Numerical Index and Daugavet Property of Operator Ideals and Tensor Products

NUMERICAL INDEX AND DAUGAVET PROPERTY OF OPERATOR IDEALS AND TENSOR PRODUCTS MIGUEL MARTÍN, JAVIER MERÍ, AND ALICIA QUERO Abstract. We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain and the range. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain, and this result provides interesting examples. We also show that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Finally, we show that if a projective tensor product of two Banach spaces has the Daugavet property and the unit ball of one of the factor is slicely countably determined or its dual contains a point of Fréchet differentiability of the norm, then the other factor inherits the Daugavet property. If an injective tensor product of two Banach spaces has the Daugavet property and one of the factors contains a point of Fréchet differentiability of the norm, then the other factor has the Daugavet property. 1. Introduction The numerical index of a Banach space is a constant that relates the numerical radius and the norm of bounded linear operators on the space. It was introduced by G. Lumer in 1968 (see [7]). Let us present the needed definitions and notation. Given a Banach space X, we write SXand BXto denote, respectively, the unit sphere and the closed unit ball of the space. By X∗we denote the topological dual of Xand L(X) will denote the Banach space of all bounded linear operators on X. The numerical range of an operator T∈ L(X)is the set of scalars given by V(T) := {x∗(Tx): x∈SX, x∗∈SX∗, x∗(x)=1}, and the numerical radius of Tis then given by v(T) := sup{|λ|:λ∈V(T)}. It is clear that the numerical radius is a seminorm on L(X)which is not greater than the operator norm. Very often, the numerical radius is actually an equivalent norm on L(X)and to quantify this fact it is used the numerical index of the space X: n(X) := inf{v(T): T∈ L(X),kTk= 1} = max{k⩾0: kkTk⩽v(T)∀T∈ L(X)}. It is clear that 0⩽n(X)⩽1; the value n(X)=1means that the numerical radius and the norm coincide, while n(X) = 0 when the numerical radius is not an equivalent norm on L(X). We refer the reader to the expositive paper [12], to Chapter 1of the recent book [10], and to Subsection 1.1 of the very recent paper [14]. Some results on numerical index which we would like to emphasize are the following. For every Banach space X,n(X∗)⩽n(X)and the inequality can be strict; n(c0) = n(`1) = n(`∞)=1, a result which is also valid for all L- and M-spaces, the disk algebra, and H∞. The numerical index behaves differently when dealing with real or complex Banach spaces. For instance, Hilbert spaces of dimension greater than or equal to two have numerical index 0in the real case and 1/2in the complex case. In general, if Xis a complex Date: May 22nd, 2020. 2010 Mathematics Subject Classification. Primary 46B04, 47A12; Secondary 46B20, 46B28, 47B07. Key words and phrases. Banach space; numerical index; numerical range; numerical radius; operator ideal; projective and injective tensor product; Daugavet property; slicely countably determined sets and operators. Research partially supported by projects PGC2018-093794-B-I00 (MCIU/AEI/FEDER, UE) and FQM-185 (Junta de Andalucía/FEDER, UE). The third author is also supported by the Ph.D. scholarship FPU18/03057 (MECD). 