Finite dimensional subspaces of L

p

William B Johnson

Department of Mathematics

Texas AM University

Col lege Station Texas USA



Gideon Schechtman

Department of Mathematics

Weizmann Institute of Science

Rehovot Israel

We discuss the nite dimensional structure theory of L in particular

p

n

the theory of restricted invertibility and classication of subspaces of

p

Contents

Intro duction

The go o d the bad the natural and the complemented

The role of change of density

n

Subspaces of

p

n

Fine emb eddings of subspaces of L into

p

p

k n

Natural emb eddings of into

r p

k n

subspaces of mdimensional subspaces and quotients of

r p

Finite dimensional subspaces of L with sp ecial structure

p

Supp orted in part by NSF DMS and DMS Texas Advanced

Research Program and the USIsrael Binational Science Foundation

Supp orted in part by the USIsrael Binational Science Foundation

Preprint submitted to Elsevier Preprint January

Subspaces with symmetric basis

Subspaces with bad gl constant

Restricted invertibility and nite dimensional subspaces of L of

p

maximal distance to Euclidean spaces

n

Restricted invertibilityof op erators on

p

Subspaces of L with maximal distance to Hilb ert spaces

p

Complemented subspaces

n

Fine emb eddings of complemented subspaces of L in

p

p

Finite decomp osition and uniqueness of complements

References

Intro duction

The good the bad the natural and the complemented

innite dimensional subspaces of There are many interesting problems ab out

L L which have nite dimensional analogues For example it has

p p

long been a central problem in Banach space theory to classify the comple

mented subspaces of L up to isomorphism the nite dimensional analogue

p

is to nd for anygiven C a description of the nite dimensional spaces which

are C isomorphic to C complemented subspaces of L A lot is known ab out

p

b oth the innite dimensional see and nite dimensional see section

versions of this complemented subspaces of L problem but in neither case

p

do es a classication seem to b e close at hand

It sometimes happ ens that the nite dimensional version of an innite di

mensional problem leads to a theory which is much more interesting than

the innite dimensional theory Take for example the problem of describing

the subspaces of L which embed isomorphically into a smaller L space

p p

namely for which there is a fairly simple answer see Now it is clear

p

that a nite dimensional subspace X of L emb eds with isomorphism constant

p

N

if N N X is suciently large The attempt to estimate into

p

well N in terms of and X or the dimension of X has led to adeep theory

see sections and A sideline of this investigation also led to deep er

understanding of how certain natural subspaces of L such as the span of a

p

sequence of indep endent Gaussian random variables are situated in L see

p

section

Besides b eing an interesting sub ject in its own right the study of nite di

mensional subspaces of L is often needed in order to understand prop erties

p

of innite dimensional subspaces For example the easiest and best way to

obtain subspaces of L p which fail GLlust cf section

p

n

is to show that random large dimensional subspaces of are bad in a certain

p

sense see section

The topic discussed herein which has the most applications is that of restricted

invertibility see section Basically the theorem says that an n b y n matrix

n

which has ones on the diagonal and is of norm M say as op erator on must

p

be invertible on a co ordinate subspace of dimension at least M n One of the

many consequences of this result is that certain nite dimensional subspaces

n

X of L contain wellcomplemented subspaces with n prop ortional to the

p

p

dimension of X

The role of change of density

Generally the structure of the L spaces is describ ed for a xed value of p

p

However pro ofs of many of the results ab out L for a xed p use the entire

p

scale of L spaces p Consider for example the pro of section

r

that a subspace X of L p is either isomorphic to a Hilb ert space

p

or contains a subspace which is complemented in L and is isomorphic to

p p

There one needs only to compare the L norm and the L norm on X

p

In pro ofs of other theorems ab out L it is necessary to change the measure

p

b efore making a comparison between the L norm and another norm Since

p

the technique of changing the measure making a change of density is used

in the pro ofs of most of the results we discuss in this article wechose to devote

