sequence

T g is square summable and kT g k ckg k c jg jdj j Thus T can b e factored through

1 1

the closure H j j of A in L j j

1 2

T A H j j

and T is absolutely summing To see that T is not compact we compute the k th comp onent

k k

2 1 2 1

of T F to b e rigorous rather we must consider lim T f

Z Z Z Z

k k k k

k k k k k

2 2 2 2

2 +1 2 1 2 +1 2 1 2 1

F F F F d F d F d F d T F F

Let E b e the set on which jF j Since is carried by E the rst integral on the right is

F d Since jF j o E as n tends to

Z Z Z

F d F d F d i Im

E E E

which is not zero Since the k th comp onents of the vectors T F n do not tend to

n 2

zero uniformly in n the vectors T F do not lie in a compact subset of and then neither

do es the image of the unit ball of A under T Thus T is not compact and by Lemma

T is not integral on A

If we analyze the pro of of Theorem we nd that it extends to any closed subspace

B of a uniform algebra A providing there is a function f B such that f A B while f A

Indeed let I b e the set of all f satisfying f A B This is a closed ideal in A We cho ose

A such that generates a normone functional on I and we pro ceed as b efore

For some time it was an op en problem known as the Glicksberg problem as to whether

a prop er uniform algebra on a compact space K can b e complemented in C K Theorem

settles the Glicksb erg problem and it do es even more

Theorem If A is a proper uniform subalgebra of C K then A is not isomorphic

to a quotient of a C J space In particular A is not complemented in C K

Indeed supp ose A is the quotient of C J If we comp ose the op erator T from

Theorem with the quotient map we obtain an op erator

C J C J Z A

that is absolutely summing hence compact by Lemma Since the pro jection is an op en

mapping the op erator T must b e compact and this contradicts Theorem

We mention some further conclusions that can b e drawn from this circle of ideas

Recall see Basic Concepts that a Banach space X has GordonLewis local unconditional

structure GL lust if roughly sp eaking its nite dimensional subspaces are well emb ed

dable in spaces with unconditional basis This o ccurs if and only if X is a complemented

subspace of a Banach lattice Thus C K has GL lust as do all L spaces p

Theorem If A is a proper uniform subalgebra of C K then A does not have

GL lust

The idea of the pro of is to lo ok for a factorization of the Paley op erator P C

through A with the help of the second adjoint of the op erator T dened in the pro of of

Theorem

U T

P C A

However in this way we only obtain an op erator quite similar to P but not P itself

We redene F to b e a weakstar limit p oint of the sequence ff g in A and we use the

realization of A as a weakstar closed subspace of C describ ed in the next section The

op erator T may still b e dened by where now g b elongs to A and the functions

b eing integrated in are the pro jections of the functions in C into L j j The

corresp ondence pZ pF mapping a p olynomial in the co ordinate function Z to pF

is of norm at most hence extends to a b ounded op erator U from C into A Now the

hyp othesis of GL lust implies by the GordonLewis theorem see DJT Theorem

that the absolutely summing op erator T factors through an L space and we obtain

U V W

1 2

T U C A L

By Bourgains extension of the Grothendieck theorem Theorem the comp osition VU

mapping C into an L space is summing hence weakly compact By the DunfordPettis

theorem the weakly compact op erator W maps weakly compact subsets of L to norm

compact subsets of Thus the comp osition WVU is compact contradicting the noncom

pactness of T U see

Along similar lines it can also b e proved that if a prop er uniform algebra A is a

quotient of a Banach space X having GL lust then X contains a complemented copy of l

The crucial observation here is that if X fails to have a complemented copy of l then every

op erator from X to L is weakly compact To see this combine the Pelczynski prop erty

of L with LT Prop osition e

In another direction Garling Ga showed that the dual A of a prop er uniform algebra

is not a subspace of the dual of a C algebra The pro of is mo deled on an earlier argument

for C and uses the basic ob jects and F app earing in the pro of of Theorem

Tight Subspaces and Subalgebras of C (K )

The classical Hankel op erator corresp onding to a function g on the unit circle op erates from

2 2

H d to H d sending f to g f P g f where P is the orthogonal pro jection from

2 2 2

L d onto H d The Hankel op erator is equivalent to the op erator f g f H from

2 2 2

H to the quotient space L H The analogue of these op erators acting on subspaces of

C K has proved a key to understanding uniform algebras

Let A b e a closed subspace of C K To each g C K we asso ciate a generalized

Hankel operator S from A to the quotient Banach space C K A by

S f g f A f A

We say that A is a tight subspace of C K if the op erators S are weakly compact for all

g C K We say that A is a compactly tight subspace of C K if S is compact for all

g C K Tightness was intro duced in CG Our discussion is based on that pap er and on

Sac Sac

We will use the representation of the bidual C of C C K as a uniform algebra

This representation is realized as follows The dual space of C is the space M K of nite

regular Borel measures on K with the total variation norm and this can b e regarded

as the direct limit of the spaces L M K The bidual C is then represented

as the inverse limit of their dual spaces L M K A simpleminded way to

express this is to say that each element F C determines for each M K a function

F L and these satisfy the compatibility condition that F F almost everywhere

with resp ect to whenever Conversely each uniformly b ounded compatible family

fF g F L determines an element of C The norm of F in C is the supremum of

the norms of F in L The multiplication in the spaces L determines an obvious

multiplication in C

Let A b e a subspace of C Recall that H A denotes the weakstar closure of A

in L The bidual A can then b e identied with the weakstar closed subspace of C

consisting of F C such that F H A for all M K The bidual of the quotient

space C A is isometric to C A and the canonical emb edding maps C A isometrically

onto C A In particular C A is a closed subspace of C A and consequently A C

is a closed subspace of C

Now the op erator S is weakly compact if and only if the image of A under S is

contained in the canonical image of C A Identifying C with its canonical image in C we

see that for g C

S is weakly compact g A A C

From this it follows that the g s for which S is weakly compact form a closed subalgebra of

C K Thus A is tight just as so on as S is weakly compact for any collection of g s that

generates C K as a uniform algebra

If A is a subalgebra of C then each space H A is an algebra and A is a

subalgebra of C In this case we obtain from the following

Theorem A uniform algebra A on a compact space K is a tight subalgebra of

C K if and only if A C K is a closed subalgebra of C K

For algebras of analytic functions tightness is related to solving a problem Roughly

sp eaking the connection is as follows Functions that b elong to an algebra of analytic

functions A are characterized as the functions f satisfying f Supp ose that is a

solution op erator for the problem which need not b e linear Let g b e a smo oth function

