Grothendieck operators on tensor products

Abstract

We prove that for Banach spaces E,F,G,H and operators T∈ℒ(E,G), S∈ℒ(F,H) the tensor product T⊗S:E⊗ ε F→G⊗ ε H is a Grothendieck operator, provided T is a Grothendieck operator and S is compact.

Full text

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 125, Number 8, August 1997, Pages 2285–2291
S 0002-9939(97)03763-5
GROTHENDIECK OPERATORS ON TENSOR PRODUCTS
P. DOMA´
NSKI, M. LINDSTR ¨
OM, AND G. SCHL ¨
UCHTERMANN
(Communicated by Palle E. T. Jorgensen)
Abstract. We prove that for Banach spaces E, F,G, H and operators T∈
L(E,G), S∈L(F,H ) the tensor product T⊗S:E⊗εF→G⊗εHis
a Grothendieck operator, provided Tis a Grothendieck operator and Sis
compact.
1. Introduction
J. Diestel and B. Faires proved in ’76 that for Banach spaces E, F, G,H,for
T∈A(E,G)andcompactS∈L(F, H ) the tensor product of Tand Sdefined as
T⊗S:E⊗εF→G⊗εHbelongs again to the operator ideal A,providedAis closed
and injective [DF]. For the ideal of weakly compact operators E. Saksman and H. O.
Tylli [ST] have obtained similar results for both the projective and injective tensor
product.
The mentioned results open a natural, and interesting in itself, question on sta-
bility of non-injective operator ideals with respect to injective tensor products.
We solve this problem for the non-injective, closed ideal of Grothendieck opera-
tors. We are interested in exactly that ideal because the corresponding problem
of tensor stability turns out to be closely related to the question of existence of
complemented copies of c0in injective tensor products even for Fr´echet spaces. In
fact, it was shown [R, p. 98] that for a large class of Banach spaces E(containing
all E=C(K)) we have that Eis a Grothendieck space (that is, weak* and weak
sequential convergence coincide on equicontinuous subsets) if and only if Econtains
no complemented copy of c0.
On the other hand, by a surprising result of Freniche [Fr1] (compare [C]), each
completed injective tensor product E⊗εFof a Fr´echet space Econtaining a copy of
c0and a Fr´echet space Fsatisfying the Josefson-Nissenzweig type theorem (that is,
weak* and strong convergence do not coincide for sequences in the dual) contains
always a complemented copy of c0. A fortiori, such a tensor product cannot be a
Grothendieck space. All infinite dimensional Banach spaces satisfy the Josefson-
Nissenzweig theorem and for Fr´echet spaces it was proved by Bonet, Lindstr¨om and
Valdivia [BLV] that this property exactly characterizes the non-Montel spaces.
This development leads to two natural questions: Let Ebe a Fr´echet space and
FaFr´echet-Montel space. When exactly does E⊗εFcontain a complemented
copy of c0and when exactly is it a Grothendieck space? Both problems can also
be interpreted in terms of tensor stability.
Received by the editors August 29, 1995 and, in revised form, January 9, 1996.
1991 Mathematics Subject Classification. Primary 47A80.
c
1997 American Mathemati cal Society
2285
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2286 P. DOMA ´
NSKI, M. LINDSTR ¨
OM, AND G. SCHL ¨
UCHTERMANN
In case of E=C(K) it follows immediately from results of Freniche [Fr2] (com-
pare [DL, Cor. 3.7]) that C(K, F ) is a Grothendieck space if and only if C(K)isa
Grothendieck space which automatically implies that C(K, F ) contains a comple-
mented copy of c0if and only if C(K) contains a complemented copy of c0.
For general injective tensor products, the known results are contained in [DL,
Th.2.3,Th.3.6]:
(i) if Fhas the approximation property, then E⊗εFcontains a complemented
copy of c0if and only if Econtains a complemented copy of c0.
(ii) if either For E00 has the approximation property, then E⊗εFis a Grothen-
dieck space if and only if Eis a Grothendieck space.
Using our injective tensor stability result for the ideal of Grothendieck operators
we are able to remove the approximation property assumption from the second
result when Eis a Banach space and Fis a Schwartz space.
Let us now fix some notations and definitions. E, F, G,H are Banach spaces.