this section to describing the change of densities that arise later It turns out

that the framework in which this technique is most naturally used is that of

an L space when is a probabilityFor us there is no loss of generalityin

p

N

restricting to that case since the space is isometric to L when is any

p

p

probabilityon fNg for which fng for each n N For such

N N

a measure we denote L by L or just L if assigns mass N to

p

p p

each integer n n N

A density on a probability space is a strictly p ositive measurable

R

function g on for which g d Such a density g induces for xed

p an isometry M M from L onto L g d dened by

gp p p

p

Mf g f Sometimes a gain is achieved b y making such a change of

density that is by replacing L by its isometric copy L g d The gain

p p

usually occurs b ecause for some subspace E of L and some value of r

p

dierent from p the space M E is b etter situated with resp ect to L g d

gp r

than E is to L For example it follows from the Pietsch factorization

r

theorem section that if an op erator T from some space X into L

has psumming adjoint then by replacing the original L space by another

L space TX is actually contained in L Formally isometrically equivalent

p

Prop osition If T X L a probability measure has psumming

adjoint then there is a change of density g and an operator T X L g d

p

so that M T is the composition of T fol lowed by the canonical injection from

gp

L g d into L g d Moreover kT k T as long as TX has ful l sup

p p

port ie thereisnosubset with so that Tx Tx ae

for every x in X

To prove this factorization theorem assume for simplicitythat is a regular

Borel measure on a compact space and that TX has full supp ort Get a

Pietsch measure for the restriction of T to C K which means that is a

R

p p p

regular probability measure on K such that kT f k T jf j d for all

p

f in C K This same inequality is true if is replace by its RadonNikodym

derivative with resp ect to so one can assume that is of the form g d

R

with g and g d It is easily checked that this g is the desired

density provided that g is strictly p ositive ae which it must be since TX

has full supp ort When TX do es not have full supp ort reason the same way

but at the end add a small constant function to g and renormalize

The next change of density result due to D Lewis gives useful information

ab out nite dimensional subspaces of L

p

Theorem Let be a probability measure and let E be a k dimensional

subspace of L p with ful l support Then there is a density g

p

so that M E has a basis ff f g which is orthonormal in L g d and

gp k

k

P

such that jf j k

i

n

For a pro of when p see The rst step is to apply Lewis lemma

N

section to get an op erator T from onto E for which T

p

T N The rest of the pro of involves checking that the choices and I

p

p

P

p

g jTe j and f kg Te satisfy the conditions of Theorem

i i i

Another pro of for the entire range p is contained in

Recall that if T X Y is an op erator T is the inmum of kT kkU k over

all factorizations T SU with U X and S Y

Theorem If E is a k dimensional subspaceofL then thereisaprojec

p

pj j

tion P from L onto E with P k

p

k k jpj

Since the BanachMazur distance from to is k section

p

Theorem implies that the distance of a k dimensional subspace of L to

p

k k

is maximized when the subspace is

p

To prove Theorem observe rst that the case p follows trivially from

p

the fact proved in section that I k for every k dimensional

E

space E When p in view of the comments made in section there

is no loss in generality in assuming the is a probability measure and that

E has full supp ort Theorem says that we can further assume that E has a

k

P

jf j k basis ff f g which is orthonormal in L and such that

i k

n

Let P be the orthogonal pro jection onto E If p a simple computation

p

shows that kP L L k k and so also P L

p p

p

L k The case p is even easier

p

For a generalization of Theorem to spaces of typ e p see

The change of density in Theorem is used in section to showthatak di

n

mensional subspace of L well emb eds into with n not to o large There what

p

p

is needed is the relation between the L and the L g d norms on M E

p gp

The relevant estimate follows from a trivial observation which we record for

later reference

Lemma Let be a probability measure and let E be a k dimensional sub

space of L which has a basis ff f g which is orthonormal in L

p k

k

P

p

ach f in E kf k k kf k if jf j k Then for e and such that

p i

n

p and kf k k kf k if p

p

There is a prettycharacterization due to Maurey I I IH of subsets

of L which are b ounded in L after a change of density

p q

Theorem Let be a probability measure p q and S a subset

of L of ful l support Then there is a density g so that M S B if

p gp

L g d

q

and only if for al l nite subsets S of S and fa x S g

x

X X

q q q q

k ja xj k ja j

x x

L

p

xS xS

Assuming the Pietsch factorization theorem for the case p and when

T X L Corollary is little more than a S TB for some op erator

X

restatement of Theorem Recall Section that when T is an op erator

q

into a Banach lattice M T denotes the q convexity norm of T

Corollary Suppose T is an operator from a Banach space X into an L

p

space Then M T T

p

A consequence of Theorem that we shall need in section is the result of

that an op erator on L is b ounded on L after an appropriate change of

p

density

Theorem If p is a probability measure and T is an operator

on L then kM TM L g d L g dk K kT k for some

p gp G

gp

density g where K is the constant in Grothendiecks inequality

G

section

If one do es not wish to makeachange to the measure g dthenby throwing

away the part of the measure space where g one gets

Corollary If p is a probability measure and T is an operator

on L then kR TR L L k K kT k for some set A with

p A A G

A where R is the restriction operator dened by R f f

A A A

Theorem is a xed p oint version of the following factorization consequence

due to Maurey of Theorem

Theorem Let beprobability measure and T beanoperator from a Banach

lattice X into L p Then there is a change of density g and an

p

operator T from X into L g d so that M T I T where I is the iden

gp p p

tity mapping from L g d into L g d Moreover kT kK M X kT k

p G

if TX has ful l support

n

For the pro of of Theorem let fx g be in X and assume that TX has

i

i

of Grothendiecks inequality see full supp ort Then using a consequence

section in the rst step we have

P P

n n

k jTx j k K kT kk jx j k

i L G i

p

i i

P

n

K kT kM X kx k

G i

i

Now apply Theorem with S the image under T of the unit sphere of X to get

the inequality in the conclusion of Theorem A trivial p erturbation argument

now gives Theorem when TX do es not have full supp ort In the general case

given the density g can b e chosen so that kT kK M X kT k

G

We now deduce Theorem in the range p from Theorem By

duality we can assume that p and also supp ose that TL has

p

full supp ort If h is a density and we apply Theorem to the adjoint of the

op erator

M T

I

p

ph

L hd L hd L

p p

we get a density g so that for every f in L

p

Z Z

pp pp

jf j g d kT k jTfj h d K

G

Set g and get densities g g so that for each n is satised with

P

p n p

This series is absolutely g h g and g g Dene g

n n

n

n

convergent in L to a function whose norm is at most one so the

pp

function g kgk g is a densitywithg which satises for arbitrary

L



f in L

p

Z Z

pp pp

jTfj g d K kT k jf j g d

G

This gives Theorem with a slightly b etter constant when TL has full

p

supp ort and hence also Theorem as stated in the range p

For p Theorem follows via interp olation from Theorem

Theorem If p a is a probability measure and T is an

operator on L then kM TM L g d L g dk akT k for

p gp

gp

some density g

p

To prove Theorem it is enough to nd a strictly p ositive function g in

p p p p

L sothat T maps the order interval g g into akT kg akT kg

p

The main point is that every op erator T on L p has a mo d

p

jk kT k and jTfj jT jjf j for all f in L ulus jT j which satises kjT

p

p

see eg the remark preceding Theorem in One then denes g

P

n n n

a kT k jT j See for details when p

n

We conclude this section with a change of density lemma due to Pisier