If f A then from the Leibnitz rule we obtain f g f g g f f g This shows

that the oneform f g is closed We apply the solution op erator and obtain a function

h f g satisfying h f g so that h f g A and S f h A Thus the action

of S on f amounts to multiplying f by g applying and pro jecting into the quotient

space C A It follows that if there is a weakly compact solution op erator for the problem

then each S is weakly compact and A is tight By the same token if there is a compact

solution op erator for the problem then A is compactly tight

If D is a b ounded strictly pseudo convex domain in complex nspace with smo oth

b oundary the problem can b e solved by means of integral op erators with Holder estimates

on the solutions so that there are compact solution op erators see Ra The argument

outlined ab ove can b e made precise It shows that the algebra AD asso ciated with any

such domain is compactly tight The connection b etween tightness and solving the problem

is not completely understo o d but our line of reasoning do es establish the following

Theorem Let D be a bounded domain in complex nspace Suppose there is

a weakly compact subset E of C D such that the equation h on D has a solution

h E for every closed smooth form on D that extends continuously to D and

satises k k Then AD is a tight subalgebra of C D If E is compact then AD

is compactly tight

If D is strictly pseudo convex with smo oth b oundary the problem solution tech

niques can b e used to show that any f H D can b e approximated p ointwise on D by

a b ounded sequence of functions in AD that extend analytically across D with uniform

convergence on D if f AD In the strictly pseudo convex case every p oint p in D

is a p eak p oint for AD that is there is f AD satisfying f p and jf j on

D nfpg Thus the following theorem applies to strictly pseudo convex domains with smo oth

b oundaries

Theorem Let D be a bounded domain in complex nspace for which the

problem is solvable as in Theorem and let be the volume measure on D Suppose that

the functions in AD that extend analytical ly across D are pointwise bounded ly dense in

1 1

H D Then AD is the direct sum of L AD and an L space and the bidual

AD is isometrical ly isomorphic to the direct sum of H D and an L space Further

if every point of D is a peak point for AD then H D C D is a closed subalgebra of

L and AD C D is isometrical ly isomorphic to the direct sum of H D C D

and an L space

The idea of the pro of is as follows Let B b e the band of measures on D that

are singular to every measure in AD and let B b e the band of measures generated

by AD There is a direct sum decomp osition M D B B with a corresp onding

a s

decomp osition AD H B L B We claim that the summand H B is

a s a

isometrically isomorphic to H D We regard H D as a subalgebra of L If F

AD there is a b ounded net ff g in AD that converges weakstar in AD to F Then

ff g converges weakstar to F in L Since the f s are uniformly b ounded they are

equicontinuous at each p oint of D and consequently any limit function is analytic on D

Thus F H D The hyp othesis of p ointwise b ounded density implies that the pro jection

F F maps AD onto H D Supp ose F AD satises F Let AD

Let g b e a smo oth function and let ff g b e a b ounded net in AD that converges weakstar

to F Then f on D Cho ose h E such that f g h on D Passing to a

subnet we may assume that h h weakly where h C D Then g f h is analytic

on D and in the limit h is analytic on D Now g f h h g F weakstar in L

Thus g F this for all smo oth functions g so that F It follows that F for all

B is Thus the algebra measures B and consequently the pro jection of F in H

a a

homomorphism H B H D is onetoone and onto Since any homomorphism of

uniform algebras that is onetoone and onto is an isometry H B is isometric to H D

The nal statement of the theorem that H D C D is isometric to a direct summand of

AD C D is equivalent to the statement that H B H D is a lo cal isometry

in the sense that it is an isometry at every p oint of D An easy way to guarantee this is to

assume that every p oint of D is a p eak p oint for AD

Another class of examples of tight algebras are the algebras RK and AK asso ci

ated with a compact subset K of the complex plane For these the solution op erator for the

problem is the Cauchy transform op erator

Z Z

h dxdy K

which is a compact op erator on C K Again the line of reasoning outlined ab ove together

with a few technical details establishes the following

Theorem Let K be a compact subset of the complex plane Then the algebras

RK and AK are compactly tight If is the area measure on K and A is either of

1 1

these algebras then A is isometric to the direct sum of L A and an L space The

bidual A is isometrical ly ismorphic to the direct sum of H and an L space Final ly

H C K is a closed subalgebra of L and A C K is isometrical ly ismorphic

to the direct sum of H C K and an L space

Here H is the weakstar closure of A in L In the case of AK the measure

can b e taken to b e the area measure on the interior of K or the harmonic measure on the

b oundary of the interior of K In the case of RK can b e taken to b e the area measure

on the set of nonp eak p oints of RK which serves in some sense as an interior for K with

resp ect to RK The pro of of Theorem is similar to that of Theorem except that

Davies theorem is used to obtain the isometric isomorphism of H B and H The

pro of of the nal statement dep ends up on estimating solutions of the problem for some

sp ecic bump functions

Summarizing we can say that very many standard uniform algebras of analytic func

tions are tight We turn to an example of a tight subspace that is not an algebra and that

has a dierent avor Let U b e the space of continuous functions f e on the unit circle

f k e of the Fourier series of f converge for which the symmetric partial sums T f

uniformly Normed by jjjf jjj sup jjT f jj the space U b ecomes a Banach space We may

n C

regard U as a subspace of a C K space as follows For each n n let b e a copy

C n

of the unit circle and let K b e the disjoint union of the s with the natural top ology

determined by declaring that as n Each f U determines F C K by

n C

setting F T f on n and F f on Then U is isometric to a closed

n n C

subspace of C K

Theorem Let U be the Banach space of functions on with uniformly con

vergent Fourier series regarded as a closed subspace of C K as above Then U is a

tight subspace of C K though U is not compactly tight Further the weakstar closure

H U d of U in L d where d is the arc length measure on coincides with the

C C

space of functions f L d such that the symmetric partial sums of the Fourier series of

f are uniformly bounded The bidual U is isometric to the direct sum of H U d and

an L space

If g C K is supp orted on one of the circles for n nite then the op erator S

n g

is nite dimensional Thus to check that U is tight it suces to show that the op erators

S and S are weakly compact where z e on each circle Let f U have Fourier

z z n C

series a e and denote the corresp onding function in C K by f T f T f f

k 0 1

With this notation

i i(k +1) ik

z f z f a e a a e a e

0 1 k k 1

2 2

Since this expression is in C K for all twotailed sequences fa g and since ja j

k k

for f U the op erator S factors through Thus S is weakly compact as is S and U

C z z z C

is tight To see that S is not compact apply S to the sequence of exp onential functions

z z

fe g

There is another way to see that S is weakly compact Since only two Fourier

co ecients app ear in each comp onent ab ove we obtain

kS f k kz f z f k kf k

L (d 2 )

This estimate shows that S is an absolutely summing op erator in fact an integral op erator

Since b ounded subsets of L are weakly compact in L S is weakly compact

A similar theorem holds for the space U of analytic functions on the unit disk with

uniformly convergent Taylor series Again we may regard U as a subspace of C K as

ab ove and U is a tight subspace In this case the weakstar closure H U d coincides

