Comment.Math.Univ.Carolinae 41,1 2000107{110 107

Boundedness of linear maps

T.S.S.R.K. Rao

Abstract. In this short note we consider necessary and sucient conditions on normed

linear spaces, that ensure the b oundedness of any linear map whose adjoint maps extreme

p oints of the unit ball of the domain space to continuous linear functionals.

Keywords: b ounded linear maps, extreme p oints, barrelled spaces

Classi cation : 46B20

Intro duction

Let X , Y be normed linear spaces and T : X ! Y be a linear map. In

this note we are interested in studying some \weak" continuity conditions on T ,

that will imply continuity. Motivation for this work comes from a recent work

of Labuschagne and Mascioni [6], where they characterize linear maps between



C algebras whose adjoints preserve extreme p oints. A small step in their work



consists of showing, using C algebra metho ds, the continuityofsuch a map. In

this note we rst show that if X and Y are normed linear spaces such that for

   

each extreme p oint y of the dual unit ball Y , y  T is an extreme p ointofX ,

1 1

then T is b ounded.

Let X denote the closed unit ball of X and let @ X denote the set of extreme

e

1 1

p oints. Since b oundedness of the set T X  in the weak top ology implies the

1

b oundedness of T , a natural question that can b e asked now is: Is T b ounded if

   

one merely assumes that for all y 2 @ Y , y  T 2 X ? Here we give necessary

e

1

and sucient conditions on X and Y so that any such linear map T is b ounded.

Main results

We rst show the continuityofT when the \adjoint" preserves extreme p oints

of the dual ball. Let LX; Y  denote the space of b ounded op erators from X to Y .

Prop osition 1. Let X , Y b e normed linear spaces. Let T : X ! Y b e a linear

   

map such that for each y 2 @ Y , y  T 2 @ X ; then T 2 @ LX; Y  .

e e e

1

1 1

  

Proof: Let x 2 X . Cho ose a y 2 @ Y such that kT xk = y T x. By

e

1



hyp othesis y  T is a functional of norm one. Thus kT k  1. That T is an

extreme p oint can b e proved using the hyp othesis and the Krein-Milman theorem

see [2, p. 148]. 

108 T.S.S.R.K. Rao

Remark 1. Op erators whose adjoints preserve extreme p oints are known as

\nice" op erators see [7] and the references listed therein. The analogous ques-

tion, \when are elements of @ LX; Y  nice op erators ?" received considerable

e

1

attention, we again refer the reader to [7] and the references listed there for more

information.

From nowonwe study conditions on X or Y that will result on the b oundedness

   

of T under the assumption y  T 2 X for every y 2 @ Y . We may assume

e

1

w.l.o.g. that Y is a Banach space. To show the weak b oundedness of T X , it

1

    

is enough to show that y  T 2 X for every y 2 Y . If Y is the convex hull

1 1

of its extreme p oints then T is b ounded without any further assumptions. This

for example is the case when Y is a nite dimensional space or the space of trace

class op erators on a complex Hilb ert space.



We recall that any in nite dimensional C algebra contains an isometric copy

of c .

0

Theorem 1. Let Y be a Banach space containing no isomorphic copy of c .

0

For every normed linear space X , every linear op erator T : X ! Y such that

   

y  T 2 X for all y 2 @ Y , is b ounded.

e

1

Proof: Let X b e a normed linear space and T : X ! Y b e a linear map such

   

that y  T 2 X for all y 2 @ Y . To show that T is b ounded it is enough to

e

1

show that for every sequence fx g  X , fT x g is a b ounded sequence

n n

1

n1 n1

    

in Y . For any y 2 @ Y , since y  T 2 X ,wehave that fy T x g is a

e n

n1

1

b ounded sequence of scalars. Therefore it follows from [3] that fT x g is a

n

n1

b ounded sequence in Y . 

Example. Let Y = c and X = spanfe g , where fe g is the canon-

n n

0

n1 n1

P

k

ical Schauder basis of c . Then by de ning T : X ! Y by T  e  =

n n

0

n=1

P

k

   

ne ,we see that T is a linear map and y  T 2 X for all y 2 @ Y

n n e

n=1

1

and T is not b ounded.

In the next prop osition we show that the Example describ ed ab oveworks as a

counterexample whenever the range space contains an isomorphic copyof c . Our

0

pro of involves the notion of an M -ideal whose de nition wenow recall from [5].

De nition. Let Z b e a Banach space. A closed subspace Y  Z is said to b e an

 1 ? 

M -ideal if Z is the ` direct sum of Y and another closed subspace N  Z .