which except for constants improves Theorem Theorem will b e used in

section

Theorem Let be aprobability measure p q andS a subset

of L The fol lowing statements are equivalent

p

i There is a constant C and a density g so that for al l measurable sets E

R

pq

and x in S k xk C g d

E L

p

E

ii There is a constant C and a density g so that M S C B

gp

L g d

q

iii There is a constant C so that for al l nite subsets S of S and subsets

P

q q

fa x S g of k sup ja j ja xjk C

x x x L

xS

p xS

Moreover thereisaconstant C C p q so that in the implication i j

C CC

j i

For a pro of of Theorem see This pap er also contains a nice pro of of

Theorem

n

Subspaces of

p

n

Fine embeddings of subspaces of L into

p

p

Let X be a k dimensional subspace of L p and let What is

p

n

the smallest n such that X emb eds in That is what is the smallest

p

n

n such that there is a k dimensional subspace Y of and an isomorphism

p

T X Y with kT kkT k Let us denote this n by N X andthe

p

maximal N X when X ranges over all k dimensional subspaces of L by

p p

N k

p

Fixing a basis in X and approximating each of its memb ers by an appropriate

simple function one sees that N X for every k dimensional X and

p

Moreover it dep ends on X only through its dimension k so that N k

p

However one gets that way a larger than exp onential in k b ound on N k

p

In this section we shall review results which give much b etter b ounds close

to the best p ossible ones

The case p is of course trivial and one can take n k even for The

case p has a nice geometrical interpretation The unit ball of the dual to a

n

k dimensional subspace of is easily seen to b e isometric to the Minkowski

k

sum of n segments in R centered at and visa versa Consequently the n

sought after is the smallest n such that every centered at zero body K in

k

R which is the Minkowski sum of arbitrarily many segments or the limit of

such b o dies these are called zonoids can be approximated by a body Z

which is the sum of n such segments in the sense that Z K Z

The history of this problem can be traced to the osprings of Dvoretzkys

theorem as discussed in There the case of X being a k dimensional Hilb ert

space which embeds isometrically in all the L spaces is treated and solved

p

k

quite satisfactory For some absolute constant C N is at most C k

p

p

for p and C pk for p This is b est p ossible except that it is

unknown whether the factor can be replaced by a smaller function of

Notice the following nonintuitive sp ecial case The k dimensional Euclidean

ball can b e approximated by abodywhichthe sum of a constant dep ending

k

on the degree of approximation times k segments in R

The rst result in this direction which did not involve Euclidean spaces was

proved in There it was shown that for p and p q

k

which is known to embed isometrically into L whenever p q

p

q

n

emb eds into for some n C p q k Later the second named author

p

found some initial results indicating in particular that the dep endence of n on

k in the general problem stated at the b eginning of this section is p olynomial

rather than exp onential as one is rst tempted to b elieve The rst pro ofs

were quite complicated and worked only for p as they used ne prop erties

of q stable random variables Later a much simpler metho d was intro duced in

N

Assuming as we may that X is already a subspace of for some nite

p

N pick randomly a few co ordinates and hop e that the natural pro jection

onto these co ordinates restricted to X is a go o d isomorphism If we do it

with no additional preparation this cannot work Indeed X may contain a

N

vector with small supp ort say one of the unit vector basis elements of

p

in which case the chance that a co ordinate in its supp ort is picked is small

of course if no such co ordinate is picked the said pro jection cannot be an

isomorphism on X The p oint is that one wants to change X rst to another

isometric copy of X in which each elementofX is spread out This can b e

done by achange of density The metho d of was used with other to ols in

and some other pap ers to pro duce the b est known results In these

results it is not known what is the right dep endence of n on and we shall

not try to emphasize what are the exact estimates one gets from the pro ofs

However the dep endence of n on k is b est p ossible except for log factors in

some places we shall pay more attentiontothis in the sequel

We now state the b est known results

p

Theorem i For p N k C p k log k

p

ii For p N k C k log k log log k

p

iii For p N k C k log k

iv For p N k C p k log k log log k

p

m

Under some conditions ensuring that X do es not contain go o d copies of

p

spaces one gets better results for p Recall that a quasinormed space X

is of typ e p with constant C for some p provided

p

n n

X X

p

kx k C x k E k

i i i

i i

for all nite sequences x x of elements of X The b est C is denoted

n

T X The space L p is of typ e p Recall also that K X denotes

p p

the K convexity constantofX ie the norm of the Rademacher pro jection in

L X See for a brief discussion of these notions although it is restricted

to the normed spaces which always have typ e p and for a more

comprehensive discussion

Theorem

i Let p q and let C Then for some constant C

C p q C and al l k dimensional subspace X of L with T X C

p q

N X C k

p

For p we have a quantitatively better estimate

ii For al l k dimensional subspaces X of L N X C K X k

Theorem i was proved for p in contains the full statement

with a dierent pro of also contains Theorem i and somewhat weaker

versions of Theorem ii iii and Theorem ii The exact Theorem

ii is contained in while Theorem ii is the main result of

p

Theorem iii follows from it since it is known that K X C log k

for every k dimensional subspace X of L see Lemma Finally Theorem

iv was proved only recently after noticing its omission while

writing this survey

Before describing the pro ofs we mention that there are several unsettled prob

lem related to Theorems and The most imp ortant one or at least the

one that attracted the most attention is whether the various log factors and

the dep endence on the typ e and K convexity are really needed It is strange

m

that the constants in the pro ofs blowupwhen X contains spaces Actually

p

as we shall see b elow in at least some of the pro ofs the worst case o ccurs when

k

Another problem is the determination of the dep endence X is isometrically

p

of N onScant attention has b een given to that in the published work A

problem we nd particularly interesting is whether there is an isomorphic

as opp ose to almost isometric version to some of the results here Here is

an instance of this problem is it true that for all p and all there

k

is a p ositive constant C C p such that whenever n k C embeds

p

n

into Some progress on this problem has recently b een achieved in

Next wewould liketosketch some of the ideas involv ed in the pro ofs of

some of the statements of Theorems and Aswe already indicated

ab ove a common feature of all the pro ofs we shall sketch is that using a

change of density we rst nd an isometric copy of X with some additional

go o d prop erties We delay stating theses prop erties and the actual change of

density that ensure them until later and denote the new space by the same

notation X We may also assume without loss of generality that X lies in a

m m

space say L where is a probability nite but large dimensional

p p

measure on fmg We denote fig We would like to show that

i

the restriction op erator to a set of relatively few of the m co ordinates is a

go o d isomorphism on X We prefer to do it iteratively by rst showing how

to nd a subset of cardinalityatmost msuch that the restriction op erator

is a very go o d isomorphism on X provided m is much larger than k We

shall then showhow to iterate this pro cedure The choice of the subset will b e

random for example it is enough to show that for an appropriate k m

m

X X

p p

Ave sup jx j jx j

A i i i i

xXkxk

i

iA

m

X

p

jx j k m E sup

i i i

xXkxk

i

Here Ave denotes the average over all subsets of fmg while E is the

A

n

exp ectation with resp ect to the natural pro duct measure on f g Indeed

if this is the case then for some A the restriction op erators to b oth A and the

