A(X) C(K) James type results for polynomials and symmetric multilinear forms

Ark. M a t . , 4 2 (2004), 1 11

@ 2004 b y I n s t i t u t Mittag-Lettter. All r i g h t s reserved

James type results for polynomials and symmetric multilinear forms

Maria D. Acosta, Julio Becerra Guerrero and Manuel Ruiz Galen(1)

A b s t r a c t . W e prove versions of J a m e s ' weak c o m p a c t n e s s t h e o r e m for p o l y n o m i a l s a n d s y m m e t r i c m u l t i l i n e a r f o r m s of finite type. V~'e also s h o w t h a t a B a n a c h s p a c e X is reflexive if a n d o n l y if it a d m i t s a n e q u i v a l e n t n o r m s u c h t h a t t h e r e e x i s t s x07~0 in X a n d a w e a k - * - o p e n s u b s e t A of t h e d u a l space, s a t i s f y i n g t h a t x* @x0 a t t a i n s its n u m e r i c a l radius, for e a c h x* in A.

1. I n t r o d u c t i o n

T h e classical J a m e s ' theorem states t h a t a Banach space is reflexive if and only if e a c h - - b o u n d e d and linear--functional attains its norm [J]. On the other hand, there are results stating t h a t in the non-reflexive case. the set of norm attaining functionals is small. For instance, if the unit ball of a separable Banach space is not dentable, then the set of norm attaining functionals is of the first Baire category, a result due to Bourgin and Stegall (see [B, T h e o r e m 3.5.5 and P r o b l e m 3.5.6]).

Kenderov, Moors and Sciffer proved t h a t for any infinite c o m p a c t Hausdorff space K , the space

C(K)

satisfies the same p r o p e r t y [KMS].

By using the weak-* topology in the dual space. Debs, Godefroy and Saint R a y m o n d showed t h a t a separable Banach space is reflexive provided t h a t the set of norm attaining functionals has a n o n - e m p t y weak-* interior [DGS, L e m m a 11].

This result was generalized by Jim6nez Sevilla and Moreno, who proved t h a t the same assertion holds for any Banach space not necessarily separable [JIM, Proposi- tion 3.2].

We shall use

A(X)

to denote the set of norm attaining functionals on a Banach space X. For the norm topology, it is known t h a t any Banach space is isomorphic to another one satisfying t h a t the interior of the set of norm attaining functionals is non-empty [AR1, Corollary 2].

(1) T h e first a n d t h i r d a u t h o r were s u p p o r t e d in p a r t by D.G.E.S., p r o j e c t no. B F M 2000- 1467. T h e s e c o n d a u t h o r was p a r t i a l l y s u p p o r t e d by J u n t a de A n d a l u c f a G r a n t F Q M 0 1 9 9 .

However, in the non-reflexive case. if X satisfies that the set

A(X)

has a non- empty interior, then the norm of X does not satisfy strong differentiability con- ditions. For instance, Jimdnez Sevilla and Moreno proved that a Banach space X satisfying the Mazur intersection property and being such that

A(X)

^{has non-}

empty interior, must be reflexive [JIM]. The parallel assertion also holds if the space is very smooth or H a h n - B a n a c h smooth [AR1], [AR2]. For other results along the same line, but assuming that the space does not contain (an isomorphic copy of) 11, see [AB1] and [AB2].

Here we shall show that an abundance of certain other functions on the Banaeh space that attain their suprema also implies reflexivity. To be more precise, we shall use as functions the simplest finite-type homogeneous polynomials, symmetric multilinear forms and numerical radii of rank-one operators.

In Section 2, we shall prove a characterization of reflexive spaces in terms of norm attaining ~t-homogeneous polynomials or norm attaining symmetric multilin-

* ... z * z* in X*. consider the polvnomial given ear forms. For n + l flmctionals x 1 . . . . . . .

by

n ,

i = 1

We prove that a Banach space X is reflexive if and only if for n fixed non-zero functionals z~, ..., x n, the set of elements z* in X* such that the above polynomial attains its norm, contains a (non-empty) weak-*-open set. We also provide a parallel result by considering the symmetrization of the n-linear forms given by

i = 1

Section 3 is devoted to the numerical radius case. Here we show that a space X is reflexive as soon as, for some non-trivial a'0 in X. it holds that the set of elements z* in the dual of X satisfying that the operator x* ~ z 0 attains its numerical radius (see the definition below) has a non-empty weak-* interior. Moreover, we show that any reflexive space admits an equivalent norm such that in this new norm rank-one operators whose image is contained in a fixed one-dimensional subspace attain their numerical radii. Therefore, we get a characterization of reflexive spaces in these terms. A Banach space is known to have finite dimension if and only if for every equivalent norm, each rank-one operator attains its numerical radius [AR3], [AR4].

