Linear and Multilinear Algebra, Vol. 55, No. 6, December 2007, 507–513
On the numerical index of vector-valued
function spaces
ELMOULOUDI ED-DARI*y, MOHAMED AMINE KHAMSIz,
and ASUMAN GÜVEN AKSOYx
yFaculté des Sciences Jean Perrin, Université d’Artois, SP 18,
62307-Lens Cedex, France
zDepartment of Mathematical Sciences,University of Texas at El Paso,
500 West University, Ave. El Paso, Texas 79968-0514, USA
xDepartment of Mathematics, Claremont McKenna College,
Claremont, CA 91711, USA
Communicated by W. Watkins
(Received in final form 13 December 2006)
Let X be a Banach space and a positive measure. In this article, we show that
nðLp ð, XÞÞ ¼ limm nðlpm ðXÞÞ, 1 p < 1. Also, we investigate the positivity of the numerical
index of lp-spaces.
Keywords: Numerical index; Numerical radius
Mathematics Subject Classification (2000): 47A12
1. Introduction
Let X be a Banach space over R or C, we write BX for the closed unit ball and SX for the
unit sphere of X. The dual space is denoted by X and the Banach algebra of all
continuous linear operators on X is denoted by B(X). The numerical range of
T 2 BðXÞ is defined by
VðTÞ ¼ x ðTxÞ: x 2 SX , x 2 SX , x ðxÞ ¼ 1 The numerical radius of T is then given by
vðTÞ ¼ sup jj: 2 VðTÞ *Corresponding author. Email: [email protected]
Linear and Multilinear Algebra
ISSN 0308-1087 print/ISSN 1563-5139 online ß 2007 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/03081080601030594
508
E. Ed-dari et al.
Clearly, v is a semi norm on B(X) and vðTÞ kTk for all T 2 BðXÞ. The numerical index
of X is defined by
nðXÞ ¼ inf vðTÞ: T 2 SBðXÞ The concept of numerical index was first suggested by Lumer [7] in 1968. Since then a
lot of attention has been paid to this constant of equivalence between the numerical
radius and the usual norm in the Banach algebra of all bounded linear operators of
a Banach space. Classical references here are [1,2]. For recent results, we refer the
reader to [3,5,6,8,10].
In this article, we show that for any positive measure and Banach space X, the
numerical index of Lp ð, XÞ, 1 p < 1 is the limit of the sequence of numerical
index of lpm ðXÞ. This gives a partial answer to Martı́n’s question [9] and generalizes
the result obtained for the scalar case [5]. We also study the positivity of the numerical
index of lp-space.
Here, Lp ð, XÞ is the classical Banach space of p-integrable functions f from into X
where ð, , Þ is a given measure
P space.p And lp ðXÞ is them Banach space of sequences
x ¼ ðxn Þn1 , xn 2 X, such that 1
space of
n¼1 kxn k < 1. Finally lp ðXÞ is the Banach
P
p 1=p
finite sequences x ¼ ðxn Þ1nm , xn 2 X, equipped with the norm kxkp ¼ ð m
kx
.
nk Þ
n¼1
2. Main results
THEOREM 2.1 Let X be a Banach space. Then, for every real number p, 1 p < 1, the
numerical index of the Banach space lp ðXÞ is given by
nðlp ðXÞÞ ¼ lim n lpm ðXÞ :
m
Proof Let m 1 and T : lpm ðXÞ ! lpm ðXÞ x ° ðT1 ðxÞ, . . . , Tm ðxÞÞ. Define the linear
operator T~ : lp ðXÞ ! lp ðXÞ as follows for x ¼ ðx1 , . . . , xm , xmþ1 , . . .Þ 2 lp ðXÞ,
~
TðxÞ
¼ ðT1 ðx1 , . . . , xm Þ, . . . , Tm ðx1 , . . . , xm Þ, 0, . . .Þ. Clearly, T~ is bounded and
~ We have also vðTÞ ¼ vðTÞ.
~ To prove this, let us first note that if x ¼
kTk ¼ kTk.