p

km

complementof A are isomorphisms and of course either A or its

km

c

complement A is of cardinality at most m

Alternatively in order to nd such an A it is enough to show that

m

X

p

jx j k m P sup

i i i

xXkxk

i

In b oth cases after iterating we get that as long as

p

l

i

Y

k m

i

k m

i

n l

X must embed into for some n m

p

So this approach reduces the problem to nding go o d b ounds on the quantity

in or For technical reasons involving splitting of atoms as explained in

Lemma b elow we may need to enlarge m to at most mbefore making

the random choice in each step This do es not eect the pro cess signicantly

m

after the random choice we end up in and this just means that instead

p

of we shall need to ensure that

p

l

i

Y

k m

i

k m

i

Before pro ceeding further we would like to point out why there is very little

hop e of eliminating the log factors altogether using this approach Consider in

P

cn log n

e with i clog n n vectors of the form x c log n

j i i

j

i

disjoint sets of cardinality c log n assuming it is an integer It is easy to

calculate and app ears in many probability b o oks sometimes as the coup on

c

collectors problem that if c is small a random choice of n log n of the

co ordinates will most likely miss at least one of the s Thus the restriction

i

c

n log n will not be an isomorphism on to a random subset of cardinality

n

X span fx g which incidentally is isometric to We did not change the

i

density here but it is not hard to see that a change of density is not going to

help to reduce the minimal m for which this pro cedure works below cn log n

for some absolute c

Wenow continue sketching the idea of the pro ofs We rst sketch a version

of the simple argument of which only gives

N k C k for p and

p

p

N k C k for p

p

m

By Lemma there isachange of density on L so that for all x X

p

p

kxk k kxk for p and

p

kxk k kxk for p

p

Splitting the atoms of this change of density we may assume in addition that

m paying by enlarging the original m to m This follows from the

i

following simple lemma

Lemma Let beaprobability measureonf mg Then thereisan

m M m a probability measure on f Mg and a partition

f g of f Mg satisfying

m

i fig m i M and

P

ii figfj g

i

j

whichhave mass The pro of of the lemma is very simple Split the atoms of

larger than m into pieces each of size larger than m This do es not add

more than m atoms to the original ones thus ending the pro of Of course

L and thus also X naturally embeds into L in particular the estimates

p p

in still hold for the image of X in L This justies the statement in the

p

paragraph preceding the statement of the lemma

Fixing an x X of norm one we get by classical deviation inequalities see

for example Prop osition iii in that for p

P P

m m

p p

P j jx j j t K exp t jx j

i i i i i

i i

p

K exp t max jx j

im i i

K exp t mk

Let t and pick a tnet N in the unit sphere of X which has at most

k

elements see eg p for an easy pro of of the existence of such a t

net Then as long as k ct log tm for some absolute constant c

m

X

p

P sup jx j t K exp t mk

i i i

xN

i

from which it is easy to get that as long as k cm

m

X

p

jx j Ckm P sup

i i i

xXkxk

i

This and implies the desired result for p The treatment for

p is similar

This sketch of the argument of was given only to illustrate the basic

metho d involved We now continue to sketch the arguments which give

the stronger statements of theorems and The point is to nd

estimates for the quantities in or which are better than We rst

relate the second quantity in to a similar one involving Gaussian variables

Let g g be indep endent standard Gaussian variables Then

m

r

m m

X X

p p

E sup E sup g jx j jx j

i i i i i i

xXkxk xXkxk

i i

This version of the contraction principle is easy to prove Replace each of

the g on the right by jg j where f g is indep endent of fg g and replace E

i i i i i

with the successive application of the two exp ectations E E Now push the

g

q

exp ectation E inside the outer jj and use the fact that E jg j

g

The problem now reduces to evaluating the quantity

m

X

p

E sup g jx j

i i i

xXkxk

i

P

m

p

Set G g jx j Then fG g is a Gaussian pro cess indexed

x i i i x

xXkxk

i

by elements of BX and we are required to estimate the exp ectation of its

supremum This is a well studied area in probability theory see eg

related to the continuity of Gaussian pro cesses This quantity or the similar

one involving the Rademacher functions in is evaluated by dierentmeans

in the pro ofs of the dierent parts of Theorems and We shall rst present

a

Sketc h of pro of of Theorem ii Consider the pro cess fH g

x xXkxk

P

m

where H g x Then

x i i i

i

E G E H and E G G E H H

x y x y

x x

for all x y X of norm at most Slepians lemma see eg p implies

now that

m m

X X

E sup g jx j E sup g x

i i i i i i

xXkxk xXkxk

i i

We shall show that after a change of density and p ossibly enlarging m to

m

m

X

k

E sup g x CKX

i i i

m

xXkxk

i

Then one concludes the pro of of Theorem ii by applying To prove

we shall use the following two prop ositions

m

Prop osition Let X bea k dimensional subspace of L with a prob

ability measure and let f f be an orthonormal basis of X consideredas

k

m

a subspaceofL Then one can split some of the atoms of to a total of

at most M m atoms getting a new pr obability measure and a natural

m M

embedding I L L satisfying

k M

X X

E sup g f k g y m E k

i i i i i

y Yky k

i i

Here Y IX and kk is the dual norm to that of X where duality is given

P P P

by h a f b f i a b

i i i i i i

Prop osition With the notation of the previous proposition

k k

p

X X

f K X E g f

i i

i

i i

We now conclude the pro of of ii by applying Lewis change of density

Theorem to get an orthonormal basis for an isometric copyofX satisfying

P

k

n f

i i

Before sketching the pro ofs of Prop ositions and wewould like to deduce

Theorem iii This follows from the following Lemma rst observed in

p

Lemma Let X bea k dimensional subspaceofL ThenK X C log k

for some absolute C

PROOF Using Lewis change of density we may assume kxk k kxk for

pp

all x X Then easily kxk kxk k kxk for all x X Letting X

p p

denote X with the L norm we get that

p

q

pp pp pp

K X k K X k K L Ck pp

p p

where the last inequality follows from the easy fact that K L K L

p pp

and a not entirely obvious application of Khinchines inequality with the

b est order of the constant in L Picking p with pp log k we get

pp

the result

We now turn to the

Sketch of pro of of Prop osition Using the contraction principle in the

rst inequalitywe get

m m

X X

g x g x max E sup E sup

i i i i i

i i

im

xXkxk xXkxk

i i

m

E D

X

max E sup g e x

i i

i i

im

xXkxk

i

m k

E D

X X

max E sup g e f a

i i j j

i i

P

im

k a f k

i j

j j

E D

P

m

Then h h are standard Gaussian vari Put h g e f

k j i i j

i

i

ables which are easily seen to b e indep endent check that E h h Thus

j l jl

m k k

E D

X X X

E sup g e f E sup a a g

i i j j j j

i

P P

k a f k a f k k

i j j

j j j j

k

X

E g f

i i

i

It remains to see that splitting the atoms wemayalsoassumethat m

i

but this follows from the splitting of atoms lemma

Pro of of Prop osition The prop osition follows easily from Lemma

nk

but we present a dierent pro of Let f g denote nk indep endent

ij

ij

Rademacher functions ie the are the co ordinate functions in the pro duct

ij

nk

probability space f g In the rst inequality below we use the central

limit theorem and in the last Khintchines inequality with the b est constant

k n k

X X X

E g f f lim E n

i i ij j

n

i i j

k n

nD E o

X X

lim n sup f f kf k

ij j L X



n

j i

k n

n o

X X X

h f x i x X E x K X lim n sup

ij j ij ij ij ij

n

j i ij

k n

o n

X X X X

n f E x x K X lim sup

ij ij

j ij

n

L L





j i ij ij

k

o n

X X X

E x K X f sup x

ij ij

ij j

L L





ij j ij

k

p

X

f K X

j

L



j

We now sketch a pro of of Theorem i We have chosen a version of the

pro of of since it is the shortest one However this pro of do es not give a

go o d dep endence on

Sketch of pro of of Theorem i We assume rst as we may that X

m

The condition of Theorem iii is easily seen to b e satised for S any

p

nite subset of the unit ball of X and C T X So we can deduce from

q

m

that theorem that without loss of generality X L for some probability

p

measure and kxk C kxk for all x X and some C which dep ends

q p

X By Theorem i it is also easy to see that we may only on p q and T

q

g

assume that m replace g with Splitting the large atoms of

i

as in Lemma and changing m to m we may assume in addition that

m for all i m

i

Put r qp and note that for x X with kxk

p

m

p r p

kf jx j g k max j jx j

i i r j j

i

j m

p r

max tfi jx j tg

i i

t

p p

with f jx j g denoting the decreasing rearrangement of f jx j g

j j j j

Using the fact that is of order m and relating the quantity in to

i

q

kxk max tfi jx jtg

q i

t

q

max tm fi jx jtg

i

t

p

q

