ON SEPARABLE BANACH SPACES
RICHARD BECKER
Given two Banach spaces B and E, withB separable, we try and describe the operators fromBtoEthat can be extended to some copy ofC([0,1]) containingB.
AMS 2010 Subject Classification: 46B04, 46B25, 47B38, 47B68.
Key words: separable Banach spaces, extension of operators, factorization of operators.
1. INTRODUCTION
The origin of this paper is the following natural question: Given two Banach spaces B and E, is it possible to describe all the operators fromB to E that can be extended to every Banach space containingB? In this paper we shall deal with the case whereB is contained in the Banach spaceC([0,1]) (C for short).
We shall place ourselves within the framework of the following definition.
B will be always assumed to be separable. (It is a well known fact that there exists a linear map ϕ:B → C such that, for everyx∈B: kϕ(x)k=kxk.)
Definition 1. Let 1 ≤ C < ∞. We denote by F(B, E, C) the set of all operators T : B → E such that there exists a linear map ϕ : B → C, with kxk ≤ kϕ(x)k ≤Ckxk, for everyx∈B, and there existsTϕ:C →E satisfying T =Tϕ◦ϕon B.
IfT ∈ F(B, E, C) we set kTkC = inf{kTϕk}, where the inf is taken over all ϕand all Tϕ as above.
We denote byF(B, E,∞) the union of all the spacesF(B, E, C).
We shall prove in our Theorem 2 that, for 1≤C <∞, F(B, E, C) is a normed space when it is equipped with k kC. We shall also equipF(B, E,∞) with a norm.
We show that, in any case, every 2-summing operator belongs to F (B, E, C) for 1 ≤ C ≤ ∞. We also show that, when E is of cotype 2, the
REV. ROUMAINE MATH. PURES APPL.,56(2011),4, 261–267
spaces F(B, E, C), 1 ≤ C ≤ ∞, are nothing but the space of all 2-summing operators.
Moreover, we discuss various cases: whenE does not containC, whenB is reflexive, when E is of finite cotype, whenE does not contain c0, whenE0 is a L1 space, whenB is a subspace ofc0.
Finally, we introduce a stronger version of our Definition 1 which yields comparable statements and leads to an open question.
2. THE RESULTS
Theorem 2. 1) F(B, E, C) is a normed space when it is equipped with kTkC.
2)For each T ∈ F(B, E, C) thenkTk ≤CkTkC.
3) For 1 ≤ C1 ≤ C2 < ∞ then F(B, E, C1) ⊂ F(B, E, C2) and, if T ∈ F(B, E, C1) thenC2kTkC2 ≤C1kTkC1.
4) F(B, E,∞) is a normed space when it is equipped with kTk∞ = limCkTkC when C → ∞, and kTk ≤ kTk∞.
The Banach spaces associated are denoted byF(B, E, C)for1≤C ≤ ∞, and we keep the notation k kC.
Proof. 1) Let T1, T2 ∈ F(B, E, C). For every ε >0 there existϕi :B → C, with kxk ≤ kϕik ≤ Ckxk, for every x ∈ B, and Tϕi : C → E, such that Ti =Tϕi◦ϕi, and kTikCi ≤ kTϕik+εfori= 1,2.
We defineϕ:B→ C([0,3]) and T :C([0,3])→E by:
•ϕ(x) =ϕ1(x) on the interval [0,1], for x∈B,
•ϕ(x) =ϕ2(x) on the interval [2,3], for x∈B, by abuse of notations,
•ϕ is linear on the interval [1,2].
Thenkxk ≤ kϕ(x)k ≤Ckxk for everyx∈B.
For everyf ∈ C([0,3]) we setTϕ(f) =Tϕ1(f|[0,1])) +Tϕ2(f|[2,3]), with an abuse of notations.
SinceT1+T2 =Tϕ◦ϕonBandkTϕk ≤ kTϕ1k+kTϕ2kthe result follows.