B(E) stands for the unit ball, while E∗denotes the topological dual. By an operator
Tfrom Einto Fwe mean a bounded linear map. Let us call an operator T∈
L(E,F)approximable if there exists a sequence of finite rank operators (vn)⊂
L(E,F) such that kT−vnkn→∞
→0 (cf. [Jh]). We refer to [Pi], [DU] and [DFl] for
background information on operator ideals, measure theory and tensor products,
respectively.
Definition 1.1. Let E,F be Banach spaces. An operator T∈L(E, F ) is called a
Grothendieck operator if every w∗-null sequence (y∗
n) is mapped by the adjoint T∗
into a weak null sequence (T∗(y∗
n)) ⊂E∗.
The ideal of Grothendieck operators GR(E,F) is not injective, since the inclusion
map ι:c0→∞is Grothendieck (note that ∞is a Grothendieck space, since w∗-
null sequences are weakly null in the dual of ∞). But the identity id :c0→c0is
not Grothendieck, since otherwise it would be weakly compact.
Definition 1.2. A subset K⊂Eis called a Grothendieck set if for all T∈L(E, c0)
the set T(K) is relatively weakly compact in c0.
We have immediately the following result (cf. [DU, p. 179]).
Lemma 1.3. Let E, F be Banach spaces, T∈L(E,F). Then the following condi-
tions are equivalent:
(a) T∈GR(E,F),
(b) ∀S∈L(F, c0):S◦Tis weakly compact,
(c) ∀A⊂Ebou nded :T(A)is a Grothendieck set.
It follows that the ideal GR is surjective and closed.
2. Main results
Let Abe a closed operator ideal and αbe a tensor norm. We define the class
Aαof all operators S:E→Fsuch that for any pair of Banach spaces E1,F
1and
any operator T∈A(E
1
,F
1)themapT⊗S:E
1⊗
αE→F
1⊗
αFbelongs to A
as well. J. Diestel and B. Faires (see [DF, Th. 1 and Th. 2]) proved that Aαis
always a closed operator ideal which is injective whenever Aand αare injective.
Analogously, it is easily seen that if Ais surjective and αis projective, then Aαis
surjective. Thus we obtain immediately:
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GROTHENDIECK OPERATORS ON TENSOR PRODUCTS 2287
Proposition 2.1. Let E, F, G,H be Banach spaces, Abe a closed operator ideal,
αbe a te ns or norm and T∈A(E, G).
(a) If S∈L(F, H )is approximable, then T⊗S:E⊗αF→G⊗αHis again in
A.
(b) If αand Aare injective, then (a) holds even for compact S.
(c) If αis projective and Ais surjective, then (a) holds even for compact S.
Proof. The ideal of approximable operators is the smallest closed operator ideal.
Similarly, the ideal of compact operators is the smallest surjective (injective) closed
operator ideal.
Since GR is surjective, we can state the following result.
Corollary 2.2. Let E,F,G,H be Banach spaces. Then T⊗S:E⊗πF→G⊗πH
belongs to GR whenever Tis Grothendieck and Sis compact.
Remark 2.3.If we apply Theorem 2.2 and Remark 2.9 of [S] the following criterion
of weak compactness in the dual space (E⊗πF)∗=L(E, F ∗) can be obtained:
Let (Tn)⊂L(E, F∗) be a bounded sequence. Then Tn
n→∞
→0weaklyifandonly
if {(hTn(x),y
∗∗i)n∈N;x∈B(E),y
∗∗ ∈B(F∗∗)}⊂c
0is relatively weakly compact.
This result or Theorem 1 in [K] can be used to give a direct proof of the above
corollary. It also follows from the proof that E⊗πFis a Grothendieck space if Eis
a Grothendieck space, Fis reflexive and every operator from Einto F∗is compact.
At this stage we mention that from P. Enflo’s famous example [E] it is an easy
consequence that there is a Banach space Efor which there is a non-approximable
but compact operator from Einto itself. In [A] F. A. Alexander obtained a similar
result for a closed subspace Eof lpwhen 2 <p<∞.
The ideal of Grothendieck operators is not injective. Thus our main aim is
to improve 2.1 in that case and to obtain injective tensor stability with compact
operators. First we reduce the problem to reflexive Fand H.