we get that the quantity in is at most Cm for some C dep ending only

on p q and T X We now use the inequality

q

n

X

s

a P t exp tkfa gk

i i i r

i

which holds for all t r and all sequences of scalars fa g Here

i

r

and is a p ositive constant dep ending only on r The inequality is s

r

a sp ecial case of a martingale inequality of Pisier see p for a pro of

or Prop osition for a discussion of this and other similar inequalities

Note that wemay assume that r qp Using we get from that

for all x X with kxk

p

m

X

p s

jx j t exp t m P

i i i

i

for some dep ending only on p q and T X From this we get as in the

q

standard argument leading to

m

X

p s

P sup jx j t exp k logt t m

i i i

xXkxk

i

s

It follows that as long as t m log t k We can nd a set of at most

m co ordinates for which the restriction op erator is an tisomorphism

m k

s

log we get that there is a set of at most m co ordi Cho osing t

m k

m k

s p

log isomor nates for which the restriction op erator is an C

m k

phism where C dep ends only on p q and T X Iterating we get that as long

q

as

l

i

Y

s p

k m

C log

i

m k

i

n l

X must embed into for some n m Note that in eachstep

p

of the iteration we get a space which is at most isomorphic to X Since the

new space has typ e q constant at most twice that of the original space wecan

continue the iteration

Now it is easy to get the conclusion

k n

Natural embeddings of into

r p

The metho ds describ ed in Section as well as all other metho ds for pro

ducing tightemb eddings are probabilistic and as such are not constructive

and do not pro duce an explicit go o d embedding The most basic question

concerning explicit emb eddings may b e to pro duce a sp ecic go o d emb edding

n m

of into with m prop ortional to n m n say We remark in passing

that for p an even integer there are sp ecic emb eddings even isometric ones

m n

with the relation b etween m and n close to the optimal one in into of

p

particular for p m n See for that

n m

A natural approach to get an explicit emb edding of into is to x a

n

natural subspace X of L whichiswell isomorphic to for example the span

n

of n indep endent standard Gaussian variables or n indep endent Rademacher

functions and nd a subspace Y with X Y L Y well isomorphic

m n m m

m

to and m small However this fails in a very strong sense under the

n

requirements ab ove the smallest m can be is C for some C dep ending

m

only on the distance of Y to Here is a somewhat stronger theorem from

m

Recall that for an op erator T X Y T inf kukkv k where the inf

is taken over all L spaces and all op erators v X L u L Y

satisfying T uv

Theorem For every K there exists a K such that if

X is the span in L of n independent Gaussian variables or n independent

Rademacher functions X Y L and the inclusion J X Y satises

n m

J K then for some m e is isomorphic to asubspaceofY In

n

particular dim Y e

The pro of of this is rather technical and we shall not repro duce it here

and contain many renements and variations of this theorem Also p

contains an exp osition of the pro of of the simplest instance of this class

of results namely the statement in the in particular part of Theorem

for X b eing the span of n indep endent Rademacher functions

k n

subspaces of mdimensional subsp aces and quotients of

r p

k

This short section deals with the question of what is the largest k suchthat

p

n

aswell as some related well emb eds into any mdimensional subspace X of

p

questions We shall not presentany pro ofs but only summarize what is known

on this sub ject

m n

Note that since X well embeds into p for m prop ortional to

p

n the answer to the question ab ove for p is not very interesting ie k

must be b ounded unless n m on We shall sa y something ab out this

case latter For p the following theorem of Bourgain and Tzafriri

basically solves the problem Let k k X K b e the maximal dimension

p

k

of a subspace Y of X which is K isomorphic to

p

Theorem Let p and Then there are positive constants

c cp C C p such that for al l m n and every mdimensional

n

subspace X of

p

p p pp

X c minfm mn g k

p

The result is best possible in the sense that for each m n there exists a

p p pp

subspace X with k X C minfm mn gp pp

p

As can be susp ected from the statement the pro of of this result is quite in

volved and very technical It uses ideas from the work of Bourgain and Tzafriri

concerning restricted invertibility some of which is surveyed Section be

low as well as from Bourgains work on sets We think it worthwhile

p

to nd a simpler pro of

As we said ab ove there can b e no similar theorem in the range p However

n

one can prove a similar theorem for quotients of p This was done

p

by Bourgain Kalton and Tzafriri in

Theorem For each p there is a constant C such that if X

p

n

is an mdimensional quotient space of then

p

p p p

k X C C m n

p p

p

for p while

k X C C m log nm

Except for the constants involved the results are best possible

Note that for m prop ortional to n the resulting dimension of the contained

k

space is also prop ortional to n For p this case was observed earlier in

p

Note also that the conclusion of this theorem is isomorphic rather than

almost isometric We do not know if one can replace the constant C in the

p

left hand side of the inequalities by of course paying by replacing the

constant in the right hand side by one dep ending on

There is also a version of Theorem for p If m n with then

n k

every mdimensional subspace X of contains a well isomorphic copyof

with k c m This was proved in for m prop ortional to n and in

in general

When m is very large there is also a version of Theorem for p It was

n

proved in that for every mdimensional subspace X of

m lognn mng k X K c minfnn

K and c are universal constants

Finite dimensional subspaces of L with sp ecial structure

p

Subspaces with symmetric basis

In this section we treat the classication of the nite symmetric basic sequences

in L p and to some extent also the classication of the nite

p

unconditional basic sequences in L p

p

Recall that a sequence x x in a quasi normed space X over R is said

n

to be K symmetric if for all scalars fa g all sequences of signs f g and all

i i

p ermutations of fng

n n

X X

k a x k K k a x k

i i i i i

i i

If we require the inequalityonly for the identity p ermutation the sequence is

called K unconditional

treats the classication of symmetric basic sequences in L The article

p

p so we only state the result from see Theorem

Theorem For every p and every constant K there is a constant

D such that any normalized K symmetric basic sequencein L is D equivalent

p

n

to the unit vector basis of R with the norm

X X

p p

kfa gk maxf ja j w ja j g

i i i

for some w

n

Of course since isometrically embeds in L any norm of the form

p

embeds with constant into L

p

For p the structure of the symmetric sequences in L is more involved

p

Let M be a Orlicz function see Section and the asso ciated Orlicz

M

sequence space It turns out that the space embeds isomorphically into

M

L if and only if the unit vector basis of is pconvex and concave and

p M

p

this happ ens if and only if M jtj is equivalent to a convex function and

M t is equivalent to a concave function on see Recall that two

functions M M R are equivalent at if there exist constants

K K and x such that for all jxj x K M x M x

K M x

With the rightquantiers a similar statement holds also for nite dimensional

n

Orlicz spaces The emb edding when it exists is as a span of indep endent

M

m

identically distributed symmetric random variables It follows that if fM g

j

j

n

K embed in L for all j and is a collection of Orlicz functions such that

p

M

j

P

p

m

n p

for all j then also R with the norm kk K embeds kk

j j

j

M

j

in L The converse is also true

p

Theorem For every constant K and for every p there is a con

n

stant D such that given any normalized K symmetric basic sequence ff g

i

i

in L there is an m and there are m symmetric functions M R

p j

p

j m for which M M jtj are convex and M t are

j j j

n

concave on and for some weights ff g is D equivalent to the unit

j i

i

n

vector basis of R with the norm

m

X

p

p

kk kk

j

M

j

j

This theorem is a consequence of the following very nice inequalityofKwapien

and Schutt

 

n n n

Theorem Let fa g R and denote by fa g the decreasing rear

ij

ij i i

Then for any p rangement of fa g

ij

p p

P P P



p

n n n

p

Ave ja j a a

i i

i

i i i in

n n

p

P

n

p

Ave ja j

i i

i

Here Ave denotes the average over al l permutations of n

The case p of Theorem app ears in The pro of for the other values

of p is quite similar and we shall sketch it b elow The starting p oint is a lemma

which gives another equivalent expression to the ones in For p

put

p

p p

n jtj if jtjn p

n

M tM t

p

p p

p pjtj if jtj n

n n

p

Note that M is an Orlicz function and that M t is a concave function on

p p



Lemma Let p and let a a a Then

n



n n

X X

p

p

 

