ON THE NUMERICAL INDEX OF REAL Lp (µ)-SPACES

ON THE NUMERICAL INDEX OF REAL Lp (µ)-SPACES

arXiv:0903.2704v2 [math.FA] 29 Jan 2010

MIGUEL MARTÍN, JAVIER MERÍ, AND MIKHAIL POPOV

Abstract. We give a lower bound for the numerical index of the real space Lp (µ) showing, in particular, that it is non-zero for p 6= 2. In other words, it is shown that for every bounded linear operator T on the real space Lp (µ), one has   Z Mp kT k sup |x|p−1 sign(x) T x dµ : x ∈ Lp (µ), kxk = 1 > 12 e |tp−1 − t| > 0 for every p 6= 2. It is also shown that for every bounded linear t∈[0,1] 1 + tp operator T on the real space Lp (µ), one has  Z 1 |x|p−1 |T x| dµ : x ∈ Lp (µ), kxk = 1 > sup kT k. 2e where Mp = max

1. Introduction The numerical index of a Banach space is a constant introduced by G. Lumer in 1968 (see [3]) which relates the norm and the numerical radius of (bounded linear) operators on the space. Let us start by recalling the relevant definitions. Given a Banach space X, we will write X ∗ for its topological dual and L(X) for the Banach algebra of all (bounded linear) operators on X. For an operator T ∈ L(X), its numerical radius is defined as v(T ) := sup{|x∗ (T x)| : x∗ ∈ X ∗ , x ∈ X, kx∗ k = kxk = x∗ (x) = 1}, and it is clear that v is a seminorm on L(X) smaller than the operator norm. The numerical index of X is the constant given by n(X) := inf{v(T ) : T ∈ L(X), kT k = 1} or, equivalently, n(X) is the greatest constant k > 0 such that k kT k 6 v(T ) for every T ∈ L(X). Classical references here are the aforementioned paper [3] and the monographs by F. Bonsall and J. Duncan [1, 2] from the seventies. The reader will find the state-of-the-art on the subject in the recent survey paper [7] and references therein. We refer to all these references for background. Let us comment on some results regarding the numerical index which will be relevant in the sequel. First, it is clear that 0 6 n(X) 6 1 for every Banach space X, and n(X) > 0 means that the numerical radius and the operator norm are equivalent on L(X). In the real case, all values in [0, 1] are possible for the numerical index. In the complex case one has 1/ e 6 n(X) 6 1 and all of these values are possible. Let us also mention that n(X ∗ ) 6 n(X), and that the equality does not always hold. Anyhow, when X is a reflexive space, one clearly gets n(X) = n(X ∗ ). Second, there are some classical Banach spaces for which the numerical index has been calculated. For instance, the numerical index
of L1 (µ) is 1, and this property is shared by any of its isometric preduals. In particular, n C(K) = 1 for every compact K and n(Y ) = 1 for every finite-codimensional subspace Y of C[0, 1]. If H is a Date: March 10th, 2009. Revised September 28th, 2009. Correction November 26th, 2009. 2000 Mathematics Subject Classification. 46B04, 46E30, 47A12. Key words and phrases. numerical radius, numerical index, Lp -spaces. First and second authors partially supported by Spanish MEC and FEDER project no. MTM2006-04837 and Junta de Andalucía and FEDER grants FQM-185 and P06-FQM-01438. Third author supported by Junta de Andalucía and FEDER grant P06-FQM-01438 and by Ukr. Derzh. Tema N 0103Y001103. 1

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MARTÍN, MERÍ, AND POPOV

Hilbert space of dimension greater than one then n(H) = 0 in the real case and n(H) = 1/2 in the complex case. Let (Ω, Σ, µ) be a measure space and 1 < p < ∞. We write Lp (µ) for the real or complex Banach space of measurable scalar functions x defined on Ω such that kxkp :=

Z

p

|x| dµ Ω

 p1

< ∞.