For

X=Ip

( l < p < o c ) , every compact operator satisfies the previous condition for the usual norm.

James type results for polynomials and symmetric multilinear forms

2. V e r s i o n s o f J a m e s ' t h e o r e m f o r p o l y n o m i a l s and s y m m e t r i c multilinear forms

Hereafter, X will be a Banach space over the scalar field K ( R or C) and X*

its topological dual. By B x and S x we will denote the closed unit ball and unit sphere, respectively. The usual norm of a bounded multilinear form A on X is given by

IIAIl~sup{lA(/1,...,x,~)l:llxili_<l, l < i < n } .

If we fix n functionals x~ .... , x,, CX*, we consider the finite type n-linear form A given by

Tt

II"

A ( x l , . . . , x , ) = z i ( z i ), x i c X ,

i = 1

and in this case, [[A[[ is just [ I i ' t IIx~ll 9 It is also clear that if A is non-trivial, then A attains its norm, that is, there are x i E S x , l<i<_n, with

x )l =

IIAII,

if and only if every functional x~ attains its norm. Therefore, if we assume that for n - 1 fixed functionals x~, ... , x n _ l E X * the n-linear form A attains the norm, for any x*~EX*, then, by using James' theorem, the space is reflexive. For this reason, we shall use n-homogeneous polynomials of the form

77

I-[ x:(x/

i = 1

and symmetric multilinear forms deriving from A instead of n-linear forms, in or- der to obtain James-type results. We shall denote by Px~...~:~ the continuous n- homogeneous polynomial on X given by

n

i = 1

Let us recall that a continuous polynomial P on X attains the norm if the supremum defining the usual norm is a maximum, that is.

IP(x0)l = IIPII for some x0 E B x .

By Sx~...:~; we denote the symmetrization of the continuous multilinear form

> e x ,

so, S~r .... 9 is the continuous s y m m e t r i c n-linear form given by

1

_{' n}

("ill 1

_--

)

where An is t h e set of all p e r m u t a t i o n s of n elements.

Some key references on t h e denseness of n o r m a t t a i n i n g polynomials a n d multi- linear forms c a n be found in [AFW], [AAP], [CK], [Ch], [aP], [a2], [ACKP] a n d [PSI.

N o w we state t h e J a m e s - t y p e result for polynomials.

T h e o r e m 2.1. A B a n a e h space X is reflexive if and only if there are n>_l and non-zero functionals x~, ..., x~ E X * so that the .weak- * interior of the set

{x* E X * : P ~ .... ;x ~ attains its n o r m } is n o n - e m p t y .

Proof. Since each p o l y n o m i a l P ~ ... ;,x* is weakly continuous, on a reflexive space t h e finite t y p e polynomials t h a t we consider a t t a i n the norm.

O n t h e other hand, let us observe t h a t for an 5' x* 9

IlPzr * II = sup Ix~(x)... x * ( x ) x * ( x ) l = sup ]x*(b)l,

Ilxll<_l bEB

where B is given by

B := {x~(x)... x ; (x):L" : x 9 B x } .

It is also clear t h a t the p o l y n o m i a l P*~...*nx" a t t a i n s its n o r m if a n d only if t h e function Ix*] a t t a i n s its s u p r e m u m on B. T h u s we can reformulate t h e a s s u m p t i o n by s w i n g t h a t t h e set

A : = {x* E X* : Ix*] a t t a i n s the s u p r e m u m on B}

has n o n - e m p t y weak-* interior.

Since we shall prove t h a t X is reflexive, we can clearly assume t h a t X is infinite-dimensional. N o w we shall give an equivalent n o r m on X such t h a t for this new n o r m A will be c o n t a i n e d in the set of n o r m a t t a i n i n g flmetionals. For this purpose, let us note t h a t a n y element y E X satisfying t h a t x * ( y ) # O , l<i<_n, c a n be expressed as

In (v)]

I l Y l l n + l X~ Y Y

Y = I l L , x;(y) 17~

L i = I

IlylI'~+I B.

FI n x~(y) _{i = 1}

James type results for polynomials and symmetric multilinear forms In view of the continuity of the scalar-valued function f given by

Ily]ln+l y E X and z~"

I(Y) - 1-i~'~ 1 x ~ ( y ) ' ' (y) # 0.

in order to prove that the set

D B = { A b : A E K , I A l < l a n d b E B }

has a non-empty norm interior~ it suffices to find some y E X satisfying I f ( y ) l < l . To this end, let us fix x0 in S x such that for all i = 1 ... n. x ~ ( x o ) r and a scalar satisfying 0 < a < II]~_l x* (x0)l, and consider y = a x 0 . Thus

Ilyll~§ a~

I f ( Y ) l - I I I ~ : ( y ) l - anll-[:~ ~x;(x0)l < 1.