 ?

It is easy to see Lemma I.1.5, [5] that @ Z = @ Y  [ @ N . We also note

e e e

1 1

1



that N is canonically isometric to Y .

Prop osition 2. Let Y b e a Banach space containing an isomorphic copyof c .

0

Then Y can b e renormed such that for the new norm on Y there is a normed linear

  

space X and a linear map T : X ! Y such that for every y 2 @ Y , y  T 2 X ,

e

1

but T is not b ounded.

Proof: Since we are interested in renorming Y , by applying Lemma 8.1 in

Chapter 2 of [1], wemay assume that Y contains an isometric copyof c . It now 0

Boundedness oflinear maps 109

follows from Prop osition I I.2.10 in [5] that we can renorm Y so that c b ecomes

0

an M -ideal in Y . We also note that c still has the supremum norm. Now let X

0

 

and T b e as in the ab ove Example. For any y 2 @ Y w.r. to the new norm

e

1

 1  ?  

either y 2 @ ` or y 2 c . Thus y  T 2 X . Also T is not b ounded. 

e

1 0

Our next theorem gives a necessary and sucient condition on the domain

space for the validity of a similar result.

Theorem 2. Let X b e a normed linear space. For every Banach space Y ,every

   

linear op erator T : X ! Y such that y  T 2 X for all y 2 @ Y is b ounded,

e

1

i X is barrelled.

Proof: Let X be barrelled and let T : X ! Y be a linear map such that

      

y  T 2 X for all y 2 @ Y . It is easy to see that fy  T : y 2 @ Y g is

e e

1 1

a p ointwise b ounded family of functionals on X . Nowbyinvoking the uniform

b oundedness theorem for barrelled spaces Theorem 9-3-4 in [8] we conclude that

T X  is a b ounded set.

1

Conversely supp ose that X is a normed linear space such that for all Banach

   

spaces Y every linear op erator T : X ! Y such that y  T 2 X for all y 2 @ Y

e

1

 

is b ounded. We shall show that every weak compact set K  X is a norm

b ounded set. It would then follow from Theorem 9-3-4 of [8] again that X is

barrelled.

 

Let K  X be a weak compact set. Take Y = C K , the Banach space of

continuous functions on K . If wenow de ne T : X ! Y by T xk =k x for



x 2 X and k 2 K , then clearly T is a linear map. Since elements of @ C K  are

e

1

given byevaluation functionals  k , k 2 K it is easy to see that T satis es the



hyp othesis and hence it is a b ounded op erator. Since T  k  = k ,we conclude

that K is a norm b ounded set. 

Remark 2. When the range space is a separable, real Banach space and contains

no copyofc , it follows from Theorem 4 in [4] that one can weaken the hyp othesis

0

    

in Theorem 1to y  T 2 X for all weak exp osed p oints y 2 Y . Similarly

1

 

when X is separable, since anyweak compact set K  X is metrizable we see



that for any k 2 K ,  k isaweak exp osed p oint.

Acknowledgment. Thanks are due to Professor Y. Abramovich for suggesting

that I consider barrelled spaces.

References

[1] Deville R., Go defroy G., Zizler V., Smoothness and renormings in Banach spaces, Pitman

Monographs and Surveys in Pure and Applied Mathematics 64, New York, 1993.

[2] Diestel J., Sequences and series in Banach spaces, GTM 92, Springer, Berlin, 1984.

[3] Fonf V.P., Weakly extremal properties of Banach spaces, Math. Notes 45 1989, 488{494.

[4] Fonf V.P., On exposed and smooth points of convex bodies in Banach spaces, Bull. London

Math. So c. 28 1996, 51{58.

[5] Harmand P., Werner D., Werner W., M -ideals in Banach spaces and Banach algebras,

Springer LNM 1547, Berlin, 1993.

110 T.S.S.R.K. Rao



[6] Labuschagne L.E., Mascioni V., Linear maps between C algebras whose adjoints preserve

extreme points of the dual unit bal l, Advances in Math. 138 1998, 15{45.

[7] Rao T.S.S.R.K., On the extreme point intersection property,\Function spaces, the second

conference", Ed. K. Jarosz, Lecture Notes in Pure and Appl. Math. 172, Marcel Dekker,

1995, pp. 339{346.

[8] Wilansky A., Modern methods in topological vector spaces, McGraw Hill, New York, 1978.

Indian Statistical Institute, R.V. College Post, Bangalore- 560 059, India

Current address:

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E-mail : [email protected]

Received January 8, 1999