2) The result follows from the relation T =Tϕ◦ϕ.
3) Letϕ1 :B → C such thatkxk ≤ kϕ1(x)k ≤ C1kxk for every x ∈B. Then ϕ2 =C2/C1ϕ1 satisfieskxk ≤ϕ2(x)k ≤C2kxkfor every x∈B.
IfT :B →E andTϕ1 :C →EsatisfiesT =Tϕ1◦ϕ1thenT2=C1/C2Tϕ1 satisfies T =Tϕ2 ◦ϕ2. It follows that kTkC2 ≤C1/C2kTkC1.
4) Follows from 2) and 4) above.
Let us note that some cases are obvious:
1) WhenBcan be embedded inCas a complemented subspace ofC. This is the case when B is a space C(K), where K is an infinite compact metric
space. (By Milutin’s Theorem and Milutin’s Lemma, see [8, H.P. Rosenthal, pp. 1551–1552].)
2) WhenE is anL∞ space, associated to aσ-finite measure, or when E is the space c0 (see [11, M. Zippin, Proposition 1.2, (1.6)]).
In these cases, for every Banach space B, every T : B → E belongs to F(B, E, C) for any C.
Here are some consequences of Theorem 2.
Proposition 3. 1) Every element of F(B, E, C) can be viewed as a continuous operator from B to E, for 1≤C≤ ∞.
2) If T ∈ F(B, E, C) and V ∈ L(E) then V ◦T ∈ F(B, E, C) and kV ◦TkC ≤ kVkkTkC.
3) If E = B the space F(B, B, C) is an algebra, for every 1 ≤ C ≤
∞. When 1 ≤ C < ∞ the map T → CkTkC satisfies CkT1 ◦ T2kC ≤ (CkT1kC)(CkT2kC) for T1, T2 ∈ F(B, B, C).
Moreover, kT1◦T2k∞≤ kT1k∞kT2k∞ for T1, T2∈ F(B, B,∞).
Proof. 1) is a consequence of 2) and 3) of Theorem 2.
2) follows from Definition 1.
3) follows from 2) of Theorem 2.
The following proposition yields information concerning the role of the constantC in the definition ofF(B, E, C).
Proposition 4. Let 1 ≤ C < ∞. If U ∈ L(B) satisfies mkxk ≤ kU(x)k ≤ Mkxk for every x ∈ B, with m ≤ M < ∞, then, for every T ∈ F(B, E, C):
T◦U ∈ F(B, E, CM/m) and kT◦UkCM/m ≤mkTkC. Moreover, mkTk∞≤ kT ◦Uk∞≤MkTk∞.
Proof. Let ϕ : B → C with kxk ≤ kϕ(x)k ≤ Ckxk for x ∈ B. Let Tϕ : C → E with T = Tϕ ◦ϕ. We have T ◦ U = mTϕ ◦(ϕ◦U/m) and kxk ≤ kϕ◦U(x)/mk ≤(CM/m)kxk.
Then,kT ◦UkCM/m≤mkTkC and kT ◦Uk∞≤MkTk∞.
Similarly,kT◦U−1kM C/m≤1/MkTkC, andkTkM C/m≤1/MkT◦UkC, thusmkTk∞≤ kT ◦Uk∞.
Remark 5. The preceding result shows that the space F(B, E,∞) does not change if B is replaced by one of its isomorphic copy.
The next result shows that, whenE is not too big, then the elements of F(B, E, C) are small in some sense.
Proposition 6. If E does not contain a copy of C then, for every T ∈ F(B, E, C) and 1≤C≤ ∞, the space T0(E0) is separable.
Proof. Let 1 ≤ C < ∞, and T ∈ F(B, E, C). By a theorem due to H.P. Rosenthal [9], if T0(E0) were not separable, then Tϕ would fix a subspace ofC isomorphic toC. ThenE would contain a copy ofC, a contradiction. The extension to F and toC =∞ follows from 2) and 4) of Theorem 2.