ðx1 , . . . , xm , . . .Þ 2 Slp ðXÞ , then there exists an element, namely xx , in Slq ðX Þ , where q is
the conjugate exponent to p, such that xx ðxÞ ¼ 1. Explicitly xx ¼ ðkx1 kp1 x1 , . . . ,
kxm kp1 xm , . . .Þ where the xk ’s are taken in SX such that xk ðxk Þ ¼ kxk k. Now, let
~ ¼ supfjx ðTxÞj
~ : x 2 Slp ðXÞ g ([4], Lemma 3.2 and
" > 0. Following the expression vðTÞ
x
Proposition 1.1) there exists x ¼ ðx1 , . . . , xm , xmþ1 , . . .Þ 2 Slp ðXÞ such that
~ " < jx ðTxÞj
~
vðTÞ
x
¼ kx1 kp1 x , . . . , kxm kp1 x Tðx1 , . . . , xm Þ :
1
m
P
p 1=p
~ " < rp vðTÞ which yields
Put r :¼ ð m
1. Then we obtain vðTÞ
k¼1 kxk k Þ
~ vðTÞ. The reverse inequality is easy. Therefore
vðTÞ
n
o vðTÞ: T 2 lpm ðXÞ, kTk ¼ 1 vðUÞ : U 2 lp ðXÞ, kUk ¼ 1
509
Numerical index of vector-valued function spaces
which yields nðlp ðXÞÞ nðlpm ðXÞÞ. Consequently nðlp ðXÞÞ lim infm nðlpm ðXÞÞ. Now we
shall prove that lim supm nðlpm ðXÞÞ nðlp ðXÞÞ. Let T 2 Bðlp ðXÞÞ. Define the sequence of
operators fSm gm as follows; for each m 1, Sm is defined on lpm ðXÞ by
Sm ðxÞ ¼ ðT1 ðx1 , . . . , xm , 0, 0, . . .Þ, . . . , Tm ðx1 , . . . , xm , 0, 0, . . .ÞÞ
x 2 lpm ðXÞ Clearly, the Sm’s are bounded and kSm k kTk for all m. We claim that
(i) kSm k ! kTk
(ii) vðSm Þ ! vðTÞ.
Indeed, we consider the sequence of operators fS~m gm defined on lp ðXÞ by
S~m ðxÞ ¼ ðT1 ðx1 , . . . , xm , 0, 0, . . .Þ, . . . , Tm ðx1 , . . . , xm , 0, 0, . . .Þ, 0, 0, . . .Þ
for all x ¼ ðx1 , . . . , xm , xmþ1 , . . .Þ 2 lp ðXÞ. It is easy to see that kSm k ¼ kS~m k, and S~m
converges strongly to T. This implies that kTk lim infm kS~m k, and it follows that
kSm k ! kTk. As in (i) we have also vðSm Þ ¼ vðS~m Þ, so it is enough to prove that
vðS~m Þ ! vðTÞ. Let " > 0 and fix u 2 SX , u 2 SX such that u ðuÞ ¼ 1. There exists
x 2 Slp ðXÞ such that
x ðTxÞ > vðTÞ "
ð1Þ
x
For each n 1, consider
xn ¼ x1 , . . . , xn1 , n u, 0, 0, . . . ; xxn ¼ kx1 kp1 xx1 , . . . , kxn1 kp1 xxn1 , p1
n u , 0, 0, . . .
where n ¼
P1
k¼n
kxk kp
1=p
. Then
xxn ðxn Þ ¼ 1 ¼ kxxn k ¼ kxn k
Moreover, kx
xn k ! 0 and kxx xxn k ! 0 where xx ¼ kx1 kp1 xx1 , . . . ,
kxn kp1 xxn , . . . . It follows that xxn ðTxn Þ ! xx ðTxÞ as n tends to infinity. Let n0 1
be such that
jxxn ðTxn Þj > vðTÞ "
ðn n0 Þ
ð2Þ
Since S~m converges strongly to T, thus for fixed n n0 , xxn ðS~m xn Þ converges to xxn ðTxn Þ
as m tends to infinity. So there is m0 n such that
jxxn ðS~m xn Þj > vðTÞ "
ðm m0 Þ
ð3Þ
This yields vðS~m Þ > vðTÞ " for all m m0 and therefore vðS~m Þ converges to v(T) as
m tends to infinity. Now, following (i) and (ii) we have nðlp ðXÞÞ lim supm nðlpm ðXÞÞ.