We use the notation ℓm p for the m-dimensional Lp -space. For A ∈ Σ, χA denotes the characteristic function of the set A. We write q = p/(p − 1) for the conjugate exponent to p and |tp−1 − t| |tp−1 − t| = max , t>1 1 + tp t∈[0,1] 1 + tp

Mp := max

(which is the numerical radius of the operator T (x, y) = (−y, x) defined on the real space ℓ2p , see [8, Lemma 2] for instance). The problem of computing the numerical indexof the Lp -spaces was posed for the first time in the seminal paper [3, p. 488]. There it is proved that n(ℓ2p ) : 1 < p < ∞ = [0, 1[ in the real case, even though the exact computation of n(ℓ2p ) is not achieved for p 6= 2 (even now!). Recently, some results have been obtained on the numerical index of the Lp -spaces [4, 5, 6, 8, 9].
(a) The sequence n(ℓm ) m∈N is decreasing. p

every measure µ such that dim Lp (µ) = ∞. (b) n Lp (µ) = inf{n(ℓm p ): m ∈ N} for  1 1 (c) In the real case, max , Mp 6 n(ℓ2p ) 6 Mp . 21/p 21/q (d) In the real case, n(ℓm p ) > 0 for p 6= 2 and m ∈ N. The aim of this paper is to give a lower estimation for the numerical index of the real Lp -spaces. Concretely, it is proved that
Mp . n Lp (µ) > 12 e

(1)

As Mp > 0 for p 6= 2, this extends item (d) for infinite-dimensional
real Lp -spaces, meaning that the numerical radius and the operator norm are equivalent on L Lp (µ) for every p 6= 2 and every positive measure µ. This answers in the positive a question raised by C. Finet and D. Li (see [5, 6]) also posed in [7, Problem 1].
The key idea to get this result is to define a new seminorm on L Lp (µ) which is in between the numerical radius and the operator norm, and to get constants of equivalence between these three seminorms. Let us give the corresponding definitions. For any x ∈ Lp (µ), we denote x# =

(

|x|p−1 sign(x) |x|p−1 sign(x)

in the real case, in the complex case,

which is the unique element in Lq (µ) such that kxkpp = kx# kqq

and

Z

Ω

x x# dµ = kxkp kx# kq = kxkpp .


With this notation, for T ∈ L Lp (µ) one has  Z  # v(T ) = sup x T x dµ : x ∈ Lp (µ), kxkp = 1 . Ω

ON THE NUMERICAL INDEX OF REAL Lp (µ)-SPACES

3


Here is our new definition. Given an operator T ∈ L Lp (µ) , the absolute numerical radius of T is given by Z  # |v|(T ) := sup |x T x| dµ : x ∈ Lp (µ), kxkp = 1 Z Ω  p−1 = sup |x| |T x| dµ : x ∈ Lp (µ), kxkp = 1 Ω

Obviously,

v(T ) 6 |v|(T ) 6 kT k



T ∈ L Lp (µ) .

Given an operator T on the real space Lp (µ), we will show that


n LC Mp p (µ) |v|(T ) and |v|(T ) > kT k , v(T ) > 6 2

C where n LC p (µ) is the numerical index of the complex space Lp (µ). Since n Lp (µ) > 1/ e (as for any complex space, see [1, Theorem 4.1]), the above two inequalities together give, in particular, the inequality (1). 2. The results We start proving that the numerical radius is bounded from below by some multiple of the absolute numerical radius. Theorem 1. Let 1 < p < ∞ and let µ be a positive meassure. Then, every bounded linear operator T on the real space Lp (µ) satisfies Mp v(T ) > |v|(T ), 6 |tp−1 − t| . where Mp = max t>1 1 + tp Proof. Since |v| is a seminorm, we may and do assume that kT k = 1. Suppose that |v|(T ) > 0 (otherwise there is nothing to prove), fix any 0 < ε < |v|(T ) and choose x ∈ Lp (µ) with kxk = 1 such that Z def

|x# T x| dµ > |v|(T ) − ε = 2β0 > 0.

Ω

Now, set A = {t ∈ Ω : x# (t)[T x](t) > 0} and B = Ω \ A. Then Z Z Z # # |x# T x| dµ > 2β0 x T x dµ = x T x dµ − Ω

B

A

and so at least one of the summands above is greater than or equal to β0 . Without loss of generality, we assume that Z def β = x# T x dµ > β0 A

(otherwise we consider −T instead of T ). Remark that Z Z

# and (2) x# T (xχB ) dµ 6 (xχB )# q kxχB kp v(T ) 6 v(T ). x T x dµ 6 v(T ) B

Ω

Now, put yλ = x + λxχB for each λ ∈ [−1, ∞). Observe that Z Z  (3) kyλ# kq kyλ kp = kyλ kpp = |x|p dµ + (1 + λ)p |x|p dµ 6 max 1, (1 + λ)p , A