WhenB is assumed to be reflexive we have:
Proposition7. IfB is reflexive then all the elements of F(B, E, C)are compact operators, for 1≤C≤ ∞.
Proof. IfB is reflexive then the unit ball B1 of B is weakly compact as well as ϕ(B1) and the result follows for the elements ofF(B, E, C), 1≤C <
∞, from a theorem due to N. Dunford and B.J. Pettis [3]. The extension to F and to C =∞ follows from 2) and 4) of Theorem 2.
We recall the definition of summing operators (see [1]).
Definition 8. LetX, Y be two Banach spaces, and 1≤p <∞. A linear operator T : X → Y is said to be p-summing if for every finite sequence x1, . . . , xn inX one has
n
X
1
kT(xi)kp 1/p
≤csup n
X
1
|hx0, xii|p 1/p
:x0 ∈X0, kx0k= 1
, wherecis some constant. The smallest possible constant is denoted byπp(T).
Here is a class of operators which is contained in F(B, E, C) for every 1≤C ≤ ∞.
Proposition 9. Let T ∈ π2(B, E). Then, for every 1 ≤ C ≤ ∞, T ∈ F(B, E, C), and kTk∞≤π2(T), and CkTkC ≤π2(T) for 1≤C <∞.
Proof. By a classical theorem (see [1, Corollary 2.16]) there are a proba- bility space (Ω,A, P), a mapa:B →L∞(Ω,A, P), and a mapb:L2(Ω,A, P)
→ E such that T = bja, where j :L∞ → L2 is the canonical inclusion. We can assume thatkbk=π2(T).
Let ϕ : B → C with kϕ(x)k = kxk for every x ∈ B. By an abuse of notations let ϕ−1 be the map from ϕ(B) to B defined by ϕ−1(ϕ(x)) = x for every x ∈ B. Then the map T1 = bjaϕ−1 is 2-summing on ϕ(B) and π2(T1) =π2(T), sincekϕ−1k= 1.
It is a classical fact, since L∞ is an injective space, that T1 extends to a 2-summing operator, denoted T2, from C toE such thatπ2(T2) =π2(T1) = π2(T). Thus kT2k ≤π2(T), andkT1k ≤π2(T).
The result follows sinceCkTkC ≤ kTk1 for every 1≤C <∞.
Here is a case whereF(B, E, C), can be determined precisely. We recall the definition of the cotype of a Banach space ([1, Chapter 11]).
Definition 10. Let X be a Banach space and 2≤ q < ∞. X is said to be of cotype q if for every finite sequence x1, . . . , xn inX we have
n
X
1
kxikq 1/q
≤c Z
n
X
1
ri(u)xi
q
du 1/q
,
where cis some constant and (ri) is the sequence of Rademacher variables.
LetCq(X) denote the best constant c.
Proposition11. IfE has cotype2then, for every1≤C≤ ∞, the space F(B, E, C) is nothing but the space of all 2-summing operators T : B → E, and CkTkC ≤π2(T)≤ACkTkC, where A is some constant, for 1≤C <∞, and kTk∞≤π2(T)≤AkTk∞.
Proof. Since E has cotype 2, by a theorem due to E. Dubinsky, A. Pel- czynski and H.P. Rosenthal [2], every operator Θ : C → E satisfies π2(Θ) ≤ AkΘk, where A is some constant. From the relation T = Tϕ ◦ ϕ, where T ∈ F(B, E, C), it follows that π2(T)≤Cπ2(Tϕ) ≤ACkTϕk. Thus π2(T)≤ ACkTkC. The result follows by Proposition 9.
Proposition 12. If E is of finite cotype q then every T ∈ F(B, E, C) is r-summing for every r > q and for 1≤C ≤ ∞. Then πr(T) ≤ArCkTkC, where Ar is some constant, for1≤C <∞ and πr(T)≤ArkTk∞.
In particular, every T ∈ F(B, E, C) is weakly compact for every 1≤C ≤ ∞.