A A

with the functions in f H d such that the partial sums of the p ower series of f

are uniformly b ounded The bidual U is isometrically isomorphic to the direct sum of

H U d and an L space The pro of see Sac dep ends on a generalization of the F

and M Riesz theorem due to Ob erlin Ob asserting that any measure on K orthogonal to

U is absolutely continuous with resp ect to d on

The Pelczynski and DunfordPettis Prop erties

As Banach spaces tight subspaces of C K share a numb er of prop erties of C K We

discuss the Pelczynski prop erty which is shared and also the DunfordPettis prop erty which

is partially shared

A Banach space X has the Pelczynski property if whenever T is an op erator from X

to another Banach space that is not weakly compact there is an emb edding c X such

that the restriction of T to c is an isomorphism The spaces C K have the Pelczynski

prop erty see Wo j I I IDx However L spaces do not have the Pelczynski prop erty

unless they are nitedimensional Reexive Banach spaces have the Pelczynski prop erty by

default

Theorem Saccone Any tight subspace of C K has the Pelczynski property

Before saying something ab out the pro of we discuss the Pelczynski prop erty in more

detail

P P

A series x in X is weakly unconditional ly convergent or a wuc series if x x

k k

converges unconditionally for all x X In this case x x converges absolutely for

each x X and the closed graph theorem shows that the op erator x fx x g is

continuous from X to In particular there is such that jx x j kx k for

all x X The preadjoint op erator T c X dened on the standard basis vectors

e of c by T e x is then seen to b e continuous and satisfy kT k Conversely

k 0 k k

any continuous op erator T c X determines a wuc series T e Thus wuc series

0 k

corresp ond to op erators from c into X

Let E b e a subset of X If E is weakly precompact in X and if x is a wuc series

in X with corresp onding op erator T then T E is weakly precompact in Consequently

T E e tends to as k that is

sup jx x j as k

x E

With a little more eort it can b e shown that if fails then there is an basic sequence

fx g in E that is a sequence that is equivalent to an basis These statements charac

terize weak compactness precisely when X has the Pelczynski prop erty We state this result

formally

Theorem The fol lowing statements are equivalent for a Banach space X

i X has the Pelczynski property

ii If E is a subset of X such that for any wuc series x in X we have x x as

k k

k uniformly for x E then E is weakly precompact

iii If E is a subset of X that is not weakly precompact then there is an basic sequence

in E

We refer to Wo j for the pro of A related result in this circle of ideas is that if X has

the Pelczynski prop erty then its dual space X is weakly sequentially complete

Now we return to Saccones theorem which is proved in Sac The crux of the

matter is to nd for a given E that is not weakly compact a wuc series in X for which

fails To do this Saccone b egins with a characterization of weak compactness due to

RCJames and eventually he throws the pro of back on some dicult work of Bourgain B

B for which there is a clear treatment in Wo j I I ID xx

Recall that an op erator T X Y is completely continuous if T maps weakly

convergent sequences in X to norm convergent sequences in Y If X is reexive then the

completely continuous op erators coincide with the compact op erators while every op erator

on X is weakly compact In contrast for X C K the completely continuous op erators

coincide with the weakly compact op erators

An isomorphism of c cannot b e completely continuous as the standard basis of

c converges weakly to Thus if X has the Pelczynski prop erty then any completely

continuous op erator from X to another Banach space is weakly compact As a corollary to

Saccones theorem we then obtain the following

Corollary If B is a tight subspace of C K then any completely continuous

operator T B Y is weakly compact

A Banach space X has the DunfordPettis property if every weakly compact op erator

from X to another Banach space is completely continuous This o ccurs if and only if whenever

the sequence fx g in X converges weakly to and the sequence fx g in X converges

weakly to then x x The spaces C K and any L space have the Dunford

Pettis prop erty If a dual Banach space has the DunfordPettis prop erty then its predual

do es also Reexive Banach spaces do not have the DunfordPettis prop erty unless they are

nitedimensional See Basic Concepts

Not every tight subspace of C K has the DunfordPettis prop erty In fact any

innitedimensional reexive subspace of C K is tight but fails to have the DunfordPettis

prop erty However the DunfordPettis prop erty does hold under hyp otheses that are some

what stronger than tightness The following statement can b e extracted from Bourgains

work in B

Theorem Bourgain Let A be a subspace of C K If S is completely

continuous for al l g C then A and A have the DunfordPettis property

The collection of g C such that S is completely continuous is a closed subalgebra

of C It is called the Bourgain algebra asso ciated with A

is completely continuous We wish to develop some criteria that guarantee that S

The following condition is a variant of the notion of a rich subspace which stems from Wo j

An op erator T from A to another Banach space is nearly dominated if there is a

probability measure on K such that if ff g is a b ounded sequence in A that converges to

in L then kT f k Trivially absolutely summing op erators are nearly dominated

If T A X is nearly dominated by and if T T in op erator norm then T is nearly

j j j

dominated by It is straightforward to show that the collection of g C such that

S is nearly dominated forms a closed subalgebra of C The p ointwise b ounded convergence

theorem can b e used to show that nearly dominated op erators are completely continuous

Theorem Let A be a subspace of C K Each of the fol lowing conditions

guarantees that S is completely continuous for al l g C hence that A and A have the

DunfordPettis property

i The subspace A is compactly tight

ii The operators S are absolutely summing for a family of functions g C that generates

C as a uniform algebra

iii The operators S are nearly dominated for a family of functions g C that generates C

as a uniform algebra

For i observe that if S is compact then S is compact hence completely continu

ous For ii we use the fact that if S is absolutely summing then S is absolutely summing

hence completely continuous The pro of under the condition iii is straightforward

Note that either of the conditions ii or iii covers the subspaces U and U discussed

A C

earlier Each of the three conditions covers the algebras RK and AK from rational

approximation theory

Absolutely Summing and Related Op erators on the

Disk Algebra

Now we consider some prop erties of the Banach space C of continuous functions on the

unit circle that are inherited by the disk algebra C A prototypical theorem along these

lines is the following see Pe

Theorem MityaginPelczynski For p every psumming operator

from the disk algebra C to a Banach space Y extends to a psumming operator from C

to Y hence is strictly pintegral

The Paley op erator shows that the statement fails at the endp oint p The pro of of

p p

the theorem dep ends on the b oundedness of the Riesz pro jection R from L d onto H d

p together with some Hardy space theory Indeed let T b e a psumming op erator

and let b e the measure on for T given by the Pietsch theorem see Basic Concepts

p p

so that T extends to a continuous op erator from the closure H of C in L to Y

Let w d b e the Leb esgue decomp osition of with resp ect to Leb esgue measure

p p p 1

d The Hardy space theory gives H H w d L Further if log w L d

p p 1

then H L and T is pintegral On the other hand if log w L d and

h explog w iHlog w is the outer function in H d such that jhj w then

1p 1p p p

Q f h Rh f pro jects L w d onto H w d and this pro jection allows us to

factor T

Q I

p p

T C L L H Y

again showing that T is strictly pintegral

Our primary fo cus will b e on Bourgains extension of the Grothendieck theorem to

the disk algebra with emphasis on the interp olatory nature of the pro ofs We will sketch

the pro ofs mo dulo the interp olation theorems which we defer to the next section