B

which obviously implies that Z



(4) yλ# T yλ dµ 6 v(T ) yλ# kyλ kp 6 v(T ) max 1, (1 + λ)p . Ω

q

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MARTÍN, MERÍ, AND POPOV

On the other hand, using that yλ# = x# χA + (1 + λ)p−1 x# χB and (2), we deduce that Z Z Z Z # # p−1 # p−1 yλ T yλ dµ = β + λ x T (xχB ) dµ + (1 + λ) x T x dµ + λ(1 + λ) x# T (xχB ) dµ Ω B B ZA # p−1 > β + λ x T (xχB ) dµ − (1 + λ) β A Z Z − (1 + λ)p−1 x# T x dµ − |λ|(1 + λ)p−1 x# T (xχB ) dµ Ω B Z

p−1 x# T (xχB ) dµ − 1 + |λ| (1 + λ)p−1 v(T ). β+λ > 1 − (1 + λ) A

This, together with (4), gives us that Z   p−1 p p−1 x# T (xχB )dµ . (5) v(T ) (1 + |λ|)(1 + λ) + max{1, (1 + λ) } > (1 − (1 + λ) )β + λ A Z Therefore, putting a = β −1 x# T (xχB ) dµ and A

f (λ) = |λ|−1




 1 + |λ| (1 + λ)p−1 + max 1, (1 + λ)p

and multiplying (5) by |λ|−1 β −1 , we obtain that 1 − (1 + λ)p−1 − a β −1 v(T )f (λ) > λ


λ ∈ [−1, ∞) \ {0} ,

for every λ ∈ [−1, ∞) \ {0}. Thus,

β −1 v(T ) 1 + f (λ) = β −1 v(T ) f (−1) + f (λ) (1 + λ)p−1 − 1 1 − (1 + λ)p−1 − a > − 1 > −1 − a + λ λ

for every λ ∈ [−1, ∞) \ {0} or, equivalently,

(1 + λ)p−1 − 1 − λ
 v(T ) > β |λ| + 1 + |λ| (1 + λ)p−1 + max 1, (1 + λ)p

for every λ ∈ [−1, ∞). Now we restrict ourselves to λ > 0 and setting t = 1 + λ, we obtain that |tp−1 − t| |tp−1 − t| 1 + tp = β t − 1 + 2tp 1 + tp t − 1 + 2tp for every t ∈ [1, ∞). Since it obviously holds that 1 1 + tp > p t − 1 + 2t 3 for each t ∈ [1, ∞), one obtains that v(T ) > β

v(T ) >

|v|(T ) − ε |v|(T ) − ε |tp−1 − t| |tp−1 − t| β > = sup sup Mp , p 3 t>1 1 + t 6 1 + tp 6 t>1

which is enough in view of the arbitrariness of ε.



Our next goal is to prove an inequality relating the absolute numerical radius and the norm of operators on real Lp -spaces. Theorem 2. Let 1 < p < ∞ and let µ be a positive measure. Then, every bounded linear operator T on the real space Lp (µ) satisfies
n LC p (µ) |v|(T ) > kT k, 2
C where n Lp (µ) is the numerical index of the complex space Lp (µ).

ON THE NUMERICAL INDEX OF REAL Lp (µ)-SPACES

5


Proof. We consider the complex linear operator TC ∈ L LC p (µ) given by
(6) TC (x) = T (Re x) + i T (Im x) x ∈ LC p (µ) . Evidently, kT k 6 kTC k. Now, consider any simple function x =

m P

j=1

aj > 0, θj ∈ [0, 2π), the sets A1 , . . . , Am ∈ Σ are pairwise disjoint,

aj eiθj χAj ∈ LC p (µ) where m ∈ N,

m P

j=1

x# ∈ LC q (µ) is given by the formula x# =

m X

apj µ(Aj ) = 1, and observe that

ap−1 e−iθj χAj . j

j=1

Then, writing αj,k =

Z

TC (χAk ) dµ =

Z

T (χAk ) dµ,

Aj

Aj

we obtain that m m m m X X Z X X p−1 iθk −iθj iθk a e α a 6 ap−1 e a e α x# TC (x) dµ = k j,k k j,k j j Ω

j=1

(7)

6

m X

ap−1 j

j=1

62

j=1

k=1

max

k=1

m m X X ak sin(θk ) αj,k ak cos(θk ) αj,k + k=1 m X

(zk )∈[−1,1]m

j=1

k=1

m X a z α ap−1 k k j,k = 2 j k=1

max

!