1 arXiv:2005.12773v1 [math.FA] 26 May 2020 2 M. MARTÍN, J. MERÍ, AND A. QUERO Banach space, then n(X)⩾1/eand all the values in the interval [1/e,1] are valid; for real Banach spaces, there is no restriction and all the values of the interval [0,1] are possible. The numerical index of Lpspaces for 1<p<∞,p6= 2, is still unknown, but it is known that n(Lp(µ)) >0in the real case for p6= 2. All these results can be found in the cited papers [10,12,14]. Some recent results can be found in [18], where the exact value of some two-dimensional `pspaces is calculated, and in [1,2,22], for instance. Different extensions of the concept of numerical index appear in [11] and [25]. There is a property somehow related to the numerical index called Daugavet property. A Banach space Xhas the Daugavet property [13] if the norm equality (DE) kId +Tk= 1 + kTk holds for all rank-one operators T∈ L(X)and, in this case, the same happens for all weakly compact operators on X. Examples of Banach spaces satisfying this property are L1(µ, Y )when the positive measure µis atomless and Yis arbitrary, C(K, Y )when the compact space Kis perfect and Yis arbitrary, or the disk algebra. Let us say that there is a relation between the Daugavet property and the numerical range of operators: an operator Tsatisfies (DE) if and only if sup Re V(T) = kTk(see [7] for instance). Classical references for Daugavet property include [13,24,26]. For very recent results, we refer the reader to [5,19], for instance. To state the results of the paper, we need to introduce some definitions and notation. Given Banach spaces Xand Y, we write L(X, Y ),K(X, Y ),W(X, Y ), and A(X, Y )to denote, respectively, the space of (bounded linear) operators, compact operators, weakly compact operators, and approximable operators (i.e. norm limits of finite rank operators), all of them endowed with the operator norm. Finally, we consider the space of all nuclear operators: an operator T:X−→ Ybetween Banach spaces is called nuclear if there exist x∗ n∈Xand yn∈Yfor every n∈Nsuch that P∞ n=1 kx∗ nk kynk<∞and Tx = ∞ X n=1 x∗ n(x)yn(x∈X). The space of all nuclear operators, denoted by N(X, Y ), is a Banach space endowed with the norm N(T) = inf (∞ X n=1 kx∗ nk kynk:Tx = ∞ X n=1 x∗ n(x)yn), where the infimum is taken over all the representations of Tas above. The projective tensor product of X and Y, denoted by Xˆ ⊗πY, is the completion of X⊗Yunder the norm given by kukπ= inf (n X i=1 kxik kyik:u= n X i=1 xi⊗yi), where the infimum is taken over all the representations of u=Pn i=1 xi⊗yi. It follows from the definition that BXˆ ⊗πY= conv(BX⊗BY). The projective tensor product of two operators S∈ L(X, W)and T∈ L(Y, Z) between Banach spaces, denoted by S⊗πT, is the unique operator between Xˆ ⊗πYand Wˆ ⊗πZsuch that (S⊗πT)(x⊗y) = Sx ⊗Ty for every x∈Xand y∈Y, which also satisfies that kS⊗πTk=kSk kTk. The injective tensor product of Xand Y, denoted by Xˆ ⊗εY, is the completion of X⊗Yunder the norm given by kukε= sup ( n X i=1 x∗(xi)y∗(yi) :x∗∈BX∗, y∗∈BY∗), where Pn i=1 xi⊗yiis any representation of u. The injective tensor product of two operators S∈ L(X, W)and T∈ L(Y, Z)between Banach spaces, denoted by S⊗εT, is the unique operator between Xˆ ⊗εYand Wˆ ⊗εZ such that (S⊗εT)(x⊗y) = Sx⊗Ty for every x∈Xand y∈Y, which also satisfies that kS⊗εTk=kSk kTk. We refer the reader to [9] and [21] for more information and background about ideals of operators and tensor products of Banach spaces. For ideals of operators, we show in Section 2that for every operator ideal Zof L(X, Y )endowed with the operator norm we have that n(Z)⩽min{n(X), n(Y). In the case of compact and weakly compact operators, NUMERICAL INDEX AND DAUGAVET PROPERTY OF OPERATOR IDEALS AND TENSOR PRODUCTS 3 we may improve this inequality to n(K(X, Y )) ⩽min{n(X∗), n(Y)}, n(W(X, Y )) ⩽min{n(X∗), n(Y)}. This