The Grothendieck theorem see Basic Concepts asserts that any op erator from an

1 2

L space to is absolutely summing A dual version of Grothendiecks theorem asserts

1 1

that any op erator from a C K space to is summing In fact we can replace in

this statement by any space of cotyp e In reading the following version of Grothendiecks

theorem recall that among the spaces L precisely those with p are of cotyp e

Theorem Every operator from C K to a space of cotype is summing

Theorem is a simple consequence of the following two lemmas

Lemma If Y is of cotype and if p then any psumming operator from a

Banach space X to Y is summing

Lemma For every Banach space Y and every nite rank operator T C K Y

we have T T kT k for p where p

p 2

If the lemmas are proved and Y is of cotyp e we combine the lemmas to obtain

T c T c T kT k for every T C K Y of nite rank whence T

2 4 2 2

c kT k The nite rank assumption is easily lifted by approximation

The rst lemma is proved by an easy concatenation of inequalities one of which is the

Khinchin inequality see Wo j I I IF x and also Basic Concepts It yields the estimate

T c C Y T where C Y is the cotyp e constant of Y and c dep ends only on

2 p q p q p

p To prove the second lemma we cho ose by the Pietsch theorem a probability measure

on K such that the op erator T acts from L to Y with norm T Also T acts from

C K to Y with the norm kT k and consequently T extends to L with at most the

same norm

(T )

T L Y

kT k

T L Y

p 1

By interp olation T acts from L to Y with norm not exceeding T kT k where

is given by the convexity condition p This proves the

lemma and with it Grothendiecks theorem

Now we turn to Bourgains version of the theorem for the disk algebra

Theorem Bourgain Every operator from the disk algebra C to a space of

cotype is summing

As previously the pro of is an easy consequence of Lemma and the following analog

of Lemma

Lemma For every Banach space Y and every nite rank operator T C Y

we have T c T kT k for p where p and c is a universal constant

p 2

For the pro of we start as in the pro of of Lemma with a probability measure

2 2

on the unit circle such that T acts from the closure H of C in L to Y with norm

T Our rst problem is that the measure need not b e absolutely continuous with

resp ect to arc length d For this we invoke the following

Absolute Continuity Principle In problems like this the singular parts of mea

sures can be disregarded

p p p

One way to justify this is to refer to the decomp osition H H w d L

used ab ove and to the Hardy space theory underlying this decomp osition However some

times other arguments are also applicable In the case under consideration we may work

with the op erators T f T K f in place of T where K is the nth Fejer kernel For them

n n n

the ab ove measure b ecomes absolutely continuous and moreover the T s may b e regarded

directly as op erators on H d

Thus we assume that w d where w is a weight w d We arrive at

the following analogs of and

(T )

T H w d Y

kT k

T H w d Y

The question now is whether we can interp olate b etween and as we did b etween

and The answer is that we can replace w by a weight v w v C such

that for this new weight the ab ove interp olation is p ossible This will follow from results

in the next section Indeed since L d is BMO regular see Prop osition there is

a ma jorant v for w such that log v b elongs to BMO and on account of Theorem the

desired interp olation holds for this ma jorant v This proves Bourgains theorem

In a standard way Bourgains theorem implies that every op erator from C or from

1 1 2

L H to l is absolutely summing Then from the relations

1 1 1 1 1 1

L H L H L H C C C

A A A c

0 0 0

1 1

see Wo j I I IE x it is also standard to conclude that L H and C are of cotyp e

0 A

See Wo j I I II x for more details

We mention another approach to the Grothendieck theorem due to Maurey This

metho d gives some information ab out op erators T C K Y where Y is a space of

arbitrary nite cotyp e In particular it applies to Y L for any p For the pro of

of the following theorem see DJT Chapter or K

Theorem Maurey For p q the class of q psumming operators

dened on C K does not depend on p and is contained in the class of q summing

operators for any

A related result due to Pisier see the ab ove references is that T C K Y is

q psumming if and only if T factors through the inclusion C K L the Lorentz

q 1

space for some probability measure on K

If Y is of cotyp e q the identity op erator of Y is q summing Thus Maureys

theorem shows that every op erator from C K to Y is q summing We recover the

Grothendieck theorem by setting q and applying Lemma

The following theorem allows us to transp ort these results to the disk algebra C

Theorem Kislyakov For an arbitrary Banach space Y and q p every

q psumming operator T C Y extends to a q psumming operator from C to Y

According to the MityaginPelczynski theorem the theorem remains true if q p

The remainder of this section is devoted to an outline of a pro of of Theorem with

some simplications compared to the exp osition in K

Lemma Under the conditions of Theorem the operator T is q summing

We break the pro of of Lemma into four steps First note that the family of q p

summing op erators grows as p decreases so we may assume that p We assume also

that T When convenient we use h i to denote the pairing b etween vectors and

q 1

functionals

Step There is a probability measure on such that

kT xk q jh ij for all x C satisfying jxj jj

To see this we use a trick invented by Pisier to prove his characterization of q psumming

op erators on C K mentioned ab ove Let

X X

C sup kT x k x x C jx tj

n j 1 n A j

(n) (n)

x C Clearly C T We cho ose and for every n nd x

A n q 1 n

P P

(n) (n) (n)

such that kT x k and jx tj C Then we cho ose Y such that

n n

j j j

j j

P P

(n) (n) (n)

hT x i and dene a functional on C by the formula and k k

n A

j j j

j j

(n) (n)

hT x i Then and k k We consider a weakstar

n n n n

j j

limit p oint of the sequence f g This is a functional on C We extend it to C with

n A

preservation of norm obtaining a measure on Since and kk lim

is a probability measure

(n)

Now let x C satisfy jxj jj We dene y y C by y x

A 1 n+1 A j

for j n and y x Then

n+1

X X

jy tj C C kT y k C sup

j n+1 n n j n+1

q q q q q

which implies that jh ij kT xk C C and in the limit jh ij kT xk

n n+1 n n

q q

Finally jh ij j h ij q jh ij from which follows

Step We apply the absolute continuity principle It may b e assumed that v d

v d and that is valid for x H d Again we may convolve with Fejer

kernels to ensure this

Step There is a v a C such that

18q

118q

kT xk C kxk kxk x H

L (a)

