Rough balls in Banach spaces

Teoria Operatorów Kraków, 22-28 September 2003

pp. 133-138

ROUGH

BALLS

IN

BANACH SPACES

ANNA PELCZAR

1.

P

reliminaries

The

title of the paper serves

as

a

pretext for considering the

families of isomor

­

phisms and

bounded

projections

on

a

Banach

spaces and on its subspaces,

as

the structure of those families

is

coded in the shape

of

the

unit

ball

in

a

Banach space.

We

start

with results depicting high regularity of

some

sequence

spaces

to finish

up with

the

” pathological”

behavior

of so-called

hereditarily

indecomposable

Banach spaces (HI

spaces). This

part

of the structural Banach

space theory

experienced

a

major

development

in last

decade

thanks

to

spectacular results concerning

HI

spaces, involving, among

others, application ofthe

Set

Theory.

For the

basic

definitions and facts see

[LT2].

Letters

X, Y, Z

will denote infinite dimensional Banachspaces. If

Y is

a

closed

infinite

dimensional subspace

ofaBanach space X we

write Y

< X. We say that

a

subspace

Y

< X is

(C-)complemented

in X,

if there

is a bounded

projection

from

X

onto

Y (of norm C). Given a

sequence

(in)

c

X

by [x

n]

we denote the

closed

linear

space

spanned

by (in).

By

(resp.

Sx) we denote

closed

unit

ball

(resp.

unit sphere) of a Banach space X. The set coo denotes the set

of

all finite scalar

sequences. In the

rest of the

article

we will consider only

” non-trivial” projections

-

projections

with infinitely

dimensional

and

codimensional image.

2.

Classical spaces

We will start with

the family

of

isomorphisms and bounded

projectionson

classical spaces.

Any Hilbert

space is

extremely regularwith

respect to these

notions.

In

the

separable

case we

have

Fact

2.1.

Any space

Y <

£2 is isometric to £2 and

1-complemented in £

2

One

maythink

about

thisfact

in terms of the shape of the

ball

in £2

as

an

extremely

regular

convexbody:

every infinite dimensional

shadow

of

theball

and

every infinite dimensional section

of

the

ball

givesthe sameconvex

body.

A very

natural question

arises

if such a regularity can

occur

in

someother

Banach

spaces.

The answer

is

”no

”.

Theorem 2.2. [LT1]

Let X be

a

Banach

spacesuch that any

of its

closed subspaces

is complemented in X.

Then X

is isomorphic to

a Hilbert

space.

We will see later

that the homogeneity of

the space (i.e. the property

that every closed

infinite

dimensional subspace

is isomorphic

to

the whole

space) characterizes

£2

aswell,

however

the proofofthis fact

requires quite sophisticated machinery.

What

if we

restrict the above regularity

properties

to some

subfamily

of

subspaces?

We

will

describe now anatural

candidate

for

such

subfamily, but

first

we

need

some

definitions.

Definition 2.3.

Let

X be aBanach

space.

A

sequence

(i

n)

C

X is called a

(Schauder)

basis

of X,

iff

for any

vector

x

€

X

there is

a

unique

sequence (a

n

) of

scalars such

that

x

=

a

nx

n. A sequence

(xn) in a Banach

space is called

a

basic sequence if

it

is

a

basis of [x

n

].

The simplest

examples

are the

unit

vector

bases

(en

) in co

and

£p, 1 < p

< oo, defined as follows: for

any

n

€ N

let

e£ — 1

for k =n and

e£

=

1 otherwise.

By

Mazur

’s

construction

(see [LT2]) any Banach

space contains

a basicsequence, but not all Banach

spaces have

a

basis (for

a

classical example

see

[E]).

Definition 2.4.

Let

(xn

),

(i/n) be basic

sequences in

a Banach

space.

We

say

that (x

n

) and (yn) are

equivalent,

ifthe mapping

T : unxn i > uni/n, (fln)n € Q)0

n€N n6N

extends to an isomorphism from

[xn

]

on

[yn].

Obviously

ifsequences (x

n) and (y

n) are

equivalent,

the spaces [x

n]

and are

isomorphic,

but the inverseimplication is false.

Definition 2.5.

Let

X be a

Banach space

with a

basis

(x

n).

A sequence (yn

) C

X

is

called a block

sequence

of

(xn) if

it

is of

the

form

yn = n

€ N,

for some

sequence

(aj of

scalars

and

some sequence

(Zn) of

intervals

of

integers

such that

max

In < minZn+

i for

any n

6

N.

A

closed

subspace

ofX spannedby an infinite

block sequence

ofthe

basis is called

a

block subspace.

It

is

easy to see that a

block

sequence of a

basis is also

a

basic

sequence. The structure

of

the family of blocksubspacesof aBanach

space describes

quite efficiently the

structure

of

all

infinite

dimensional closed

subspacesof

X,

as thefollowinglemma shows:

Lemma

2.6. [LT2]

Let

X be a

Banach space with

a

basis. Then

for any

Y

<

X

and any

e > 0 there

is

a block

subspace

Z

< X

such

that

Sz

C

Sy

+eBx■

What will

happen

if

we

restrict

the previously

considered

regularity

properties to

the

class

of block

subspaces?