(zk )∈{−1,1}m

m X j=1 m

where the last equality follows from the convexity of the function f : [−1, 1] m m X X a z α ap−1 f (z1 , . . . , zm ) = . k k j,k j j=1

m X a z α ap−1 k k j,k , j k=1

−→ R defined by

k=1

On the other hand, for any finite sequence (zk ) ∈ {−1, 1}m, putting m X aj zj χAj ∈ Lp (µ), y(zk ) = j=1

one has ky(zk ) k = 1 and that Z Z # y T (y(z ) ) dµ = ℓ (zℓ ) Ω

= =

m m X X a z T (χ ) ap−1 z χ dµ k k A j A j k j

Ω j=1 m XZ

k=1

Aj

m p−1 X ak zk T (χAk ) dµ aj zj

j=1 m X

ap−1 j

j=1

ap−1 j

k=1

j=1

>

m X

This, together with (7), implies that 2|v|(T ) > 2 >2

max

zℓ ∈{−1,1}

max

zℓ ∈{−1,1}

Z

Aj k=1

Z

Z

m X ak zk T χAk dµ

Ω

m X

Aj k=1

m m X X p−1 ak zk αj,k . aj ak zk T χAk dµ = j=1

# y T (y(z ) ) dµ ℓ (zℓ )

m X j=1

m X

ap−1 j

k=1

k=1

Z ak zk αj,k > x# TC (x) dµ . Ω

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MARTÍN, MERÍ, AND POPOV

Since the set of all simple functions is dense in LC p (µ), it follows from [1, Theorem 9.3] that the above inequality implies that

C 2|v|(T ) > v(TC ) > n LC  p (µ) kTC k > n Lp (µ) kT k.
It remains to notice that n LC p (µ) > 1/ e (as happens for any complex Banach space, see [1, Theorem 4.1]), to get the following consequence from the above two theorems. Corollary 3. Let 1 < p < ∞ and let µ be a positive measure. Then, in the real case, one has

where Mp = max t>1

|tp−1 − t| . 1 + tp


Mp n Lp (µ) > 12 e

Since, clearly, Mp > 0 for p 6= 2, we get the following consequence which answers in the positive a question raised by C. Finet and D. Li (see [5, 6]) also posed in [7, Problem 1].
Corollary 4. Let 1 < p < ∞, p 6= 2 and let µ be a positive measure. Then n Lp (µ) > 0 in
the real case. In other words, the numerical radius and the operator norm are equivalent on L Lp (µ) .
It is a particular case of [6, Theorem 2.2] that n Lp (µ) = inf n(ℓm p ) for every infinite-dimensional m

2 2 Lp (µ)-space. For finite-dimensional spaces, n(ℓm p ) 6 n(ℓp ) for every m > 2 since ℓp is an ℓp -summand m 2 on ℓp and we may use [9, Remark 2.a]. On the other hand, it is clear that n(ℓp ) 6 Mp (since Mp is the numerical radius of a norm-one operator on the real ℓ2p , see [8, Lemma 2] for instance). It then follows that
n Lp (µ) 6 Mp
for every 1 < p < ∞ and every positive measure µ such that dim Lp (µ) > 2. We do not know whether the above inequality is actually an equality.

Acknowledgments: The authors would like to thank Rafael Payá for fruitful conversations concerning the matter of this paper.

References [1] F. F. Bonsall and J. Duncan, Numerical Ranges of operators on normed spaces and of elements of normed algebras, London Math. Soc. Lecture Note Ser. 2, Cambridge, 1971. [2] F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Note Ser. 10, Cambridge, 1973. [3] J. Duncan, C. McGregor, J. Pryce, and A. White, The numerical index of a normed space, J. London Math. Soc. 2 (1970), 481–488. [4] E. Ed-dari, On the numerical index of Banach spaces, Linear Algebra Appl. 403 (2005), 86–96. [5] E. Ed-dari and M. Khamsi, The numerical index of the Lp space, Proc. Amer. Math. Soc. 134 (2006), 2019–2025. [6] E. Ed-dari, M. Khamsi, and A. Aksoy, On the numerical index of vector-valued function spaces, Linear Mult. Algebra 55 (2007), 507–513. [7] V. Kadets, M. Martín, and R. Payá, Recent progress and open questions on the numerical index of Banach spaces, Rev. R. Acad. Cien. Serie A. Mat. 100 (2006), 155–182. [8] M. Martín and J. Merí, A note on the numerical index of the Lp space of dimension two, Linear Mult. Algebra 57 (2009), 201–204. [9] M. Martín, and R. Payá, Numerical index of vector-valued function spaces, Studia Math. 142 (2000), 269–280.

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(Martín & Merí) Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, E-18071 - Granada (SPAIN) E-mail address: [email protected], [email protected] (Popov) Department of Mathematics, Chernivtsi National University, str. Kotsyubyns’koho 2, Chernivtsi, 58012 (Ukraine) E-mail address: [email protected]