We write C for the closure of the convex hull of D B . Therefore, C is bounded, closed, convex and balanced, with 0 belonging to the norm interior of it; and so, it is the unit ball of an equivalent norm on X.

Because of the assumption, the set A is contained in the set of norm attain- ing functionals for this new norm, and A has non-empty weak-* interior. Since a functional x* attains its norm if and only if Rex* attains its norm, the previous assumption implies that there is a weak-*-open set of (XR)* of norm attaining functionals oil XR. Hence, by using [JIM, Proposition 3.2]. X is reflexive. []

The analogous version for symmetric multilinear forms can be stated as follows.

T h e o r e m 2.2. A B a n a e h space X is reflexive if and only if there are n>_l and x~, ... ,x*~EX*\{0} so that the weak-* interior of the set

{x* E X * : S ~ .... ;,x* attains its n o r m } is n o n - e m p t y .

Proof. One can proceed as in Theorem 2.1, if we note that given a functional x * E X * , the symmetric ( n + l ) - l i n e a r form S:~r .... ;,x* attains its norm if and only if the function Ix*l attains the supremum on B. where

{ 1 ~-~, ( 1 2 I * ) }

B : = ( n + i ) ! ~ ( ~ ( ~ ) ) ~ ( ' + ~ ) : ~'~ c B x 9

ere A,~+i \ i - - 1

The set B clearly contains the subset given by

n

Again, we assume t h a t X is infinite-dimensional. Now, it suffices to fix X o E S x

. rt .

with x i (xo)r for all i and O < a < l H i = l x i (xo)t. The element y=axo belongs to the norm interior of D B since

a 1-I,=1 ~; (Y) 71 a

y = ~ 9 y E ,,.

H,=I x~ (Xo)

llytl n+l 1"Ii=1

x~(xo)

^B.

We finish by using the argument in the proof of T h e o r e m 2.1. []

There are simple examples of spaces and polynomials of the type we considered here, such t h a t the polynomial attains its n o r m but not all the functionals involved a t t a i n the norm (see, for instance [Ru, E x a m p l e 4]). One can define analogous examples of symmetric multilinear forms [Ru]. Therefore, J a m e s ' t h e o r e m cannot be directly applied in the proofs of T h e o r e m s 2.1 and 2.2. Some parallel results on the reflexivity of the space of all homogeneous polynomials have been studied by several authors (see [R], [AAD], [MV] and [JM]).

Remark. Following similar arguments as in T h e o r e m s 2.1 and 2.2. one can prove t h a t a Banach space X is reflexive provided t h a t there are n elements z~, ..., x~'~E X * \ { 0 } and a bounded, closed and convex subset A C X with n o n - e m p t y norm interior satisfying t h a t

{x* E X* : [ R ~ r I attains its s u p r e m u m on A}

o r

{x* E X* :lSx; .... ~x* I attains the s u p r e m u m on A}

contains a (non-void) weak-*-open subset of X*.

3. T h e n u m e r i c a l r a d i u s t y p e r e s u l t

In order to state the third version of J a m e s ' theorem, let us recall t h a t the numerical range of an o p e r a t o r T E L ( X ) (the Banach algebra of all bounded and linear operators on X ) is the bounded set of scalars

v(:r) := {x*('r~): (x,**) ~ n ( x ) } ,

where I I ( X ) := { (x, x*) E S x x S x . :x* (x) = 1}, and the numerical radius of T is given by

v(T) := sup{l~l :• e V(T)}

James type results for polynomials and symmetric multilinear forms

(see [BD]). T h e operator T is said to attain its numerical radius when the s u p r e m u m defining v(T) actually is a m a x i m u m . For some known results about numerical radius attaining operators, one can see [BS], [C], [P], [A1] and [AP].

In the version of J a m e s ' theorem posed here, we shall use only rank-one op- erators. If x E X and x * E X * , then x * ~ x will denote the operator in X defined

by

(x*~x)(y) := x*(y)x, y e X .

T h e o r e m 3.1. Let X be a Banach space and suppose that there exists xoE X \ { 0 } such that the subset

{x* E X* : x* | attains its numerical radius}

has a non-empty weak-* interior. Then X is reflexive.

Proof. If one writes B for the set

B := {~*(x0)y: (y, y*) e n ( x ) } ,

then for all x* E X * we have t h a t

v(z*@xO) = sup Ix*(y)y*(zo)I = sup Iz*(b)l

(y~y*)en(X) bEB

and the rank-one o p e r a t o r x* | attains its nmnerical radius if and only if the function ]x*] attains the s u p r e m u m on B. According to the assumption, the set

{x* E X * : Ix* ] attains the s u p r e m u m on B}

has a n o n - e m p t y weak-* interior.