To see this we assume that kxk It suces to show that kT xk C kxk We

L (a)

shall deduce this from by a careful choice of Denote by H the harmonic conjugation

op erator As in the pro of of Prop osition there exists a v a C such that for

b a we have jHbj C b and jb iHbj C b We put log jxj so

0 0

that and also C jxj since jxj We dene successively

4 4

b iH b

expA

b iHb

where the constant A will b e chosen momentarily Since is the quotient of functions

with values in the right halfplane it omits the negative axis and we cho ose the branch of

whose argument ranges b etween Then Re jj and j j C jj Now

j j C so Re C We set A C and then jj expA Re

0 3 3

expAC jxj Thus we may apply with this and v d

Z Z Z

q 8

jja d C kT xk j jv d C jj a d

4 2

8 2 2 4 2 4 2 2

Now jj a j j b b H b Using the L estimate for H we obtain

Z Z Z Z Z

8 8 4 2 8 4 2 4 2

jxja d jxj a d C jj a d d b d C b H b

Taking th ro ots we obtain the required estimate for kT xk

Step The result of the preceding step shows that T acts from the interp olation

space H H a of the real metho d to Y with an appropriate norm estimate see a

18q 1

calculation in the pro of of Theorem b in BL If we were dealing with the L scale

the interp olation space would b e readily identiable as a Lorentz space L L a

18q 1

8q 1

L a Even in the case at hand on account of the ab ove choice of a we are able to

interp olate similarly in the scale of weighted Hardy spaces by Theorem This leads to

the estimate

8q 1 10q

kT xk C kxk C kxk x C

L (a) L (a)

Thus T is q summing and Lemma is established

Next we need to intro duce vector co ecients For a Banach space X we consider

the space C X of all X valued continuous functions f on the unit circle that extend

analytically to the unit disk that is that satisfy f z z d for n An op erator

T C X Y is said to b e q p X summing if

X X

1q 1p

kT x k C sup kx tk

j j

for any nite collection fx g in C X

j A

Lemma If T C X Y is q p X summing then there is a probability

10q

110q

measure on such that kT xk C kxtk dt

Indeed it is routine to carry through the ab ove pro of of Lemma For this note

that some functions will remain scalarvalued as for instance the function in The

condition on x and in b ecomes kxtk jtj for t

For any op erator T C Y we dene the op erator T fx g fT x g on sequences

A j j

p q

of functions in C Evidently T is q psumming if and only if T maps C to Y It

A A

is quite easy to see that even more is true

Lemma If T is q psumming then T is q p summing

Now we apply Lemma to T and pro ceed as in the pro of of the MityaginPelczynski

theorem As b efore we may assume that the measure is absolutely continuous and even

that ad with log a L d Then T has the factorization

p 10q p q

T C H a Y

where the rst mapping is the identity emb edding and the second is the extension of T by

continuity To complete the pro of we need a pro jection

Lemma If p s and log a L d then there is a projection Q from

s p s p

L a onto H a having the form Qff g fQf g for a projection operator Q acting

j j

on scalarvalued functions

We take Q to b e the pro jection Q in the pro of of the MityaginPelczynski theorem

Theorem The b oundedness of Q follows from standard techniques

It is now easy to establish Theorem in the case p If p in then T

p q

extends to some op erator U C Y of the form U fx g fS x g where S acts from

j j

C to Y The b oundedness of U means that S is q psumming Clearly S extends T

It remains to treat the case where p The facts already proved and Maureys

Theorem show that for r q the class of q r summing op erators from C to Y

do es not dep end on r It suces to extend this statement to r For this let T C Y

b e of nite rank and let r s q Then we have

(T )

q s

s q

T C Y

(T )

q 1

1 q

T C Y

By the remark after the pro of of Lemma where H spaces are involved but this do es

s 1

not matter to o much we can interp olate as if we had C and C This shows that

T C T T for some Since the norms and are

q r q s q 1 q r q s

equivalent we obtain the desired result

Interp olation of HardyTyp e Subspaces

Several times in Section we had to interp olate either b etween weighted Hardy spaces

p p r

H a d or b etween Hardy spaces of vectorvalued functions H To cover b oth cases

we consider the measure space m where dm d is normalized arclength

measure on the unit circle and is some xed nite measure space Since we wish

to use the full range p we will refer to quasiBanach spaces where appropriate

A lattice of measurable functions on m is a quasiBanach space X of measurable

functions such that if f X g is measurable and jg j jf j then g X and kg k C kf k

X X

Note that this is not the same as a Banach lattice as dened in Basic Concepts whose

elements are measurable functions Since we treat the term as an inseparable unit lattice

ofmeasurablefunctions there should b e no confusion The examples we have in mind are

p p r

the spaces L w dmd and the spaces L dm L of measurable functions xt such

r p

that y t jxt j d is in L dm

Let N b e the Smirnov class of analytic functions on the unit disk see Du Pr

which we identify with their b oundary value functions on the circle For our purp oses the

+ p

class N could b e replaced by H We call a function on the circle analytic if it b elongs

to N If X is a lattice of measurable functions on m we dene its analytic

subspace X to b e the set of functions f X such that f N for almost all In

the case of functions of one variable as when is a p oint mass we have L H

In the sequel we also imp ose on X the following conditions

i if f X then log jf t j dmt ae on

ii if f in X then log jf t j dmt in measure

n n

iii if f X there exists g X such that jf j jg j kg k C kf k

X X

and log jg j L m for aa

These conditions serve to exclude various degenerate p ossibilities Under these conditions it

is easy to prove for instance that X is closed in X

Now let X and Y b e lattices of measurable functions The fundamental problem we

consider is to determine when interp olation prop erties of the couple X Y are inherited by

the couple X Y We shall deal with real interp olation only

A A

We remind the reader of the denition of the real interp olation spaces X X

0 1 q

for a couple X X of compatible quasiBanach spaces By compatible we mean that X

0 1 0

and X are linear subspaces of some ambient space so we may dene the K functional

K x t X X for t and x X X by

0 1 0 1

K x t X X inf fkx k tkx k x x x x X x X g

0 1 0 0 1 1 0 1 0 0 1 1

For and q we dene the interp olation space X X to consist of

0 1 q

x X X such that t K x t X X b elongs to L dtt and we dene the norm of x in

0 1 0 1

X X to b e the norm of t K x t X X in L dtt Actually the sp ecic expressions

0 1 q 0 1

for the K functional and the norm will not play a role for us

Let Y X and Y X b e closed subspaces We say that the couple Y Y is

0 0 1 1 0 1

K closed in X X if there is C such that any decomp osition y x x of an element

0 1 0 1

y Y Y with x X can b e mo died to a decomp osition y y y with y Y and

0 1 i i 0 1 i i

ky k C kx k i In this case we have

i i i i

Y Y Y Y X X

0 1 q 0 1 0 1 q

with equivalence of norms and the interp olation prop erties of the couple X X and its