result allows us to present some interesting examples as the existence of a real Banach space Xsuch that n(X) = 1 while n(K(X, Y )) = n(W(X, Y )) = 0 for every Banach space Y. In particular, n(X)=1 while n(K(X, X)) = n(W(X, X)) = 0. For tensor products of Banach spaces, we prove in Section 3that the numerical indices of Xˆ ⊗πYand Xˆ ⊗εYare less than or equal to the minimum of n(X)and n(Y). As a consequence, and just using representation theorems, we get some consequences for the space of approximable operators and for the space of nuclear operators: n(A(X, Y )) ⩽min{n(X∗), n(Y)} and, in the case where X∗or Yhas the approximation property, n(N(X, Y )) ⩽min{n(X∗), n(Y)}. Finally, we study in Section 4the Daugavet property of tensor products of Banach spaces. We show that when Xˆ ⊗πYhas the Daugavet property and BYis a slicely countably determined set (see the definition at the beginning of the section), then Xhas the Daugavet property. We also provide with the analogous result in the case where the space Y∗has a point of Fréchet differentiability of the norm. For injective tensor products, we do not know if the result with the hypothesis of slicely countably determined unit ball is true or not, but there is a positive result when the space Yhas a point of Fréchet differentiability of the norm. 2. Numerical index of some operator ideals of L(X, Y ) Given two Banach spaces Xand Y, we first study the relationship between the numerical index of some operator ideals of L(X, Y )and the numerical indices of the spaces Xand Y. Recall that an operator ideal in L(X, Y )is a closed subspace Zcontaining all finite-rank operators and satisfying that ATB ∈ Z whenever A∈ L(Y),T∈ Z, and B∈ L(X). Proposition 2.1. Let X,Ybe Banach spaces, then nL(X, Y )⩽min{n(X), n(Y)}. Moreover, the same happens to every operator ideal Z⩽L(X, Y )endowed with the operator norm, that is, n(Z)⩽ min{n(X), n(Y)}. To give the proof of the proposition, we need the following lemma which is well known and can be deduced, for instance, from [6, Corollary 2.1.2]. Lemma 2.2. Let X1,X2be Banach spaces and suppose that there is an isometric embedding Φ: L(X1)−→ L(X2)satisfying Φ(IdX1) = IdX2. Then, n(X2)⩽n(X1). Proof of Proposition 2.1.We first show that nL(X, Y )⩽n(X). Fixed J∈ L(X), we define the map ΦJ:L(X, Y )−→ L(X, Y )by ΦJ(T) = T◦Jfor every T∈ L(X, Y )and observe that kΦJk=kJk. Indeed, the inequality kΦJk⩽kJkis evident. To prove the reverse one, given ε > 0, we find xε∈SXsatisfying kJxεk>kJk − εand then we take x∗ ε∈SX∗such that x∗ ε(Jxε) = kJxεk>kJk − ε. We fix y0∈SYand define the rank-one operator Tε∈ L(X, Y )by Tε(x) = x∗ ε(x)y0for every x∈X, which satisfies kTεk= 1 and kΦJ(Tε)k=kTε◦Jk⩾k[Tε◦J](xε)k=kx∗ ε(Jxε)y0k>kJk − ε. Therefore kΦJk⩾kJk, and hence the mapping J7−→ ΦJis an isometric embedding from L(X)to LL(X, Y )carrying IdXto IdL(X,Y ), so the inequality nL(X, Y )⩽n(X)follows by Lemma 2.2. The inequality nL(X, Y )⩽n(Y)can be proved analogously, using ΨS(T) = S◦Tinstead of ΦJ. To prove the moreover part it suffices to observe that if T∈ Z ⊂ L(X, Y )and J∈ L(X), then ΦJ(T) = T◦Jbelongs to Zfor every T∈ Z, as Zis an operator ideal. So the map J7−→ ΦJis an isometric embedding from L(X)to L(Z)carrying IdXto IdZ, and the result follows again by Lemma 2.2. For the inequality involving n(Y), the argument is analogous, considering now that ΨS(T) = S◦T∈ Z for every T∈ Z and so the map S7−→ ΨSis an isometric embedding from L(Y)to L(Z)carrying IdYto IdZ. 4 M. MARTÍN, J. MERÍ, AND A. QUERO We can get a stronger result for the numerical indices of K(X, Y )and W(X, Y ). To do so, we recall that Kw∗(X∗, Y )denotes the space of compact operators that are weak∗-weakly continuous from X∗into Yendowed with the usual operator norm. This space was originally introduced by L. Schwartz [23] as the ε-product of the spaces Xand Y. It is well-known that Kw∗(X∗, Y )≡ Kw∗(Y∗, X)and that K(X, Y )can be identified with Kw∗(X∗∗, Y )using the mapping T7−→ T∗∗. Analogously, Lw∗(X∗, Y )denotes the space of operators that are weak∗-weakly continuous from X∗into Y. Finally, we recall that W(X, Y )can be identified with Lw∗(X∗∗, Y ). We refer the reader to [20,23] for background on this type of spaces. Theorem 2.3. Let X,Ybe Banach spaces, then the following hold: (a) nLw∗(X∗, Y )⩽min{n(X), n(Y)}. (b) nKw∗(X∗, Y )⩽min{n(X), n(Y)}. (c) nW(X, Y )⩽min{n(X∗), n(Y)}. (d) nK(X, Y )⩽min{n(X∗), n(Y)}. Proof. To prove (a), for J∈ L(X)we define the operator ΨJ:Lw∗(X∗, Y )−→ Lw∗(X∗, Y )given by ΨJ(T) = T◦J∗for every T∈ Lw∗(X∗, Y ). Observe that it is well-defined because J∗is weak∗-weak∗ continuous. Moreover, reasoning as in the proof of Proposition 2.1 we get kΨJk=kJk. Therefore, the mapping J7−→ ΨJis an isometric embedding from L(X)to LLw∗(X∗, Y )carrying IdXto IdLw∗(X∗,Y )so the inequality nLw∗(X∗, Y )⩽n(X)follows from Lemma 2.2. The proof of nLw∗(X∗, Y )⩽n(Y)can be done analogously. Indeed, for J∈ L(Y)define the operator ΓJ:Lw∗(X∗, Y )−→ Lw∗(X∗, Y )given by ΓJ(T) = J◦Tfor every T∈ Lw∗(X∗, Y ), which is well-defined because Jis weak-weak continuous. As before, it is easy to check that kΓJk=kJkso the mapping J7−→ ΓJis an isometric embedding from L(Y)to LLw∗(X∗, Y )carrying IdYto IdLw∗(X∗,Y ). Therefore, the inequality nLw∗(X∗, Y )⩽n(Y)follows from Lemma 2.2. Let us prove (b). To show that nKw∗(X∗, Y )⩽n(X)it suffices to observe that ΨJ|Kw∗(X∗,Y ), the restriction of ΨJto Kw∗(X∗, Y ), lies in LKw∗(X∗, Y )and satisfies  ΨJ|Kw∗(X∗,Y ) =kJk. Therefore, the mapping J7−→ ΨJ|Kw∗(X∗,Y )is an isometric embedding from L(Y)to LKw∗(X∗, Y )carrying IdXto IdKw∗(X∗,Y )and Lemma 2.2 gives the result. To prove nKw∗(X∗, Y )⩽n(Y)one can proceed as in (a) or use what we just proved and the identification Kw∗(X∗, Y )≡ Kw∗(Y∗, X). (c) follows from (a) using the identification W(X, Y )≡ Lw∗(X∗∗, Y ). (d) follows from (b) using the identification K(X, Y )≡ Kw∗(X∗∗, Y ). As a consequence of [4, Examples 3.3] and Theorem 2.3 we have the following interesting examples. Examples 2.4. (a) There exists a real Banach space Xwith n(X)=1and nK(X, Y )=nW(X, Y )= 0 for every Banach space Y. In particular, n(X)=1and n(K(X, X)) = n(W(X, X)) = 0.Indeed, the real space Xgiven in [4, Examples 3.3.a] satisfies n(X)=1and n(X∗)=0so nK(X, Y )=nW(X, Y )= 0 for every Yby Theorem 2.3. (b) There exists a complex Banach space Xwith n(X)=1and nK(X, Y )=nW(X, Y )= 1/e for every Banach space Y. In particular, n(X) = 1 and n(K(X, X)) = n(W(X, X)) = 1/e.The complex space Xgiven in [4, Examples 3.3.b] satisfies n(X)=1and n(X∗)=1/e, so it works by Theorem 2.3 and the fact that every complex Banach space has numerical index less than or equal to 1/e. To obtain the analogue of Theorem 2.3 for the numerical index of the space of approximable operators and also to get an analogous result for nuclear operators, we will use their representation as suitable tensor products in the next section. We emphasize a consequence of the results for the case when the ideal spaces have numerical index one. NUMERICAL INDEX AND DAUGAVET PROPERTY OF OPERATOR IDEALS AND TENSOR PRODUCTS 5 Corollary 2.5. Let X,Ybe Banach spaces. (1) If n(L(X, Y )) = 1, then n(X) = n(Y)=1. (2) If n(K(X, Y )) = 