We shall prove t h a t the set D B contains an open ball. If this condition holds, then the closed convex hull of D B . let us say C. is closed, convex, bounded and contains a ball centered at zero. Hence, C is the closed unit ball of an equivalent n o r m on X. For this new norm, by using the assumption, the set of norm attaining functionals contains a weak-*-open set, and so, by using again t h a t this condition implies the corresponding assumption for XR, it follows from [JIM, Proposition 3.2 l t h a t X is reflexive.

For 0 < r < ~ let us check t h a t

8 9 C D B . Each element y E 89 Xo + r B x satisfies

1 l+r"

(1) 0 < ~ - r < IlYll -<

Let us choose y * E X * so t h a t

(y/llyll,y*)eII(X).

T h e n

y*(z0) i1@11 E B.

Furthermore, as a result of the choice of y, r and (1), we have the inequality

(2) Ly*(xo)l >_ 2(llylt-r) >_ 1--4r > 0.

T h e element y can be trivially decomposed in the form

Ilyll 9 ~!Iyll B

(3) y _ y . ( x 0 ) ~ * ( ~ 0 ) ~

^o

Finally, by using (1) and (2) it holds t h a t

[[Y[[ < 8 9 < 1

ly*(xo)l- 1-4r

and so, in view of (3), y belongs to D B . []

Let us finally observe t h a t the previous result generalizes, see JAR3, T h e o r e m 1]

where it was proven t h a t a Banach space is reflexive if every rank-one operator attains its numerical radius. Moreover. the proof given here is much simpler.

We shall now obtain a partial converse of the previous result, which is sharp, as we will observe later.

P r o p o s i t i o n 3.2. Let X be a reflexive Banach space and x 0 E X \ { 0 } . Then there is an equivalent norm on X such that, f o r any element x* in X * , the operator x* | attains its numerical radius.

Proof. W e can clearly assmne t h a t d i m X > 2 . Hence, we can decompose X = K x o | for some closed linear subspace M e { 0 } of X. Let us consider the space Y isomorphic to X given by

Y ---- K32o @1 +~I

whose dual can be identified as

Y* = K x ~ G ~ 3 I * .

where x~ E X* satisfies x~ ( x 0 ) = 1. We shall check t h a t for any x*E X*, the operator x* | attains its numerical radius on Y.

Let us consider the set

B := {Y*(xo)y: (y, y*) E II(Y)}

James type results for polynomials and symmetric multilinear forms

and assume t h a t x0 is in the unit sphere of Y. ARv element; y in the unit sphere of Y can be written as

y = sxo + m

for some scalar s and some element r n E M satisfying }s}+ }}mll=l. T h e element y* = ~x 8 +m*,

where A E K , m * E M * ,

I~xi=ll~*ll=t, xs=lsl and m*(~)=llmll,

satisfies t h a t (y, y*) GII(Y) and ]y* (x0)J= i, so y E D B .

We have checked t h a t S y c D B and so B y = D B . Therefore,

~( x * | ) = sup [~* (b )l = IIx* II

b ~ B

and x*| attains its numerical radius if and only if x* attains its norm. Since Y is reflexive, it follows t h a t any o p e r a t o r of the form x* Ex0 attains its numerical radius. []

By using T h e o r e m 3.1 and Proposition 3.2. we arrive, in fact, at the following characterization.

C o r o l l a r y 3.3. A Banach space is reflexive if and only if f o r some O r there is an equivalent norm on X such that x* •xo attains its numerical radius fo'P any element x* in the dual space of X .

Let us note t h a t in any infinite-dimensional Banach space there is an equivalent n o r m such t h a t at least one rank-one operator does not attain its numerical radius (see T h e o r e m 3.1, JAR3, Example] and JAR4. T h e o r e m 3]). Therefore, if the Banach space is reflexive but not finite-dimensional, renorming may be necessary in order to obtain the s t a t e m e n t in Proposition 8.2.

[A2]

[AAP]

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James type results for polynomials and symmetric multilinear forms 11

[JM]

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[Ru]

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Received September 30, 2002 Maria D. Acosta Universidad de G r a n a d a Facultad de Ciencias

D e p a r t a m e n t o de An~lisis Matem~tico ES-18071 G r a n a d a

Spain

email: d a c o s t a ~ u g r . e s Julio Becerra Guerrero Universidad de G r a n a d a E. U. Arquitectura T~cnica

D e p a r t a m e n t o de Matem~tica Aplicada ES-18071 G r a n a d a

Spain

email: juliobg~ugr.es Manuel Ruiz Galgm Universidad de G r a n a d a E. U. Arquitectura Tdcnica

D e p a r t a m e n t o de Matem~tica Aplicada ES-18071 G r a n a d a

Spain

email: mruizggugr.es