Lemma 2.4. Let E, F,G,H be Banach spaces, T∈L(E, G)and S∈L(F, H)
is compact. Then there exist reflexive Banach spaces G1,H
1and operators S1∈
L(E,G1),S
2∈L(G
1
,H
1),S
2compact, S3∈L(H
1
,H),suchthat
T⊗S=(idG⊗S3)◦(T⊗S2)◦(idE⊗S1).
Proof. Every compact S∈L(F, H) admits a compact factorization through a re-
flexive Banach space according to a result of T. Figel and W. Johnson [Fi, Jo] (see
also [DU, p. 260]). Then the proof is straightforward.
We write Bo(B(E∗)) for the Borel sets on B(E∗) w.r.t. the w∗-topology. If
m:Bo(B(X∗)) →Fis a vector measure of bounded variation, then kmkis the
variation norm. Let us recall the representation of the dual of E⊗εF,providedF
is reflexive.
Definition and Lemma 2.5. Let E, F be Banach spaces with Freflexive.
PI(E,F)⊂L(E, F)are the Pietsch-integral operators, defined as:
T∈PI(E, F)⇔∃m:Bo(B(E∗)) →Fvector measure of bounded variation
∀x∈E:T(x)=ZB(E
∗
)
x(x
∗
)dm(x∗).
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2288 P. DOMA ´
NSKI, M. LINDSTR ¨
OM, AND G. SCHL ¨
UCHTERMANN
We equip PI(E,F)with the integral norm, i.e. kTkPI := inf{kmk;∀x∈E:T(x)=
RB(E
∗
)x(x
∗
)dm(x∗)}(cf. [DFl, p. 522]).ThenPI(E,F ∗)is isometric isomorphic
to (E⊗εF)∗by the identity T(x⊗y)=hy, RB(E∗)x(x∗)dm(x∗)i(cf. [DFl, p. 522]).
Notation. Let E,Fbe Banach spaces with Freflexive, and let (z∗
n)⊂
B((E⊗εF)∗). According to 2.5 for all n∈Nwe choose a vector measure mn:=
mn(z∗
n):Bo(B(X∗)) →Fof bounded variation, satisfying:
i) limn→∞ |km
nk−kz
∗
n
k|=0,
ii) ∀e∈E,f ∈F, n ∈N:z∗
n(e⊗f)=hf, RB(E∗)e(e∗)dmn(e∗)i.
Furthermore we define a finite scalar-valued measure µ(·):=µ((z∗
n))(·):=
P
n∈N
2
−n
var(mn(z∗
n),·), where var denotes the variation of the corresponding
measure. Then mnis absolutely continuous w.r.t. µfor all n∈N.
We write
LH1:= (f∈L1(µ, H∗
1):∀e∈E:ZB(E
∗
)
e(e
∗
)f(e
∗
)dµ(e∗)=0
)
for a subspace H1⊂H. For a Banach space Hwe denote by qH:L1(µ, H∗)→
L1(µ, H∗)/LHthe canonical quotient map. If µ=µ(z∗
n)andH
1⊂H,thenlet
r
H
1:L
1
(µ, H∗
1)/LH1→(E⊗εH1)∗be the canonical injection.
Theorem 2.6. Let E, F,G,H be Banach spaces. If T∈GR(E,G)and S∈
L(F, H)is compact, then T⊗S:E⊗εF→G⊗εHis Grothendieck.
Proof. By Lemma 2.4 we assume that F, H are reflexive. W.l.o.g. let kTk,kSk≤1.
Let (x∗
n)⊂B((G⊗εH)∗)bew
∗
-converging to 0. First we consider the map
T⊗id :E⊗εH→G⊗εH. For a finite dimensional subspace H1⊂H, according
to 2.1, we have that for (z∗
n):=(T⊗id)∗(x∗
n) the restriction
((z∗
n)|E⊗H1)n→∞
−−−→ 0weakly.(1)
Consider now id ⊗S:E⊗εF→E⊗εH.Forn∈Nlet hn∈L1(µ, H∗)bethe
density of mnwith respect to µ:= µ(z∗
n). To show that (id ⊗S)∗((z∗
n)) is weakly
null (then we are done), we have to show that for all g∈B((E⊗εF)∗∗ ):
hg(id ⊗S)∗(z∗
n)i=ZB(E∗)
hq∗
F◦r∗
F(g),S∗◦h
nidµ
=ZB(E∗)
hS◦q∗
F◦r∗
F(g),h
nidµ →0.