Indeed, for a given " > 0, we find T 2 SBðlp ðXÞÞ such that
nðlp ðXÞÞ þ " > vðTÞ
510
E. Ed-dari et al.
Since vðTÞ ¼ limm vðS~m Þ, there exists m0 such that
nðlp ðXÞÞ þ " > vðS~m Þ ðm m0 Þ
But vðS~m Þ ¼ vðSm Þ nðlpm ðXÞÞkSm k, and kSm k ! kTk ¼ 1, so there exists k0 m0
such that
nðlp ðXÞÞ þ " > nðlpm ðXÞÞð1 "Þ ðm k0 Þ
This implies nðlp ðXÞÞ lim supm nðlpm ðXÞÞ and completes the proof of Theorem 2.1. g
It is well known that nð X Þl1 ¼ inf2 nðX Þ [9]. This shows that, in particular,
nðl1 ðXÞÞ ¼ nðXÞ ð¼ limm nðl m
1 ðXÞÞÞ. So, Theorem 2.1 is also valid for p ¼ 1.
THEOREM 2.2 Let ð, , Þ be a -finite measure space. Then, for every Banach space X
and every real number p, 1 p < 1,
nðLp ð, XÞÞ ¼ nðlp ðXÞÞ
Proof Let us first prove that nðLp ð, XÞÞ nðlp ðXÞÞ. For this we adapt the proof due to
Javier and Martin for the scalar case (unpublished result). Indeed, if is not atomic,
Lp ð, XÞ is isometric to Lp ð, XÞ p Lp ð, XÞ, so they have the same numerical index.
Let T ¼ ðT1 , T2 Þ 2 Bðl2p ðXÞÞ and define the operator S on Lp ð, XÞ p Lp ð, XÞ by
Sð f1 , f2 Þð!Þ ¼ Tð f1 ð!Þ, f2 ð!ÞÞ. POne can check easily P
that kTk ¼ kSk. Moreover,
n
1=p
1=p
v(T) ¼ v(S). Indeed, let f1 ¼ m
x
ð1
=ðA
Þ
Þ,
f
¼
) be simple
2 P i¼1 yi ð1Bi =ðBi Þ
i¼1 i Ai P i
m
n
p
p
p
functions in Lp ð, XÞ with kð f1 , f2 Þk ¼ i¼1 kxi k þ i¼1 kyi k ¼ 1. For each i we
can find
yi in SX such that xi ðxP
i Þ ¼ kxi k and yi ðyi Þ ¼ kyi k. If we set
Pm xi and
n
1=q
p1 p1 g1 ¼ i¼1 kxi k xi ð1Ai =ðAi Þ Þ and g2 ¼ i¼1 kyi k yi ð1Bi =ðBi Þ1=q Þ, we have
clearly ðg1 , g2 Þ 2 SLq ð, X Þq Lq ð, X Þ and < ðg1 , g2 Þ, ð f1 , f2 Þ >¼ 1. Moreover,
Z
jðg1 , g2 ÞðSð f1 , f2 ÞÞj jðg1 ð!Þ, g2 ð!ÞÞðTð f1 ð!Þ, f2 ð!ÞÞÞjdð!Þ
Z
vðTÞ k f1 ð!Þkp þ k f2 ð!Þkp dð!Þ ¼ vðTÞ:
Following [4], we have vðSÞ vðTÞ. For the reverse inequality, let ðx1 , x2 Þ 2 Sl2p ðXÞ . Take
A 2 with ðAÞ > 0 and consider ð f1 , f2 Þ ¼ ðx1 ð1A =ðAÞ1=p Þ, x2 ð1A =ðAÞ1=p ÞÞ. From
what we have just seen ðg1 , g2 Þ ¼ ðkx1 kp1 x1 ð1A =ðAÞ1=q Þ, kx2 kp1 x2 ð1A =ðAÞ1=q ÞÞ 2
SLq ð, X Þq Lq ð, X Þ and < ðg1 , g2 Þ, ð f1 , f2 Þ >¼ 1. Moreover,
Z
ðkx1 kp1 x , kx2 kp1 x ÞðTðx1 , x2 ÞÞ ¼ ðg1 ð!Þ, g2 ð!ÞÞSð f1 , f2 Þð!Þdð!Þ vðSÞ
1
2
This yields vðTÞ vðSÞ. Consequently fvðTÞ: T 2 Sl2p ðXÞ g fvðSÞ: S 2 SLp ð, XÞp Lp ð, XÞ g
which yields nðLp ð, XÞ p Lp ð, XÞÞ nðl2p ðXÞÞ. So
nðLp ð, XÞÞ nðl2p ðXÞÞ
511
Numerical index of vector-valued function spaces
Now, for any integer m 1, with the same work as above, we obtain
nðLp ð, XÞÞ nðlpm ðXÞÞ
It follows from Theorem 2.1 that
nðLp ð, XÞÞ nðlp ðXÞÞ
If is atomic then Lp ð, XÞ is isometric to Lp ð, XÞ p i2I X lp for a suitable set I
and an atomless measure . With the help of Remark 2 [9], we also have
nðLp ð, XÞÞ nðlp ðXÞÞ. The reverse inequality nðLp ð, XÞÞ nðlp ðXÞÞ follows with the
same technique used in [5] for the scalar case.