0 1

sub couple Y Y are identical Our basic problem can b e formulated as follows

0 1

Problem When is the couple X Y K closed in X Y

A A

We shall see that this happ ens fairly often

We b egin with a useful duality result Assume that X and X are Banach spaces

0 1

and and that X X is dense in b oth X and X Then in a natural way the spaces X

0 1 0 1

X are included in X X and consequently form a compatible couple If Y X as

0 1 i i

such that that is the set of L X the annihilator of Y in X ab ove we denote by Y

i i i

L on Y

Lemma The couple Y Y is K closed in X X if and only if the couple

0 1 0 1

Y Y is K closed in X X

0 1 0 1

The pro of is left to the reader see Pi K

If X is a Banach lattice of measurable functions on m it often happ ens

R R

that under the duality hf g i f g dmd X is also a lattice of measurable functions on

the same measure space We will assume that this is the case and further that b oth X and

X satisfy the conditions iiii ab ove Then one easily sees that as in the classical case of

the H spaces on the circle we have X Z X where Z is the co ordinate function on

Thus Lemma relates interp olation prop erties of the couples X Y and X Y

A A A A

X and Y b eing two lattices as ab ove

The class of BMO functions will play an imp ortant role in what follows Recall that

a function f on is in BMO if f u Hv where u v L As usual we disregard

the constant functions and dene kf k to b e the inmum of kuk kv k over all such

BMO

representations where k k is the norm in L mo dulo the constants

Lemma Let w be a measurable function on Then log w BMO if and

only if there exist constants C and a function f such that w C f C w

and jHf j C f Moreover C and are control led in terms of k log w k and vice

BMO

versa

Proof Supp ose log w BMO Then log w u Hv where is a real constant

u and v are real functions v and kuk kv k k log w k We pick so

BMO

small that kv k and we set F expiv iHv e cosv i sinv and

1 u 1

f e Re F w e cosv Since F is analytic in the unit disk and F is real

we have Im F HRe F Since j sin v j cos v we have jIm F j Re F and consequently

jHf j f The estimates cos v lead to w C f C w

Conversely given f as in the lemma we put G f iHf Since jHf j C f

the values of G lie in a sector in the right halfplane and the principal branch of log G is

analytic Writing log G log jGj i arg G we have j arg Gj tan C and Harg G

C f we see that log f log jGj log jGj so log jGj BMO Since f jGj

log jGj is b ounded and log f BMO Finally log f log w is b ounded so log w BMO

with BMO norm b ounded in terms of C and

From the pro of we see that we can always reduce at the exp ense of increasing C

and changing f

A weight is a function w on such that log w L dm for aa

p p

For p we denote by L w the usual space L w dmd with norm denoted by

kf k though we shall denote by L w the space of functions f on such that f w

is b ounded with the norm

kf k ess supfjf jw g

With this notation L w L w under the nonweighted duality hf g i f g dmd

w by H w p We denote L

For a weight w we say that log w is uniformly or C uniformly in BMO if the

function log w is in BMO for almost all with BMO norm b ounded by C In this case

the analog of Lemma holds where the function f can b e chosen to dep end measurably

on the parameter We will use this extended version of Lemma

A quasiBanach lattice of measurable functions X on is said to b e BMO regular

if for every x X there exists u X such that jxj u kuk C kxk and log u is C

X X

uniformly in BMO where C dep ends only on X The function u will b e referred to as a

BMO majorant of x

As an easy consequence of the extended version of Lemma we have the following

Lemma A lattice X is BMO regular if and only if there are C such that

for every x X there exists u X with jxj u kuk C kxk and jHu j C u

for aa At the expense of increasing C we can take to be arbitrarily smal l

For a quasiBanach lattice of measurable functions X and we dene X

to b e the space of functions f such that jf j X with quasinorm

kf k k jf j k

p p

Thus if X L then X L Clearly X is BMO regular if and only if X is Our main

examples of BMO regular spaces will b e based on the following prop osition

Prop osition If the operator H acting in the rst variable is bounded on X

then X is BMO regular

Proof It suces to show that Y X is BMO regular We verify the conditions

formulated in Lemma Taking y Y we put y jy j y jHy j for n and

0 n+1 n

v y where is a xed small constant Then jy j v kv k C ky k and

n Y Y

jHv j v

Lemma If log w is uniformly in BMO then L w is BMO regular for

Proof If p and w we may apply Prop osition with The

case p and w arbitrary then follows easily from the denitions For p the

w as supjxjw shows that kxk w is a BMO ma jorant of denition of the norm in L

x L w

p r

Lemma The space L dm L is BMOregular for p r

Proof The case where r is a consequence of Prop osition and the case where

r of its pro of Indeed given x we construct a BMO ma jorant for y ess sup jx j

in L dm and then treat this ma jorant as a function of two variables

It is easy to nd other examples of BMO regular spaces on the basis of the same

ideas A less trivial example is the space L dm s see K While we could

have cited this example and of course Theorem when interp olating in that pro of

s 1 s

can b e based also on the duality L dm L dm for s where the latter space

is BMO regular by Lemma Thus to interp olate in we can refer to Corollary

Now we state our main interp olation result We are assuming that X and Y are

Banach lattices of measurable functions on and when applicable that X and Y

are also lattices of measurable functions on all satisfying the conditions iiii ab ove

Also the density of X Y in X and Y is assumed when needed

Theorem If X and Y are BMOregular then X Y is K closed in X Y

A A

There is some evidence see Ka in favor of the conjecture that BMOregularity is

a selfdual prop erty However this has not yet b een veried in the general case Thus we

combine Lemma with Theorem to obtain more information

Corollary In any of the fol lowing three cases X Y is K closed in X Y

A A

and X Y is K closed in X Y

A A

a X and Y are BMOregular

b X and Y are BMOregular

c X and Y are BMOregular

Cases a and b of the corollary are direct consequences of Theorem and Lemma

Case c is not needed in Section so we leave it as an exercise

We pass to the pro of of Theorem Let w w b e two weights whose logarithms

0 1

are C uniformly in BMO

Lemma If f N for aa and f g h with g L w h L w

0 1

k k C kg k then f with H w H w and kk

w w 0 1 w

1 0 0

where C is determined by C C khk

Clearly Theorem is a consequence of this lemma Given f x y with f

X Y x X y Y we nd BMO ma jorants for x and y in their resp ective spaces and

A A

apply the lemma to these ma jorants as the weights

To prove Lemma we rst assume that f H w H w Then the

0 1

statement to b e proved is precisely the K closedness of the couple H w H w in

0 1

1 1

L w L w By duality Lemma it suces to check that the couple H w H w

0 1 0 1

1 1 1 1 1

is K closed in L w L w So let z a b H w H w where a L w

0 1 0 1 0

b L w We must replace a and b with functions roughly of the same size but analytic