n n

fa g a a fa g

i i i i

i i

M M

n n n n

i in



n

PROOF Note that M pt pM tforallt Consequentlyifkfa g k

i

i

M

then

p

X X

p p

p

p

n a pa p

i

i

n n p

a n a n

i i

P P

p

It follows that a pa so that

i i

a n a n

i i

n

n

X

a and fi a ngn

i i

i

p

P P



p

n n

a It now follows from that a n n

i

i

in i

p

P P



p

n n

a To prove the other side inequality assume a n n

i

i

in i

Then



n

X X X

p p p

p p

a a n a n

i i i

in

a nin n a

i i

P P P

n n

and since a a a a

n i i i

a n

i i

i

n n

It follows that for n

P P P



p

p p

n

p

a pa n M a p

i i

a n a n

i

i

i i

n n

p

p p

n

Since M t M t this concludes the pro of

We now turn to a

n

sketch of the pro of of Theorem Let ff g be a symmetric basic

i

i

m

Then up to a universal constant sequence in l

p

p

P P P

n m n

p

k a f k Ave a f k

i i i i

i k i

p

P P

p

m n

p

Ave a f k

i k

i

k k i

p p

n

where kff k g k

i

k i

M

p

By Theorem and Lemma the last expression is equivalent with constants

dep ending on p onlyto

m

X

p

p p p

p n

jf k j g k kfa

j

i

k ij k

M

p

n

k

Put for k m

n

X

p

p

N t M tjf k j

k p i

k

i

p

It is easy to check that the N are Orlicz functions with N t concave and

k k

that

p p p p

p n n n

kfa jf k j g k kfa g k kfa g k

j i

i i

k ij i i

M N

M

k

p

k

p

Where M tN jtj From this and it is easy to conclude the pro of

k k

The main result of states that every unconditional basic sequence in L

p

p is equivalent to a blo ck basis of a symmetric basic sequence in L

p

The blo ck basis can b e chosen to b e with equal co ecients and consequently

the emb edded space is also complemented This will not be used here Of

course if the unconditional basic sequence is nite also the containing sym

metric sequence can be taken nite and the constants of the symmetricity

and of the equivalence can be controlled by the unconditional constant Re

n

call that given a sequence of n Orlicz functions M fM g the mo dular

i

i

P

n

n

space is R with the norm kxk infft M x t gUsing

i i

i

M M

and Theorem one can now easily prove the following theorem

Theorem For every constant K and for every p there is a

constant D such that given any normalized K unconditional basic sequence

n n

ff g in L there is an m m Orlicz function sequences M fM g

i p j ji

i i

p

j mandpositive constants and such that M jtj areconvex

j ji ji

n

and M t are concave and ff g is D equivalent to the unit vector basis

ji i

i

n

of R with the norm

m

X

p

p

n

kfx g kxk k

j i ji

i

M

j

j

p

Of course if M t are K equivalent to convex functions and M t are

ji ji

K equivalent to concave functions then any norm as in emb eds into L

p

with constant dep ending on K only

Subspaces with bad gl constant

Recall rst that the GordonLewis constant glX of a Banach space X is

dened to b e

glX supf T T X T g

where denotes the summing norm see section and T X

Y inf fkAkkB kg Here the inf is taken over all L spaces L and over all

decomp ositions T AB with B X L A L Y

An easy but very useful theorem of Gordon and Lewis or p

says that the unconditional constant of every basis of X is at least glX

n

When p there is an abundance of bad subspaces of By bad

p

we mean here lacking go o d unconditional bases or even the weaker prop erty

n

of small GordonLewis constant Recall that L is L over the measure space

p

p

consisting of n points and endowed with the uniform probability measure

Theorem Therearepositive constants c c such that if X is any subspace

n

n n

of L p satisfying dimX cn and kxk kxk for al l x X

L L

p

 

p

then glX c n

Theorem was proved by Figiel and Johnson in A somewhat weaker the

orem still ensuring the abundance of subspaces with large glX was proved

earlier by Figiel Kwapien and Pelczynski We refer to p for the

pro of of Theorem for p The case p follows easily since the

n n p

BanachMazur distance between L and L is n

p

n

Of course for some c a random subspace X of L of dimension cn satises

n n

the second assumption of Theorem ie kxk kxk for all x X see

L L

 

or This is why we claim that there is an abundance of subspaces of

n

L satisfying the conclusion of Theorem

p

uniformly b ounded for any subspace X of L Nev For p glX is

p

ertheless there are nite dimensional subspaces of L which have only bad

p

n

unconditional bases see This implies that sup fub cX X g

p

as n but no estimates are known

Restricted invertibility and nite dimensional subspaces of L of

p

maximal distance to Euclidean spaces

n

Restricted invertibility of operators on

p

Motivated by some problems ab out the structure of nite dimensional sub

spaces of L that will be discussed in section in Bourgain and

p

Tzafriri proved Theorem ab out the restricted invertibility of op erators

n

on Qualitatively this result says or rather implies that a b ounded op er

p

n

ator on which has ones on the diagonal must b e invertible on a co ordinate

p

subspace of prop ortional dimension even after pro jecting back into the co

ordinate subspace In order to state Theorem we intro duce the following

n

be the span in of the notation Given a subset of f ng let

p p

unit vector basis vectors fe i g and let R be the natural co ordinate

i

n

pro jection from onto

p p

Theorem Let p For each there is so that if

p

n

T is an n by n matrix considered as an operator on with zero diagonal

p

then for each there is a subset of f ng of cardinality at least

n so that kR TR k kT k Consequently if kT k then kR I

p p p p

T R k

p

kT k

p

The case p as well as the case p which follows by duality of

Theorem was proved earlier bythe second author and indep endently

by Bourgain p Bourgains argumentgives more than what is stated in

Theorem namely that there exists a splitting of f ng

k

T k into k k disjoint sets so that for each i k kR TR k k

i i

Whether this strengthening of Theorem remains valid for other values of

p is op en For p this matrix splitting question is particularly interesting

b ecause it is equivalent to the KadisonSinger problem whether every pure

state on has a unique extension to a pure state on B For a discussion

of the matrix splitting problem on and more on the connection b etween the

BourgainTzafriri work and the KadisonSinger problem see

n

For a pro of due to K Ball of the matrix splitting result for which gives

n

the estimate k see Notice that in the case of there is no

loss of generality in treating only matrices with nonnegative entries b ecause

n

as op erators on T and jT j have the same norm Berman Halp ern Kaftal

and Weiss indep endently used ideas similar to those used by Ball to prove

n

an splitting result for matrices with nonnegative entries which of course

n

do es not give a splitting result on for general matrices It is amusing and

n

instructive to note that the matrix splitting result for formally implies the

n

p via matrix splitting result for nonnegative matrices on for all

p

a change of density argument of L Weis Here is the idea First after a

change of density a p ositive op erator T on L a probability is nicely

p

b ounded on L This follows from the fact that T maps the order interval

f f for some f into the order interval kT kf kT kf

which in turn follows via iteration and summing as in the pro of of from

the inclusion T g g TgTg which is valid for all g b ecause T

is a p ositive op erator Secondly by working b oth with T and T one can

get that after a suitable change of density a p ositive op erator T on L

p

a probability is nicely b ounded on both on L and on L Now

n

sp ecialize this to a p ositive op erator on The ab ove discussion shows that

p

n

this op erator can be mo deled as a p ositive op erator T on L for some

p

n n

probability on f ng in such a way that kT L L k and

n n n n

kT L L k do not exceed kT L L k Since we

p p

n n n

know the splitting result for op erators on and the result for follows

p

by interp olation

n n

The pro of of Theorem for uses in a more serious way By duality it

p

is enough to prove the case p so we restrict to this range of p The

natural approach to prove Theorem is to show that if the set is chosen

at random from among the subsets of f ng having cardinality n for

small enough then with big probability kR TR k kT k Thisis

p p

unfortunately obviously wrong consider the right shift op erator However

n n n n

regarding as L so that the injection I L L has norm one it is

p

p p p

true that for most such choices of the op erator W I R TR has norm

p

p p

not exceeding kT k the factor is natural it go es away when we

p

regard W as an op erator from L into L For a pro of which is nice and not

p

particularly dicult see Prop osition What is remarkable is that

the p case in Theorem then follows immediately by an application

n

of Grothendiecks inequality via Prop osition Indeed W maps L into

K kW k where K a Hilb ert space and thus has summing norm at most

G G

is Grothendiecks constant see section Applying Prop osition we

n

conclude that W DU for some op erator U on L of norm at most K kT k

G

n n

and some norm one diagonal op erator D L L The op erator D is

n

multiplication by some function g which has norm one in L hence U is

R TR f R TR f

dened by the formula Uf whence k k K kT k kf k for

G

g g

q q

n

all f in L Since kg k Pjg j that is jg j j for

at most n co ordinates j Throwing away any of these which are in the

set and calling the resulting set we have that has cardinalityat least

p

 