g
COROLLARY 2.3 Let ð, , Þ be a -finite measure space. Then, for every Banach space
X and every real number p, 1 p < 1
nðLp ð, XÞÞ ¼ lim n lpm ðXÞ m
3. On the positivity of the numerical index of lp-space
It was proved that the numerical index of lpm , p 6¼ 2, m ¼ 1, 2, . . . cannot be equal to 0
this is equivalent to that the numerical radius and the operator norm are equivalent on
Bðlpm Þ, p 6¼ 2 (see Theorem 2.3 [6]). In this section we shall also prove that both norms
are equivalent on Bðlp , lpm Þ.
THEOREM 3.1 For every real number p 1, p 6¼ 2 and every integer m, the numerical
radius is equivalent to the operator norm on Bðlp , lpm Þ.
Here lp is real and lpm is identified with its natural embedding in lp.
Proof
Let T ¼ ðtik Þ 2 Bðlp , lpm Þ. We first have
!1=q
!1=q 1 1 X
X
q
q
t1k tmk kTk ,...,
k¼1
k¼1
p
1
X
k¼1
!1=q
jt1k jq
1 X
tmk q
þ þ
!1=q
:
k¼1
j
m
j
j
Consider
P fT g 2 Bðlp , lp Þ defined by T ek ¼ Tek for k 6¼ j and T ðej Þ ¼ 0. Then for
x¼ 1
x
e
2
S
we
have
lp
k¼1 k k
xx ðT1 xÞ ¼ "1 jx1 jp1
1
X
k¼2
t2k xk þ þ "m jxm jp1
1
X
k¼2
tmk xk
ð"j 2 f1, 1gÞ
512
E. Ed-dari et al.
Take x1 ¼ "1 21=p with "1 2 f1, 1g we obtain
!
1
1 1=q X
x ðT xÞ ¼ 2
t1k xk
x
k¼2
(
)
1
1
X
X
þ "1 "2 jx2 jp1
t2k xk þ þ "m jxm jp1
tmk xk vðT1 Þ
k¼2
k¼2
Since "1 is arbitrary in f1, 1g then
2
1
1
1
X
X
X
p1
p1
t1k xk þ "2 jx2 j
t2k xk þ þ "m jxm j
tmk xk vðT1 Þ:
k¼2
k¼2
k¼2
1=q And in particular
2
1
X
t1k xk vðT1 Þ
k¼2
1=q for all ðx2 , . . . , xm , . . .Þ 2 lp such that
1
X
1
jxk jp ¼ :
2
k¼2
That is
1
1 X
t1k yk vðT1 Þ
2 k¼2
8ðy2 , . . . , ym , . . .Þ 2 Slp
which yields
1=q
1X
jt1k jq
vðT1 Þ:
2 k6¼1
The same work as above shows that
X
1
2
!1=q
q
jtjk j
vðTj Þ
k6¼j
for j ¼ 1, 2, . . . , m: Now let Rj ¼ T T j then we have
vðT j Þ vðTÞ þ kRj k:
And following (*) we obtain
1
X
k¼1
!1=q
jtjk j
q
2 vðTÞ þ kRj k þ jtjj j
ðÞ
Numerical index of vector-valued function spaces
513
which yields
kTk 2mvðTÞ þ 2
m
X
j¼1
kRj k þ
m
X
jtjj j:
j¼1
Now let fTn g be a v-Cauchy sequence in Bðlp , lpm Þ. Since vðTn Pm Þ ¼ vðPm Tn Pm Þ vðTn Þ
where Pm is the operator projection on lpm (see [5] p. 4), and using the fact that in finite
dimensional space lpm both norms are equivalent, then each Rjn ¼ Tn T jn converges in
operator norm to some Rj. Therefore fTn g is kk-Cauchy. This completes the proof of the
Theorem 3.1.
g
It is still unknown if the numerical radius and the operator norm are equivalent on
the Banach space Bðlp Þ, p 6¼ 2 which gives a complete answer to the question of
C. Finet and D. Li.
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