in the rst variable

Let v b e a BMO ma jorant for b in the BMO regular space L w Setting w w v

1 0

we apply Lemma and the remark after its pro of to nd a function k and a constant

such that w C k C w and jHk j C k Fixing an integer n we dene

k iHk

1n n n

max jajv F G F

k iHk

We claim that z Gz Gz is the required decomp osition

Indeed the summands are analytic in the rst variable and it suces to estimate the

norms Since jHk j C k we see that jF j C C and jGj C C whence

1 1 2 2

jGz j C jaj C jbj C v C jbj By the choice of v we obtain the required inequality

3 2 3 2

kGz k C kbk

1w 4 1w

1 1

We have k Gbk C kak We estimate the quantity k Gz k

1w 2 1w 1w

0 0 0

jH k j j k iH k j

j F j

jk iHk j k

Since also jF j C jbj v and w C k we see that

Z Z Z

1 n

j F j w j F j w C j Gjjbjw C

6 0 5

Z Z

1 1

C w jH k j

Z Z

1 1

C w jH k j

Since H acts on L dmd and k C w it follows that the second integral in parentheses

is dominated by the rst Now if jj v and jajv jajv

otherwise Therefore

Z Z Z

w jajv w jajw

and we are done

It remains to get rid of the asumption f H w H w This is done by a

0 1

standard approximation argument based on the Hardy space theory Supp ose only f N

For u log jf j dene G expu iHu and F f G so that jF j

ae Thus f FG is the innerouter factorization of f Set u jf j j w and

j 0

G expu iHu Then jG j jGj and G G in measure Set

j j j j j

f FG H w then f f in measure and jf j jf j jg j jhj By the rst

j j 0 j j

part of the pro of f where H w H w k k C kg k

j j j j 0 j 1 j w w

0 0

and k k C khk For some subnet of the integers we have and

j w w j j

1 1

weakstar Simultaneous convex combinations of the s and s can b e chosen to converge

j j

ae to and and this guarantees that f with the appropriate estimates for jj

and j j

Bourgain Pro jections

Let w b e a weight on the unit circle We say that w admits an analytic projection if there

p p

is an op erator Q that pro jects L w onto H w for all p at once and together

with Q is of weak typ e relative to w

Z Z

c c

w fjQf j g jf jw w fjQ f j g jf jw f L w

where we denote w e w Here the adjoint Q is calculated relative to the duality

hf g i f g w dm

Op erators of this sort served as the main technical to ol in Bourgains work extending

Grothendiecks theorem to the disk algebra Bourgain B proved that for every integrable

weight u there exists a weight w admitting an analytic pro jection and satisfying w u and

R R

w C u where C is a universal constant The existence of such a pro jection do es not

imply the K closedness in the scale H w if the extreme exp onents p and p

are involved However it still implies certain nice interp olation prop erties of this scale

sucient for instance for proving Lemma

Though here we have used dierent simpler techniques Bourgains pro jections re

main interesting in themselves We shall show that a weight w admits an analytic pro jection

if and only if log w BMO Bourgains ma jorization result quoted in the preceding para

graph then follows from the fact that L d is BMO regular

We need a technical notion A twotailed sequence of functions H j

is called an analytic decomposition of unity subordinate to a weight w if there exists a

constant c such that

18 j

i j j w c j

18 j

ii j j cw

iii j j c

Roughly sp eaking the functions b ehave like the characteristic functions of the sets where

j 1 j

w The exp onent is convenient technically but in principle may b e replaced

by any see K

Theorem For a weight w the fol lowing conditions are equivalent

log w BMO

there is an analytic decomposition of unity subordinate to w

w admits an analytic projection

p p

there is an operator Q projecting L w onto H w for two dierent values of p

We fo cus on the implications The implication is trivial and

is proved in KX Corollary

To prove that we need a lemma

Lemma Suppose u is a weight such that log u BMO Then log u BMO

and the BMO norm of log u is control led by the BMO norm of log u

Proof By Lemma there exist C and f such that uC f C u

and jHf j C f We put g f then jHg j C g and u C g

C u Thus the BMO norm of log u is controlled by the BMO norm of log u and

consequently we have similar control over the BMO norm of log uu Finally note

that u uu u

Now to show let log w BMO For any we intro duce two weights

16 8

u w u w Then k log u k k log u k C C k log w k

0 1 0 BMO 1 BMO BMO

by Lemma Since u u by Lemma we nd g H u h H u such

0 1 0 1

that g h jg j C u khk C u where here and b elow all constants are determined

0 1

by k log w k

BMO

We do this for each n Z and denote the resulting functions by g and h

n n

Next we put g g h h then

n n n+1 n+1 n

n 16 n 1 8

j j c minf w w g

We claim that f g is the required analytic decomp osition of unity Indeed iv is clear

the convergence of the series ae easily follows from and the sum telescop es Again

18 n

by j j w C which is i We verify ii condition iii is proved similarly Let

k k +1

e f w g then again by

X X X X

n k 2n 2k n n 18

j j C

e e n

k k

k n k n nZ nZ

Changing the order of summation we see that the latter expression is dominated by

C w

Now we sketch the pro of of the implication Let f g b e an analytic decom

p osition of unity for w We write with inner and outer and put

j j j j

4 4

Qf Rf

j j

j Z

where R is the Riesz pro jection Then Q is the required op erator We only check the weak

typ e prop erty of Q the weak typ e for Q is similar and the L w b oundedness

of Q for p is simpler Clearly the values of Q are analytic functions and Q xes

analytic functions b ecause

Lemma An operator T acting from a subset of L to measurable functions

is of weak type if and only if it satises the estimate

12 12 12

jT f j jg jd C kf k kg k kg k

1 1

L ()

L () L ()

Proof hint To prove the if part take g where E fjT f j g is an arbitrary

set of nite measure

Now we check the estimate of Lemma for Q using the fact that it is true for R

Z Z

2 4 12 12

j j jRf j jg jw jQf j jg jw

X X

12 12

j 4 12 12 4 j

k kf k g k j j g j C kg k jRf C

1 j j 1

j j

L L

j j

X X

12 12

12 12 12 j j 12

1 1

kf k C kg k kg k kf k C kg k kg k

j j 1 1

L L

L (w ) L (w )

j j

See K for more information on Bourgain pro jections To illustrate their usefulness

we mention an application to conformal mapping

Let G b e a Jordan domain with rectiable b oundary The analogues for G of the

p p

classical H spaces are the spaces E G consisting of the analytic functions f on G for

jf z j jdz j is nite where f g is a xed sequence of rectiable curves that which sup

tend to G in a natural sense The domain G is a Smirnov domain if the derivative g of

the conformal mapping g of the unit disk onto G is outer In this case we may identify the

p p

scale E G with the scale H jg j of weighted Hardy spaces on the disk

For a Smirnov domain G with conformal map g it is imp ortant to know when log jg j