K kT k n and kR TR k

G f

That completes the outline of the pro of of Theorem in the Hilb ertian case

p whichby itself has many applications see and It is however

the other cases of Theorem that have applications to the structure theory

of nite dimensional L spaces The pro of we sketched for p do es not

p

carry over b ecause op erators from L into L need not b e p summing when

p

p they are ssumming for all s p but this is not sucient

Lemma is used to get around this problem

Lemma Suppose r p areprobabilities X is a subspace

of L T X L and U X L are operators so that

p p

kUxk x X kTxk

r

T rpkU k where rp is the L norm of a pstable random Then

r p

variable which is normalized in the L norm

In view of Corollary to prove Lemma it suces to verify the estimate

p

M T rpkU kwhichobviously follows from the the following inequal

ityvalid for all nite sets of vectors x in X

i

X X

p p p p

k jTx j k rpk jUx j k

i i p

i i

P P

p p

Toprove write jTx j E j f Tx j where the f are indep endent

i i i i

i i

pstable random variables with E jf j andinterchange the exp ectation and

i

the L norm to see that this is estimated from ab ove by

X

r r

kE j f Ux j k

i i r

i

P P

r r p p

But E j f Ux j rp jUx j so is dominated by the right

i i i

i i

side of

The main to ol for proving Theorem is

Prop osition For each there is so that if

n

r and S is an operator on L n n r with kS k then there

is a subset of f ng of cardinality at least n so that for al l x

kR Sxk Ckxk kSxk

r r

where C is a numerical constant

n

Here we are using as usual the L normalization The imp ortant thing is that

p

r

the factor C in is a gain over the trivial factor

The p case of Theorem follows easily from Lemma Prop osition

and the p case of Theorem Indeed Corollary says that to prove

n

Theorem it is enough to consider norm one op erators on L which have

p

n

norm at most J as op erators on L The p case in Theorem then

p

n

says that it is enough to consider norm one op erators on L whichhave norm

p

n n

at most as op erators on L Now if S is an op erator on L with

p

kS k and kS k apply Prop osition to get Dene T R SR

p

n

considered as an op erator from L into L so that both the domain and

p p

range are L spaces of a probability and set U C R SR considered

p

p

n n n n

as an op erator from L into L L L Then for say r we

p

p p p p

n

n

kUxk Lemma have from Prop osition that for all x in R kTxk

L

L

r



n

T rpkU k Crp Changing back to the L then gives that

p p

p

n

normalization that is regarding R SR as an op erator on L wehavethat

p

p

pro of of Theorem is R S R Crp The completion of the

p

now just as in the p case

For a pro of of Prop osition see Section Here is the idea By the

same kind of reasoning that works in the rst part of the pro of of Theorem

most choices of a subset of fng of cardinality n make it true that

kR Sxk CkSxk for all vectors x whose supp ort has cardinality at most

r

n an estimate for comes out of the pro of For a general vector x

apply this to x where A is the set of the n largest co ordinates of jxj and

A

use the smallness of kS k to take care of x x

A

Corollary gives a metho d to be exploited in the next section for building

n

complemented copies of in subspaces of L

p

p

Corollary Let p and If T is a norm one oper

n

so that ator from into L and there are disjoint measurable sets A

i p

p

k Te k for each i n then thereisanoperator S from L into

A i p p

i

n

with kS k and a subset of fng of cardinality at least

p

p

so that ST is the identity of Consequently TR S is a projection of L

j p

p

p

onto span fTe g

i i

Here the function is taken from Theorem For the pro of of Corollary

p

take norm one vectors h supp orted on A so that hTe h i kTe k

i i i i i p

n

P

n

k Te k hf h ie Then kS k Dene S from L to by Sf

A i p i i p

i p

i

and ST has ones on the diagonal when represen ted as a matrix with resp ect

n

to the unit vector basis for Therefore by Theorem there is a subset of

p

fng of cardinality at least so that R ST R is a p erturbation

p

ST R R S of the identity op erator on Set S R

p

Subspaces of L with maximal distance to Hilbert spaces

p

In this section we give some applications to the isomorphic structure theory

of nite dimensional subspaces of L p of the restricted invert

p

ibility theorem from the previous section First however we mention what

is known in the isometric and almost isometric theories It is classical see

space must be that a subspace of L which is isometric to an L

p p

complemented in L This was extended in for p and for

p

pto the almost isometric setting

Theorem For each p there is so that if p and X

p

is a subspaceof L which is isomorphic to an L space then X is

p p

C complemented in L Moreover C as

p p p

Although Theorem is stated for general subspaces X it reduces easily to

the case where X is nite dimensional which is the case treated in and

A simpler pro of of the p case is given in Section

Conversely it is classical that a complemented subspace of L is isometric

p

to a L space see but the almost isometric version of this result is

p

op en even for nite dimensional subspaces Here a result for nite dimensional

spaces do es not seem to imply formally a result for innite dimensional spaces

It was mentioned already in section that an ndimensional subspace of

jpj n

L has Banac hMazur distance at most n to The converse to

p

this is also true for p see Section as noted in the p

case is essentially done in For p it is op en whether there is an

n

which has BanachMazur ndimensional subspace of L not isometric to

p

p

jpj n

distance n to

Weturnnow to the isomorphic theory Here and in the sequel we shall sp eak

qualitatively ab out nite dimensional spaces similar to the way one sp eaks

ab out innite dimensional spaces In order for our statements to have content

statements should b e quantied so as not to dep end on dimension but there

may be dep endence on other parameters It takes only one example to illus

trate why this convention is followed by Banach space lo cal theorists One of

the main results we shall discuss is the following

Theorem If X is an n dimensional complementedsubspaceofL

p

n

p whose distanceto is of maximal order then X contains a subspace Y

k

of pr oportional dimension say k which is isomorphic to and complemented

p

in L

p

The meaning of this statement is that there are functions f p C and

g p C dened for p and C so that if X is

an ndimensional C complemented subspace of L for some n and p and

p

n jpj

dX n then for some k f p Cn there is a k dimensional

k

subspace Y of X with dY g p C and Y is g p Ccomplemented

p

in L

p

Theorem is also true for p but in a stronger form the comple

mentation assumption is not needed Given the metho d of the main step

in the pro of of Theorem which we sketc h below is Theorem

For p the complementation assumption in Theorem is essential this

follows easily from Theorem Whether the complementation assumption is

needed when pisopen

n

A criterion for building a complemented copy of in a subspace X of L

p

p

was given in Corollary The idea for getting an op erator T taking values

in X which satises the hyp othesis of the corollary is rst to nd vectors f

i

i n in the unit sphere of X for which the norm of the square function

n

P

n n r r n

S ff g where S ff g jf j for r and S ff g

i r i i i

i i i

i

max jf j is nearly extremal that is the norm of S ff g is prop ortional

i i

in

p

to n Once one has such a sequence one uses an extrap olation argument

Indeed for p we shall outline an argument which deals with a

somewhat more general situation Supp ose f i n are in L and

i p

P

n

n p

S ff g n kf k Set p and f f kf k Then

i i i i i

i i

P

n

n p p p

ff g jf j kf k jf j using the factorization S

i i i i

i i

P

n

n p

kf k kS ff g k n

i i p

i i

pp n n

kS ff kf k g S ff g k

p i i i p

i i

pp n n

kS ff kf k g k kS ff g k

p i i i

i p i p

P

n

n

kf k kS ff g k

i i

i i p

n

S ff g and setting A A A we get Letting A jf j

i j j ij i j j

i

n

P

p p

disjoint sets A so that n k f k Thus k f k is larger

i A i p A i p

i i

i

p pp

than for at least n values of i

and that f i n are norm one vectors in Next supp ose that p

i

n p

L for which kS ff g k Cn Set pThen

p i

i

p n n n

n kS ff g kkS ff g S ff g k

p i i i

i i i

p n n n

k k kS f k C n kS ff g f g kS ff g

i i i

i i i

p

Thus there are disjoint sets A so that k f k is larger than C

i A i p

i

p

p

for at least C n values of i

To derive Theorem from Corollary we need the sequence ff g for which

i

k

k f k is b ounded away from zero also to b e dominated by the basis This