The answer

involves

the following

family

of sequence

Banach

spaces:

Definition 2.7.

For

any

1 <

p

< oo

we

define

ll(an)||p =

co

=

{(a

n) C

K

:

(a

n) converges to

0 } , IKan)!!«,

= sup{|a

n|

:n

€ N}

It

is easy to

verify

that

(co,

||

■

l]oo), || • ||

P), for

1

< p <

oo,

are

separable

Banach

spaces.

Moreover those spaces are characterized in

terms of

isomorphisms and bounded

projections

by the followingwell known M.Zippin

theorems:

Theorem

2.8. [LT2] A Banach

space

with a

basis

(x

n

)

is

isomorphic to

co or

i

p, for 1 < p < oo, iff

any normalized

block sequence (yn

)

of (i

n)

is equivalent

to (rr

n

).

Theorem 2.9.

[LT2]

A Banach

space

with a basis

(i

n)

is

isomorphic

to co

or i

p,

for

1 <

p

< oo, iff

for

any

permutation

7r

of

N

and any

normalized block sequence (y

n

)

of there

is

abounded projection on

X onto

[y

n

].

Notice

thatthe

first theorem characterizes

spaces i

p and

co

in terms

ofisomorphism, but

of

a

specific form,

given by

equivalence,

and

permuting

the

basis in

the

second

theorem

is indispensable - there is

a Banach

space

with a basis,

not

isomorphic to or

co,

with

every its

blocksubspace complemented.

3. Unconditional bases and

HI

spaces

In the previous

section

we presented

spaces

with extremely rich families of

non­

trivial bounded

projections and

isomorphisms. Now

we consider

the other edge.

Let

us

weaken

the properties

concerning

the

existence

of

bounded projections

presented in theprevioussection. Thereasonable

requirement on

thefamilyof

non-trivial

bounded

projections should

be the boundedness ofthe

canonical projections attached

to the basis.

Definition 3.1. A

basis

(i

n) of

a Banach

space

X

is

called

unconditional if

for

any /

C

N

the

mapping

P • 1 ► (^n)n £ ^-00

nEN n€/

extends to a bounded projection from

X

onto

[a?n]ng7-

It is easy

to

notice

that a

basis (x

n)

is unconditional iff

for any vector of X

its expansion 52

anXn converges unconditionally. A

block

sequence ofan unconditional

basis is

also a

unconditional

basic sequence. The

unit

vector bases of

co,

£

p,

for

1 < p

<

oo,

and

many

other classical

bases

are

unconditional. One of

the

exceptions is

the

summing

basis

(e

n) in

co:

for any n

G

N we

have

e„ = 1,

for

k < n and

e„

=0 otherwise. However,

co

has an

unconditional basis

(theunit

vector

basis). Thespace C[0,1] has no unconditional

basis,

but contains all separable Banach space,

hence

also some

unconditional sequences.

Therefore for a

long

time the following

question remained

open:

Problem. Does any Banach

space contain an unconditional

basic

sequence?

Notice that a

space without unconditional basic

sequences

cannot

contain

any of

the classical

sequence spaces

as

co

or

£p,

ad even

such examples

were not

known up

to

70’

s,

when

B.Tsirelson

constructed a first truly

non-classical

Banach

space. The definition ofthe

norm

on the

Tsirelson

space

is implicit

-

given

as the

solution

ofan equation. TheTsirelsonspace still

enjoys

veryregularproperties

-

ithas

unconditional

basis,

it is

reflexive and quasi-minimal

(the definition

is

given

later) -

but it

does not

contain any

of Hp

or

cq

.

And

it

served as a base for the

Banach spaces

with some

surprising

properties,

discovered in 90

’

s. For extensive

study of

the

Tsirelson

space

see [CS].

Definition3.2. A

Banach

space

X is called indecomposable if any

bounded

projection on X

has a rangeof finite dimension orcodimension

in

X.

A Banach spaceX

is

calledan HI

space

(hereditarily

indecomposable) space if

for any

Y

<

X, Y

is indecomposable.

Notice

that

in particular

an HI space contains no unconditional

basic sequences, since

it admits

no

non-trivial

bounded projection. The first such space

was

con­

structed by

W.T.Gowers, B.Maurey

[GM],

who

built in

a

highly ingenious

way on the Tsirelson and Schlumprecht spaces

in

search

of

a space without

unconditional

sequence. Later the variations of their

construction

came:

an

uniformly convex (su

­ perreflexive)

HI space,

an

asymptotically

unconditional

HI

space, an

asymptotically

¿1 HI space,

an

HI

space

with

no reflexive

subspace etc.

It turned

out that HI space lacks notonly non-trivial

bounded projections

but

in

fact

non-trivial

bounded

opera­

tors

-

up

tostrictly

singular

perturbations.

In

particular

an

HIspace

is

not

isomorphic

to any of

its subspaces. In

the

complex

casewehavethe following:

Theorem

3.3. [GM]

Let

X be a

complex

HIspace.