1, then n(X∗) = n(Y) = 1. (3) If n(W(X, Y )) = 1, then n(X∗) = n(Y)=1. One may wonder whether the inequalities obtained for the numerical indices of operator ideals are equalities in general. The following example shows that this is not the case, even for finite-dimensional spaces. Example 2.6. There exist finite-dimensional Banach spaces Xand Ywith n(X∗) = n(Y)=1and nL(X, Y )=nF(X, Y )=nK(X, Y )=nW(X, Y )<1.Indeed, consider X=`4 ∞and Y=`4 1, which have numerical index 1, and observe that nL(X, Y )<1by [15, Proposition 2.4, Lemma 3.2]. However there are cases in which the equality holds for the spaces of compact and weakly compact operators. Remark 2.7. Let Kbe a compact Hausdorff space, and let Xbe a Banach space. Then, nK(X, C(K))=nW(X, C(K))=n(X∗). Indeed, the space K(X, C(K)) can be identified with C(K, X∗)(see [8, Theorem VI.7.1]) and we have nC(K, X∗)=n(X∗)by [17, Theorem 5]. The equality nW(X, C(K))=n(X∗)holds by [16, Corollary 3]. In the next result we give other conditions for which the equality is satisfied for the space of compact operators. Proposition 2.8. Let Xbe a Banach space such that n(X∗∗∗)=1and let Zbe an isometric predual of `1. Then the space K(X, Z)∗∗ has numerical index one. Therefore, so do K(X, Z)∗and K(X, Z). In particular, nK(c0)=nK(c0)∗=nK(c0)∗∗=nL(`∞)= 1 and nK(`1, c0)∗∗= 1. Proof. Since Zhas the approximation property, K(X, Z)≡X∗ˆ ⊗εZ. Since Z∗has the approximation property and the Radon-Nikodým property, we can apply [9, Theorem 16.6] to obtain K(X, Z)∗≡(X∗ˆ ⊗εZ)∗≡ X∗∗ ˆ ⊗π`1. Therefore, K(X, Z)∗∗ ≡(X∗∗ ˆ ⊗π`1)∗≡ L(X∗∗, `∞). Now, by using the identification between `∞ and C(βN), where βNis the Stone–Čech compactification of N, and the one between Cw∗(βN, X∗∗∗)and LX∗∗, C(βN)(see [8, Theorem VI.7.1]), we obtain that nL(X∗∗, `∞)=nCw∗(βN, X∗∗∗)⩾n(X∗∗∗)=1, where the inequality is given by [16, Proposition 7]. Then nK(X, Z)∗∗= 1 as desired. The other statements follow straightforwardly.  3. Numerical index of tensor products Our goal here is to study the numerical index of projective and injective tensor products of Banach spaces. It is known that n(Xˆ ⊗εY)and n(Xˆ ⊗πY)cannot be computed as a function of n(X)and n(Y). Indeed, it is shown in [17, Example 10] that there exist Banach spaces Xand Ywith n(X) = n(Y)=1and such that n(Xˆ ⊗εX)<1,n(Yˆ ⊗πY)<1, and n(Xˆ ⊗πX) = n(Yˆ ⊗εY)=1. Therefore, our results will be inequalities, as in the previous section. Our first result on tensor products follows immediately by Proposition 2.1 and the identifications (Xˆ ⊗πY)∗≡ L(X, Y ∗)≡ L(Y, X∗)(see [9, Proposition 3.2], for instance). Corollary 3.1. Let X,Ybe Banach spaces. Then n(Xˆ ⊗πY)∗⩽min{n(X∗), n(Y∗)}. Our main result in this section is the following pair of inequalities. Theorem 3.2. Let X,Ybe Banach spaces. Then the following hold: (a) n(Xˆ ⊗πY)⩽min{n(X), n(Y)}, (b) n(Xˆ ⊗εY)⩽min{n(X), n(Y)}. 6 M. MARTÍN, J. MERÍ, AND A. QUERO We introduce some notation in order to present an interesting tool to calculate numerical radii which we will use in the proof of the theorem. Given a Banach space X,δ > 0, and T∈ L(X), we write vδ(T) := sup|x∗(Tx)|:x∈BX, x∗∈BX∗,Re x∗(x)>1−δ. Lemma 3.3 ([11, Lemma 3.4]).Let Xbe a Banach space. For T∈ L(X), we have that v(T) = inf δ>0vδ(T). Moreover, if A⊂BXsatisfies that conv(A) = BXand B⊂BX∗satisfies that convw∗(B) = BX∗, then the same equality holds if we replace BXand BX∗by Aand Brespectively in the definition of vδ(T), that is, v(T) = inf δ>0sup|x∗(Tx)|:x∈A, x∗∈B, Re x∗(x)>1−δ. Proof of the Theorem 3.2.