A i p

i p

can be accomplished in the following way

Let p and let P L L be the pro jection with range X

p p

k

Assume that for some k prop ortional to n ff g is a sequence in the ball

i

i

L satisfying k f k for some disjointsets of the range of P in

p A i p

i

k p k

fA g Putg P jf j sgn f Note that fg g is dominated bythe

i i i i A i

i i i

k p

basis and in particular the norm of S fg g is dominated by k and that

i

p

R

p

the norms of the g s are b ounded away from zero since kg k g f

i i i i

It follows that for a prop ortion of the g s there are disjoint B s for which

i i

kg k are b ounded away from zero Now Corollary applies to pro duce a

i B

i

subspace of X of prop ortional dimension isomorphic to of its dimension and

p

complemented in L The case p follows by duality We remark

p

that this part of the argument whichiswhatwe had in mind when we wrote

is considerably simpler than the one presented in

Finally we briey indicate how to nd in any ndimensional subspace X of

n n

L of maximal order distance from a sequence ff g with kS ff gk

p i i

i

P

n

p

of order n kf k Assume for example that p By a

i

i

theorem of Kwapien Th the typ e constant of X is of order

p

n and b y a theorem of Tomczak Th this typ e constant

is attained up to a universal constant on n vectors ie there are vectors

P

n

n p

ff g in X satisfying kS ff gk n kf k

i i i

i i

Complemented subspaces

n

Fine embeddings of complemented subspaces of L in

p

p

Let X be a k dimensional subspace of L p and assume there is a

p

pro jection of norm K from L onto X Let and denote by P X K

p p

n

the smallest n such that X emb eds in as aK complemented

p

subspace ie the smallest n such that there is a k dimensional subspace Y

n

pro jection of an isomorphism T X Y with kT kkT k and a

p

n

of norm at most K from onto Y Also denote by P k K the

p

p

maximal P X K when X ranges over all k dimensional subspaces of L

p p

k

If p lo oking at the case of X shows that at least for some K

p p

dep ending on p P k K cpmaxfk k g where p pp It

p

turns out that except for logarithmic terms one can achieve this b ound

p p C p

Theorem P k K C p K maxfk k glog k

p

For p one gets a b etter result

Theorem P k K C K k log k

m

The pro ofs followthe metho d of Section Given a k dimensional X

p

m m

and Y P p and a pro jection P on with range X we let

p p

following the scheme of Section we nd a subset f mg of the

right cardinality such that

m

X X

p p

mjj jx j jx j x X kxk

i i p

i

i

m

X X

p p

mjj jy j y Y ky k jy j

i p i

i

i

m

X X

mjj x y x y x X y Y kxk ky k

i i i i p p

i

i

For p causes of course a problem Fortunately enough in this case

the relevant inequality stating that the restriction to is a go o d isomorphism

m

when restricted to Y follows immediately from and the fact

m

that a restriction to a subset is a norm one op erator on Finally it is easy

to see that and imply the desired result

Finite decomposition and uniqueness of complements

The class of nite dimensional well complemented subspaces of L is a rich

p

class at least in the range p Itfollows from the previous section

that the same is true for the well complemented k dimensional subspaces of

n

as long as k is smaller than a sp ecic power of n The situation for larger

p

a well complemented k is not known In particular it is not known whether

n k

subspace of of prop ortional dimension is well isomorphic to a space It

p p

n

is also not known whether is primary ie whether X Y implies

p

p

k

that either X or Y is well isomorphic to some here we mean of course that

p

the isomorphism constant should dep end on the norm of the pro jection on X

with kernel Y and p only It is true however that if Y is small enough

k

of then X is well isomorphic to an space There are two known instances

p

this statement with two dierent notions of smallness These were proved in

following somewhat weaker results in for the rst theorem and for

the second

n k

Theorem Assume p X Y dY K and there is a

p

n nk

projection P from onto X with kernel Y and kP k L Then dX

p p

C C K L p

n

Theorem Let p and put p maxfp p g Assume X Y

p

p C p n

log n and that there is a projection P from with dimY k n

p

nk

onto X with kernel Y and kP k L Then dX C C L p

p

The pro ofs relay on a simple but powerful decomp osition metho d applied

to a path of complemented subspaces of L together with a prop osition

p

concerning uniqueness of complements

We b egin with the decomp osition metho d from which improved on the

original nite dimensional decomp osition metho d intro duced in

Lemma Let p Let Z i m be Banach spaces and

i

for i m let Y be a K complemented subspace of Z Assume also

i i

P

m

dY Y L i m Then Y Z is CKLisomorphic to

i i p i

i

P P

m m

s

i

Y Z In particular if Z i m and s s then

m p i i i

i p i

s s

Y is CKLisomorphic to Y

m

p p

The pro of is very simple Let X b e the complementofY in Z i m

i i i

Then

Y Y Y Y

m

C B B C C C B B

m

C B B C C C B B

X

C B B C C C B B

Y Z C B B C C C B B

p i p p p p

p p p p

C B B C C C B B

i

A A A A

X X X

m

We now shift the top row to the right

Y Y Y Y

m m

C B C B C B C B

C B C B C B C B

C B C B C B C B

C B C B C B C B

p p p p

p p p p

C B C B C B C B

A A A A

X X X

m

to get

m m

X X

Z Z Y Y

p i p i m

i i

This picturesque pro of can easily b e justied

m

To prove Theorem we assume that p and k We build a

k k

path Y Y of spaces connecting Y with Y with dY Y

m m i i

p

This can be done in several ways but since we also want Y to be well

i

s

i

isomorphic to a well complemented subspace of with s as small as p ossible

i

p

mi

i

we take Y to b e the sum of copies of

i p

mi

emb eds as a complemented subspace with constants Given any

r mip

i

dep ending only on p and into with r The fact that

i

p

this holds for some is exp osed for example in To get it for all

v uv

of intro ducing of course a represent as the sum of u copies

p

constant dep ending on u and p emb ed each summand complementably in an

appropriate space and take the sum of these u spaces as the containing

p p

space

p

s i mip

i

It follows that Y well embeds complementably into for s

i i

p

P

m

k s sk

Lemma implies then that for s s

i

i p p

k n p

The assumption that is w ell complemented in implies that n k for

p

some b ounded away from zero this is a result of It is now easy to see

k nk n

that with the rightchoice of s n k and

p p

k n nk

It remains to show that if also X then X This follows

p p

from the following simple uniqueness of complement result of in the form

proved in In the statement denotes a direct sum of subspaces and the

isomorphism constants implicit in the notation of the conclusions dep end

only on the constants for the and the pro jections in the hyp otheses

Prop osition Assume Z Y X H G with H X and assume

H Y W Then X G W

In particular if for i Z Y X H G with Y Y G G

i i i i i i

H H X and H Y W Then X X

i i

To end the pro of of Theorem we only have to show that X the comple

k n k

ment of Y in contains a subspace well isomorphic to and well

p

p

complemented Since n is of order at least k the general theory of Eu

clidean sections as exp osed in implies that for some X contains

k

a subspace U well isomorphic to U is automatically well complemented



k

see or p Now nd another copy of in the complement of

U in X and iterate

The pro of of Prop osition is very easy Put F X G Then

Z Y X Y F H and consequently G Y F Thus X F H

F Y W G W

The pro of of Theorem follows the same outline as that of the pro of

of Theorem Given a k dimensional well complemented subspace Y of L

p

p we nd a path Y Y Y of well complemented subspaces of

m

k

L with dY Y Y well isomorphic to and m log k We then

p i i m

use Theorem to embed each of Y in a well complemented fashion in a low

i

space and continue in a way similar to the pro of of Theorem dimensional

p

To build the path of spaces Y apply rst the change of density of Theorem

i

and then that of Theorem to get that without loss of generality ky k

p

p

n ky k for each y Y and the pro jection P from L onto Y is also

p

well b ounded with resp ect to the L norm Put

i

Y f y y y Y g L L i m

i p p

k

Then clearly dY Y i m Y is well isomorphic to and it

i i m

only remains to show that eachofY is well isomorphic to a well complemented

i

subspace of L

p

Since P is b ounded in both L and L Y is well complemented in Z

p i i

i

fz z y L g L L Finally by a theorem of Rosenthal see

p p p

also Z is well isomorphic to a well complemented subspace of L

i p

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