Proof. Since E has cotype q, by a theorem due to B. Maurey [6], every operator Θ : C → E satisfies πr(Θ) ≤ ArkΘk, where Ar is some constant.
From the relation T = Tϕ ◦ϕ where T ∈ F(B, E, C) we deduce πr(T) ≤ Cπr(Tϕ)≤ArCkTφkand πr(T)≤ArCkTkC.
It is classical that anr-summing operator is weakly compact. We finish the proof using 2) and 4) of Theorem 2.
Proposition 13. If E does not contain a copy of c0 then every element of F(B, E, C) is weakly compact for 1≤C≤ ∞.
Proof. By a classical theorem, due to A. Pelczynski [7], if an operator defined on C is not weakly compact it fixes a copy of c0. The result follows applying 2) and 4) of Theorem 2.
In some cases we are in a position to determine precisely the normkTkC for some T, B, and E.
Proposition 14. If E0 is an L1 space then every compact operator T : B →E belongs to F(B, E, C), for every C, andkTk1=kTk.
Proof. Letϕ:B →E withkϕ(x)k=kxk for everyx ∈B. By abuse of notations let ϕ−1:ϕ(B)→B the map defined by ϕ1(ϕ(x)) =x.
By a theorem due to J. Lindenstrauss [4], for everyε >0, the mapT◦ϕ−1, defined on ϕ(B), extends to C as a mapTε :C →E withkTεk ≤(1 +ε)kTk.
Since T =Tε◦ϕ the result follows.
Proposition 15. Let B be a subspace of c0 and E =C(K) where K is a compact Hausdorff space. Then every compact operator T :B →E belongs to F(B, E, C), for every C, and kTk ≤ kTk1 ≤2kTk.
Proof. Let ϕ : c0 → C such that kϕ(x)k = kxk for every x ∈ c0. By abuse of notations let ϕ−1 :ϕ(c0)→c0 be the map such that ϕ−1(ϕ(x)) = x for everyx∈c0. By a theorem due to J. Lindenstrauss and A. Pelczynski [5], for every ε > 0, the map T ◦ϕ−1, defined on ϕ(B), extends to an operator Tε:ϕ(c0)→ C(K) such that kTεk ≤(1 +ε)kTk.
By a theorem due to Sobczyk [10], there is a projection p from C onto ϕ(c0) withkpk ≤2. The mapTε◦p:C → C(K) satisfieskTε◦pk ≤2(1 +ε)kTk and T = (T◦p)◦ϕon B. The result follows.
Here is a stronger version of our Definition 1.
Definition 16. Let 1 ≤ C < ∞. We denote by G(B, E, C) the set of all operators T : B → E such that, for every linear map ϕ : B → C, with kxk ≤ kϕ(x)k ≤ Ckxk, for every x ∈ B, there exists Tϕ : C → E with T =Tϕ◦ϕonB and such that sup{kTϕk} ≤ ∞, where the sup is taken on all such ϕ.
We denote by|kTk|C the smallest numberK such that kTϕk ≤K for all such ϕ.
It is easy to see that these spaces are Banach spaces and thatC1 ≤C2 implies G(B, E, C2) ⊂ G(B, E, C1). It is clear that G(B, E,1) ⊂ F(G, E,1).
With the same proofs as above we can prove that:
1) These spaces contain the space of 2-summing operators.
2) When E is of cotype 2 these spaces are identical to the space of 2- summing operators from B toE.
3) WhenE is the space c0 these spaces are identical to the space of all (continuous) operators fromBtoE(see [11, M. Zippin, Proposition 1.2, (1.6)]).
Remark 17. It is not sure that Proposition 15 remains true for these spaces. Indeed, in the proof of this proposition we use an arbitrary map ϕ from c0 to C. Within the framework of the spaces G(B,C(K), C) it would be necessary to take an arbitrary map ϕfrom B toC. So the question is open.
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