is in BMO See Po for a detailed discussion The following theorem is an immediate

consequence of Theorem

Theorem The fol lowing statements are equivalent for the conformal mapping g

of the unit disk onto a Smirnov domain G

log jg j BMO

p p

There is an operator projecting L G onto E G for al l p and having

weak type

p p

There is an operator projecting L G onto E G for two dierent values of p

Theorem together with a theorem of G David D yields a pro of of the known

fact that if the arclength measure on the b oundary of G satises a Carleson condition then

log jg j BMO The Carleson condition is that the length of G D is b ounded by cr for

any small disk D of radius r By Davids theorem this implies that the Cauchy integral

over G is a b ounded op erator on L G for p In particular condition ab ove

holds and from we obtain log jg j BMO Note that the Cauchy integral op erator need

not have weak typ e and the Bourgain pro jection in statement has the form

Finally we note that the theory discussed in Section can b e carried over nearly

word for word to the Hardy spaces related to a weakstar Dirichlet algebra as can the

implications in Theorem See SW for background on weakstar Dirichlet

algebras

Perturbation of Uniform Algebras

Often a Banach function space rememb ers almost nothing ab out the set on which the func

tions are dened For example if p is xed the spaces L for separable and atomless

are all isometric The reason is that up to isomorphism in a prop er sense there are no sep

arable atomless measures other than Leb esgue measure on Something similar o ccurs

in the context of the spaces C K The celebrated Milyutin theorem see Wo j I I ID x

asserts that for any uncountable compact metric space K C K is linearly homeomorphic

to C On the other hand if we do not change the norm of C K by to o much we

keep K in sight The AmirCamb ern theorem see Ja asserts that if T is a linear isomor

phism of two C K spaces such that kT k kT k then the underlying compact spaces

are homeomorphic

This leads us to consider linear isomorphisms of uniform algebras that are not to o

far from b eing isometries For an exp osition of this area see Ja The idea of nearness of

two uniform algebras can b e given several equivalent formulations but we fo cus only on the

Banach space notion of nearness to an isometry We say that two uniform algebras are

isomorphic if there is a linear isomorphism T b etween them that satises kT k kT k

that is if the BanachMazur distance b etween them is less than log Algebras that

are isomorphic in this sense are isometrically isomorphic as Banach spaces and for these

we have the following

Theorem Nagasawa Na If two uniform algebras A and B are isometrical ly

isomorphic as Banach spaces then they are isometrical ly isomorphic as uniform algebras

In particular their maximal ideal spaces are then homeomorphic as are their Shilov

b oundaries As a consequence if K and K are compact subsets of the complex plane such

1 2

that the algebras AK and AK are linearly isometrically isomorphic then there is a

1 2

homeomorphism of K onto K that maps the interior of K conformally onto the interior of

1 2 1

K On the other hand deformation of the compact set K leads to linear isomorphisms of

AK that are close to b eing isometries For example consider the scale of annuli G fr

jz j g and the asso ciated algebras AG Since these annuli are conformally distinct no

two of the algebras AG can b e isometric On the other hand the linear op erator T from

A to A dened by

r s

n=1

X X X

n n n

a T a z a z z

n n n

n= n= n=0

is a linear isomorphism of A onto A and further kT k kT k as s r

r s

We say that a uniform algebra A is stable if there is such that any uniform

algebra B that is isomorphic to A is actually isometrically isomorphic to A Thus

the algebras C K are stable while the annulus algebras are not R Ro chb erg Ro proved

in that the disk algebra C is stable and he went on to study the p erturbations of the

algebras AK for K a nitely connected subset of the complex plane with smo oth b oundary

or more generally for K a nite b ordered Riemann surface The avor of his work is given

by the following result

Theorem Ro chb erg Let K be a nite bordered Riemann surface Then for

smal l any uniform algebra B that is isomorphic to AK has the form AJ

for a compact bordered Riemann surface J that is a deformation of K by a quasiconformal

homeomorphism with dilatation tending to with

For exp ository accounts of these results see Ro Ro More recently Jarosz Ja

was able to prove that the nonseparable algebra H is also stable

Meanwhile there is currently no known compact set K in the complex plane with

nonempty interior such that C K can b e shown to b e linearly nonisomorphic to the disk

proved by algebra Conformal mapping theory and the relation C C C

A A A c

Wo jtaszczyk see Wo j I I IE x suggest that a compact set K for which C K or P K

or RK is prop er but is not isomorphic to the disk algebra should not b e to o simple if it

exists

The Dimension Conjecture

It is natural to ask how linear top ological prop erties of an algebra of analytic functions of

several complex variables reect the geometry of the underlying domain The oldest problem

along these lines is to determine what eect the numb er of variables dimension has

n m

Dimension Conjecture If G and G are bounded domains in C and C respec

1 2

tively with n m then the spaces AG and AG are not linearly homeomorphic nor

1 2

are H G and H G

1 2

We discuss briey some results related to the dimension conjecture The main refer

ences are B and Pe x

It is most natural to examine the dimension conjecture rst for the p olydisks and

n m

the balls B In B it was proved that A is not linearly homeomorphic to A

if m n The invariant distinguishing the spaces in fact their duals is the b ehavior of

certain vectorvalued multiindexed martingales on the measure space with

natural ltration the martingales in question must have some additional complex analytic

structure The metho d in B yields the following

Theorem Let U U V V be strictly pseudoconvex domains with

1 n 1 m

C smooth boundary the dimension may vary from one domain to another If m n then

AV V does not embed in AU U as a closed subspace

1 m 1 n

Theorem includes the previously known result that the spaces AB and A

are not linearly isomorphic for m n It had b een shown that A do es not emb ed in

a direct sum of an L space and a separable space see Pe x whereas AB is such

a direct sum by Theorem

Very little is known b eyond Theorem Currently it is not even known whether

the ball algebras AB are mutually nonisomorphic for m They are all distinct from

AB A C The latter space is a subspace of C with separable annihilator

1 A

whereas AB for m do es not emb ed in C K as a subspace with separable annihilator

see Pe x

n n

The series fH g seems to b e quite similar to fA g however in general the

metho d of B is not applicable to H It is known only that H diers from

H for n Again see B Pe for pro ofs

1 n

For comparison we describ e the situation concerning the Hardy spaces H and

H B The spaces of the rst series are not isomorphic to one another see B B

whereas those of the second series are all isomorphic see Wol More recently it was shown

that for any strictly pseudo convex domain with smo oth b oundary the corresp onding space

1 1

H is isomorphic to the classical H in the disk see Ar

Finally note that in contrast to the current state of aairs in one complex variable it

is p ossible to nd in several complex variables many examples of nonisomorphic spaces AG

where the underlying domains G have the same dimension For instance from Theorem

it follows that A is not linearly isomorphic to AB B

2 2

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