(a). We prove first n(Xˆ ⊗πY)⩽n(X). Given S∈ L(X)with kSk= 1, we consider the operator T=S⊗πIdY∈ L(Xˆ ⊗πY)which satisfies that kTk=kSkk IdYk= 1. Since BXˆ ⊗πY= conv (BX⊗BY)and (Xˆ ⊗πY)∗=L(Y, X∗), by Lemma 3.3 we can estimate the numerical radius of Tas v(T) = inf δ>0˜vδ(T), where for δ > 0, ˜vδ(T) := sup |hΦ, Tzi| :z∈BX⊗BY,Φ∈BL(Y,X∗),RehΦ, zi>1−δ. Fixed δ > 0, we claim that ˜vδ(T)⩽vδ(S). Indeed, fix z=x⊗y∈BX⊗BYand Φ∈BL(Y,X∗)such that RehΦ, zi= RehΦ(y), xi>1−δ, define x∗= Φ(y)∈BX∗, and observe that Re x∗(x) = RehΦ, zi>1−δ. Then, |hΦ, Tzi| =|hΦ, Sx ⊗yi| =|hΦ(y), Sxi| =|x∗(Sx)|⩽vδ(S) which gives ˜vδ(T)⩽vδ(S). Then we get that v(T)⩽v(S)and, as kTk=kSk= 1, we deduce that n(Xˆ ⊗πY)⩽n(X). By repeating this process using this time the identification (Xˆ ⊗πY)∗≡ L(X, Y ∗), we also obtain that n(Xˆ ⊗πY)⩽n(Y). (b). We prove n(Xˆ ⊗εY)⩽n(X). Given S∈ L(X)with kSk= 1, we consider T=S⊗εIdY∈ L(Xˆ ⊗εY) which satisfies that kTk=kSkk IdYk= 1. Since B(Xˆ ⊗εY)∗= convw∗(BX∗⊗BY∗)and BXˆ ⊗εY={z∈X⊗Y:kzkε⩽1}, we use the following to estimate the numerical radius of T(again by by Lemma 3.3): v(T) = inf δ>0¯vδ(T) where ¯vδ:= sup {|z∗(Tz)|:z∗∈BX∗⊗BY∗, z ∈X⊗Ywith kzkε⩽1,Re z∗(z)>1−δ}. Given δ > 0, we claim that ¯vδ(T)⩽vδ(S). Indeed, fixed z=Pn i=1 xi⊗yi∈X⊗Ywith kzkε⩽1and z∗=x∗ 0⊗y∗ 0∈BX∗⊗BY∗with Re z∗(z) = Re Pn i=1 x∗ 0(xi)y∗ 0(yi)>1−δ, we consider x=Pn i=1 y∗ 0(yi)xi∈BX which satisfies kxk=     n X i=1 y∗ 0(yi)xi     ⩽sup ( n X i=1 y∗ 0(yi)x∗(xi) :x∗∈BX∗)⩽kzkε and Re x∗ 0(x) = Re Pn i=1 x∗ 0(xi)y∗ 0(yi)>1−δ. Hence we can write |z∗(Tz)|= hx∗ 0⊗y∗ 0, n X i=1 Sxi⊗yii = n X i=1 x∗ 0(Sxi)y∗ 0(yi) =|x∗ 0(Sx)|⩽vδ(S). Then, we deduce that ¯vδ(T)⩽vδ(S)as claimed. From this, we get that v(S)⩾v(T)⩾n(Xˆ ⊗εY). Therefore n(Xˆ ⊗εY)⩽n(X). The inequality n(Xˆ ⊗εY)⩽n(Y)follows by symmetry.  Let us observe that it is not possible to improve Theorem 3.2 to get the numerical index of the dual of the factors in the right-hand side. NUMERICAL INDEX AND DAUGAVET PROPERTY OF OPERATOR IDEALS AND TENSOR PRODUCTS 7 Example 3.4. Let X1=C[0,1],X2=L1[0,1] and let Ybe a Banach space with n(Y)=1and n(Y∗)<1 (use [4, Examples 3.3] for instance). Then, X1ˆ ⊗εY≡C([0,1], Y ), so n(X1ˆ ⊗εY) = 1 by [17, Theorem 5], while n(Y∗)<1. On the other hand, X2ˆ ⊗πY≡L1([0,1], Y ), so n(X1ˆ ⊗πY) = 1 by [17, Theorem 8], while n(Y∗)<1. Nevertheless, the next inequality for the numerical index of the dual of an injective tensor product holds. Corollary 3.5. Let X,Ybe Banach spaces. If X∗or Y∗has the approximation property and Xor Yhas the Radon-Nikodým property, then n(Xˆ ⊗εY)∗⩽min {n(X∗), n(Y∗)}. Proof. The result is an immediate consequence of Theorem 3.2 as the identification (Xˆ ⊗εY)∗≡X∗ˆ ⊗πY∗ holds under the hypotheses (see [9, Theorem 16.6]).  The next consequence is an inequality for the numerical index of spaces of approximable operators similar to the one given in Theorem 2.3 for compact and weakly compact operators. Corollary 3.6. Let X,Ybe Banach spaces. Then nA(X, Y )⩽min {n(X∗), n(Y)}. Proof. It follows from Theorem 3.2.b as A(X, Y )≡X∗ˆ ⊗εY(see [9, Examples 4.2]).  For the space of nuclear operators we may also give some interesting inequalities. Corollary 3.7. Let X,Ybe Banach spaces. If either X∗or Yhas the approximation property, then the following hold: (a) nN(X, Y )⩽min{n(X∗), n(Y)}. (b) nN(X, Y )∗⩽min{n(X∗∗), n(Y∗)}. Proof. (a). Since X∗or Yhas the approximation property, we have that N(X, Y )≡X∗ˆ ⊗πY(see [9, Corollary 5.7.1]) and the result follows from Theorem 3.2.a. (b). Corollary 3.1 gives the result using the equality N(X, Y )∗=X∗ˆ ⊗πY∗. Finally, we may give a result analogous to Corollary 2.5 for the results of this section. Corollary 3.8. Let X,Ybe Banach spaces. (1) If n(Xˆ ⊗πY)∗= 1, then n(X∗) = n(Y∗)=1. (2) If n(Xˆ ⊗εY) = 1, then n(X) = n(Y)=1. (3) If n(Xˆ ⊗πY) = 1, then n(X) = n(Y)=1. (4) If nA(X, Y )= 1, then n(X∗) = n(Y)=1. (5) If n(Xˆ ⊗εY)∗= 1, then n(X∗) = n(Y∗)=1. (6) If nN(X, Y )= 1, then n(X∗) = n(Y) = 1. (7) If nN(X, Y )∗= 1, then n(X∗∗) = n(Y∗) = 1. 