Then

every bounded operatoron

X is of

the

form

Xld

+

S, where

A

g

C and S is a

strictly

singular

operator.

Definition 3.4. A

bounded operator

S in a Banach

space is called strictly

singular,

if

for

any Y

<

X

the

restriction S|y is not an isomorphism.

This

theorem

leads

inevitably to the

following structural

question:

A ’’

scalar + compact

”

problem. Is

there

a Banach

space such

that

any bounded operator on

the

space

is ofthe form Xld

+

K with K

compact?

Let us notice here that it was not by accident that the

first

constructed space

without unconditional

sequences

shared

amuch

stronger property

(HI). W.T.Gowers

proved

the following

theorem

using Ramsey

technics in

the

framework

of

Banach

spaces:

Theorem

3.5.

Gowers’ dichotomy [Gl]. Any Banach space contains

either

an

HI subspace

or an

unconditionalsequence.

In

fact

this dichotomy

is a

corollary

of a much

more general dichotomy, concern­

ing games

in a Banach

space.

It

seems that

this

result

concerns

only

spaces

with

highly irregular

behavior; however, Gowers’ dichotomy,

combined with a

result

of R.Komorowski

and N.Tomczak-Jaegermann [KT-J]

brought a

positive

answerto the

homogeneous

spaceproblem posed by S.Banach:

Theorem

3.6.

Let

X be a homogeneous Banach space

(i.e.

isomorphic

to any

of

its closed

infinite

dimensional

subspaces). Then

X is

isomorphic

to

¿2-

4.

F

urtherquestions

The

Gowers

dichotomy leads to

the following

research program:

Program.

[G3] Identify

a

family

of

disjoint classes ofBanach spaces, defined in terms

of richness of

the

family of bounded

operators

on

a

space, such

that any

Banach

space

contains

a closed infinite dimensional

subspace

from one

ofthese classes.

W.T.Gowers proposed

the followingclassification:

Theorem 4.1. [G3]

Any infinite

dimensional Banach

space X

has

an

infinitely

dimen­

sional closed subspace

Y with one of

the following properties, which are

mutually exclusive

and all

possible:

(1)

K

is

an

HI

space (example: Gowers-Maurey space),

(2)

Y has an unconditional basis and every isomorphism

between

block subspaces

ofa

space

is a strictlysingular perturbationofthe

restriction of some invert­

ible diagonal

operator on the space

(example: variation

on Gowers-Maurey scheme),

(3) Y is

strictly

quasi-minimal

(example: Tsirelson

space)

(4) Y

is minimal

(examples:

¿p,

1 < p

<

oo, co,

the

dual

to Tsirelson space,

Schlumprecht space).

Let us

explain

the

notions used

above:

Definition

4.2.

A

Banach

space

X

is

called:

(1)

minimal, if any closed infinite

dimensional

subspace

Y

of

X contains

afurther subspace

Z

isomorphic

to X,

(2)

quasi-minimal,

if for

any infinite

dimensional subspaces

Y, Z of X there are

further infinite-dimensional

subspaces

Y' < Y,

Z'

< Z

with Y',

Z'

isomor­

phic,

(3) strictly

quasi-minimal,

if it

is quasi-minimal

and

contains

no minimal

sub­

space.

Since -

as we have seen in

the previous

section

- the

spaces

tp and cq

can

be characterized in terms of

richness

of the

family of bounded

operators (isomorphisms and non-trivial

projections),

it would be

interesting

to

split

the

last item

in Gowers’

list so

that

it

ends

on the

family of £

ps and cq. In

order to

do so

one

could

consider again

the properties of

basic

sequences.

Definition

4.3. A basis

(x

n

)

ofa

Banach space

X

is called subsymmetric

if for

any

increasingsequence (z„)

of integers

thesequence (xin)n

is equivalent

to

(z

n)n

.

Notice

that this

notion is

not

hereditary with respect to

block

sequences,

i.c.

a

block

sequence of

a

subsymmetric basis

does not need to be subsymmetric. We have a following result:

Theorem 4.4.

[P] Let

X be a Banach

space

such that

any

subspace

Y

< X contains a subsymmetric sequence. Then

X contains

a

minimal

subspace.

Notice

that there arcminimal spaces without

subsymmetric

sequences (asthe dual

to Tsirelson

space) and minimal spaces

with subsymmetric basis

containing no

or

c

0 (as Schlumprecht

space).

The

results

presented

above

lead

to

the following questions:

Isthere

any

minimal space

with

asymmetric

basis

not isomorphic

to co

or ip :

1

<

p< oo?

Let

X with

a basis be a

’’

block subsequence homogeneous

”

space, i.e.

any

nor

­

malized block sequence contains a subsequence

equivalent to

the

basis

of X. Is

X

isomorphic

to co

or :

1 < p <

oo?

The

answer

is

”

yes” under

some

additional

assumption

such

as

if

the

equivalence is

uniform

[FPR].

R

eferences [CS]

[E]

(FPR]

[Gl]

[G2]

[G3]

[GM]

[KT-J]

[LT1]

[LT2]

[M]

[O]

[P]

[T-J]

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