4. Daugavet property and tensor products In this section we study the relationship between the Daugavet property and tensor products. A sight to Corollary 3.8 may lead to think that an analogous result can be true for the Daugavet property, that is, if Xˆ ⊗πYor Xˆ ⊗εYhas the Daugavet property, do Xand Yinherit this property? The answer is negative in general since, for instance, L1([0,1], Y ) = L1[0,1] ˆ ⊗πYand C([0,1], Y ) = C[0,1]ˆ ⊗εYhave the Daugavet property for every Banach space Y, regardless that Yhas the Daugavet property or not. Our goal here is to show some cases in which the Daugavet property of a tensor product passes to one of the factors. To state our results, we need the definition and basic properties of the concept of slicely countably determined 8 M. MARTÍN, J. MERÍ, AND A. QUERO sets introduced in [3], where we refer for background. Let Abe a bounded subset of a Banach space X. A countable family {Vn:n∈N}of subsets of Ais called determining for Aif the inclusion A⊆conv(B)holds for every subset B⊆Aintersecting all the sets Vn. Recall that a slice of Ais a nonempty intersection of A with an open half space, and for x∗∈X∗and δ > 0, we write Slice(A, x∗, δ) := {x∈A: Re x∗(x)>sup Re x∗(A)−δ}. The set Ais said to be slicely countably determined (SCD in short) if there exists a countable family of slices which is determining for A. Examples of SCD sets are the Radon-Nikodým set and those sets not containing basic sequences equivalent to the basis of `1[3]. A bounded linear operator T:X−→ Ybetween two Banach spaces Xand Yis an SCD-operator if T(BX)is an SCD set, so examples of SCD-operators are the strong Radon-Nikodým ones and those not fixing copies of `1[3]. Finally, let us comment that a set A is SCD if and only if conv(A)is SCD [10, Proposition 7.17]. Consequently, if Ais SCD then so is every set Csatisfying A⊂C⊂conv(A). The main result of this section is the following one which deals with projective tensor products. Theorem 4.1. Let X,Ybe Banach spaces. Suppose that BYis an SCD set and Xˆ ⊗πYhas the Daugavet property. Then, Xhas the Daugavet property. We need the following preliminary result which shows that the projective tensor product of an SCD- operator and a rank-one operator is again an SCD-operator on a projective tensor product. Lemma 4.2. Let X,Ybe Banach spaces, let S∈ L(X)be a rank-one operator and let T∈ L(Y)be an SCD-operator. Then S⊗πT∈ L(Xˆ ⊗πY)is an SCD-operator. Proof. We may and do assume that kSk=kTk= 1. In order to prove that [S⊗πT](BXˆ ⊗πY)is SCD it is enough to prove that S(BX)⊗T(BY)is SCD as S(BX)⊗T(BY)⊂[S⊗πT](BXˆ ⊗πY)=[S⊗πT](conv(BX⊗BY)) ⊂conv (S(BX)⊗T(BY)) . Since Sis a rank-one operator, there exist x0∈SXand Γ⊂Ksuch that S(BX) = Γ{x0}(Γequals either BKor its interior). So we can write S(BX)⊗T(BY) = {x0} ⊗ ΓT(BY). Observe that ΓT(BY)is SCD since T(BY)is SCD and T(BY)⊂ΓT(BY)⊂T(BY). Therefore, for each n∈Nwe can find Vn= Slice(ΓT(BY), y∗ n, εn)such that the sequence {Vn:n∈N} is determining for ΓT(BY). Now fix x∗ 0∈SX∗satisfying Re x∗ 0(x0) = 1 and, for each n∈N, define ϕn∈(Xˆ ⊗πY)∗=L(X, Y ∗)by ϕn(x) = x∗ 0(x)y∗ nfor every x∈X. Let us prove that the slices Sn={x0} ⊗ Vn= Slice ({x0} ⊗ ΓT(BY), ϕn, εn) (n∈N) form a determining sequence for {x0} ⊗ ΓT(BY). Indeed, if B⊆ {x0} ⊗ ΓT(BY)intersects all the Sn, then Bmust be of the form {x0} ⊗ B2with B2⊂ΓT(BY)satisfying B2∩Vn6=∅for every n∈N. Since Vnis determining for ΓT(BY), this implies that ΓT(BY)⊂conv(B2)and thus {x0} ⊗ ΓT(BY)⊂ {x0} ⊗ conv(B2)⊂conv(B) which shows that the sequence {Sn}is determining for {x0} ⊗ ΓT(BY) = S(BX)⊗T(BY). We are ready to show that the Daugavet property passes from the projective tensor product to one of the factors if the other one is SCD. Proof of Theorem 4.1.Fix a rank-one operator S∈ L(X)and consider T=S⊗πIdY∈ L(Xˆ ⊗πY)which satisfies kTk=kSkand is an SCD-operator by Lemma 4.2. Since Xˆ ⊗πYhas the Daugavet property, T satifisfies the Daugavet equation by [3, Corollary 5.9]:  IdXˆ ⊗πY+T = 1 + kTk= 1 + kSk. By the definition of Twe have  IdXˆ ⊗πY+T =k(IdX+S)⊗πIdYk=kIdX+Sk NUMERICAL INDEX AND DAUGAVET PROPERTY OF OPERATOR IDEALS AND TENSOR PRODUCTS 9 and so kIdX+Sk= 1 + kSk, as desired.  We do not know whether the corresponding result for the injective tensor product is true or not. But we have the following positive result in the same line. Proposition 4.3. Let X,Ybe Banach spaces such that Xˆ ⊗εYhas the Daugavet property. Suppose that the norm of Yis Fréchet differentiable at a point y0∈SY. Then, Xhas the Daugavet property. We need the following characterization of the Daugavet property which appears in the seminal paper [13]. Lemma 4.4 ([13, Lemma 2.2]).Let Xbe a Banach space. Then the following assertions are equivalent: (i) Xhas the Daugavet property; (ii) for every x∈SX,x∗∈SX∗and ε > 0, there is y∈Slice(SX, x∗, ε)such that kx+yk>2−ε; (iii) for every x∈SX,x∗∈SX∗and ε > 0, there is y∗∈Slice(SX∗, x, ε)such that kx∗+y∗k>2−ε. Proof of Proposition 4.3.Since the norm of Yis Fréchet differentiable at y0∈SY, there is a unique y∗ 0∈SY∗ which is strongly exposed in BY∗by y0, that is, (4.1) ∀ε > 0∃δ > 0: y∗∈BY∗,Re y∗(y0)>1−δ=⇒ ky∗ 0−y∗k< ε. Given x∗ 0∈SX∗and x0∈BX, we consider u0=x0⊗y0∈BXˆ ⊗εYand ϕ0=x∗ 0⊗y∗ 0∈S(Xˆ ⊗εY)∗. Since Xˆ ⊗εYhas the Daugavet property, by Lemma 4.4, fixed ε > 0, we may find ϕ∈Slice B(Xˆ ⊗εY)∗, u0, δsuch that kϕ0+ϕk>2−ε. As B(Xˆ ⊗εY)∗= convw∗(BX∗⊗BY∗), we may suppose that ϕ=x∗⊗y∗with x∗∈BX∗ and y∗∈BY∗. On the one hand, from ϕ∈Slice B(Xˆ ⊗εY)∗, u0, δit follows that x∗∈Slice(BX∗, x0, δ)and y∗∈Slice(BY∗, y0, δ). On the other hand, we can write 2−ε < kϕ0+ϕk⩽kx∗ 0⊗y∗ 0+x∗⊗y∗ 0k+kx∗⊗y∗ 0−x∗⊗y∗k⩽kx∗ 0+x∗k+ky∗ 0−y∗k. But ky∗ 0−y∗k< ε by (4.1), so we deduce that kx∗ 0+x∗k>2−2ε. Now, Xhas the Daugavet property by Lemma 4.4. We can obtain a result similar to the previous one for the projective tensor product which does not follow from Theorem 4.1. Proposition 4.5. Let X,Ybe Banach spaces such that Xˆ ⊗πYhas the Daugavet property. Suppose that the norm of Y∗is Fréchet differentiable at a point y∗ 0∈SY∗. Then, Xhas the Daugavet property. Proof. Since y∗ 0∈SY∗is a point of Fréchet differentiability, there is a unique y0∈SYsatisfying: (4.2) ∀ε > 0∃δ > 0: y∈BY,Re y∗ 0(y)>1−δ=⇒ ky0−yk< ε. Given x0∈SXand x∗ 0∈BX∗, we consider u0=x0⊗y0∈SXˆ ⊗πYand ϕ0=x∗ 0⊗y∗ 0∈B(Xˆ ⊗πY)∗. Since Xˆ ⊗πY has the Daugavet property and BXˆ ⊗πY= conv(BX⊗BY), fixed ε > 0, we may find u∈Slice BXˆ ⊗πY, ϕ0, δ of the form u=x⊗ywith x∈BXand y∈BYsuch that ku0+uk>2−ε. On the one hand, from u∈Slice BXˆ ⊗πY, ϕ0, δit follows that x∈Slice(BX, x∗ 0, δ)and y∈Slice(BY, y∗ 0, δ). On the other hand, we have 2−ε < ku0+uk⩽kx0⊗y0+x⊗y0k+kx⊗y−x⊗y0k⩽kx0+xk+ky−y0k. But ky−y0k< ε by (4.2), so kx0+xk>2−2ε. Now, Xhas the Daugavet property by Lemma 4.4. Acknowledgement: The authors thank Abraham Rueda Zoca for many conversations on the topic of this manuscript.