(2)
The following arguments are devoted to proving this. We define g:= S◦q∗
F◦
r∗
F(g). Then g∈L∞(µ, H), since His reflexive. Further, ghas relatively compact
range, since Sis compact. We assume that (2) is not true. Then
∃(hnk) subsequence ∃ε>0 : inf
k∈NZB(E∗)
hg, hnkidµ
>ε.(3)
For the sake of simplicity assume that (hn) satisfies (3). Since ghas relatively
compact range, there is an increasing sequence of finite Bo(B(E∗))-partitions (πk),
such that
kEπk(g)−gk∞→0and∀n∈N:kE
π
k(h
n
)−h
n
k
1→0.(4)
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GROTHENDIECK OPERATORS ON TENSOR PRODUCTS 2289
We define Σ0:= σ(Sk∈Nπk). Since His reflexive, for all k∈Nthe sequence
(Eπk(hn)) is relatively weakly compact in L1(µ, H∗). Hence, for all k∈Nthere
is an mk∈L1(µ, H∗), so that Eπk(hn)→mkweakly (for at least going to a
subsequence by a diagonalization argument). (πk) is increasing, thus, (mk,π
k)isa
bounded martingale, which converges in the L1(µ, H∗)-norm to an M∈L1(µ, H ∗)
(note that the (hn) are bounded and H∗has the RNP as a reflexive space). We
show now that for all G∈L∞(µ|P0,H) with relatively compact range:
∃subsequence (hnj) such that(5)
∀δ>0∃N∈N∀j≥N:|hG, M i−hG, hnji| <δ.
Proof of (5).Ghas relatively compact range, thus there exists an increasing se-
quence of finite Bo(B(E∗),w
∗)-partitions (πk(G)) such that πk⊂σ(πk(G)) for k∈
Nand kG−Eπk(G)(G)k→0 (cf. [DU, p. 67, Lemma 1]). Let (hnj) be a subsequence
with Eπk(G)(hnj)→mk(G)∈L1(µ, H∗) weakly for all k∈N(subsequence argu-
ment like above). Thus, since πk⊂σ(πk(G)), it follows Eπk(Eπk(G)(hnj)) →mk
(mkas above). Again (mk(G)) is a martingale. Hence, as above, there exists an
M(G)∈L1(µ, H∗), such that mk(G)→M(G). We have
M=M(G).
Let A∈Sk∈Nπk.Then
Z
A
M(G)dµ −ZA
Mdµ= lim
k→∞ ZA
mk(G)−mkdµ =0,
since (mk(G)) and (mk) are martingales and there is a k0∈N, such that A∈πk0⊂
σ(πk(G)). Hence, for all B∈Σ0:RBM(G)dµ =RBMdµ.Thustoprove(5)we
first note that it suffices to demonstrate (5) for all G=Eπk(G)G(k∈N), since G
has relatively compact range and M,hn,n∈N, are measurable w.r.t. Σ0.Butthen
(5) follows by:
hEπk(G)G, M i−hE
π
k
(G)
G, hnji=hEπk(G)G, mk(G)i−hE
π
k(G)
G, Eπk(G)hnji
=hEπk(G)G, mk(G)−Eπk(G)hnji.
For a finite dimensional subspace H1⊂Hwe consider the canonical restriction
operator restH1:L1(µ, H∗)/LH∗→L1(µ, H ∗
1)/LH∗
1. Then according to (1) we
have:
∀z∗∗ ∈(E⊗εH1)∗∗ :ZB(E∗)
hq∗
H◦rest∗
H1◦r∗
H1(z∗∗),h
nidµ →0.(6)
Since q∗
H◦rest∗
H1◦r∗
H1(z∗∗) has relatively compact range for all z∗∗ ∈(E⊗εH1)∗∗
(H1is finite dimensional), (5) and (6) imply:
∀z∗∗ ∈(E⊗εH1)∗∗ :hz∗∗,r
H
1◦restH1◦qH(M)i=0.
Note that since M∈L1(µ|Σ0,H∗
) we may assume that q∗
H◦rest∗
H1◦r∗
H1(z∗∗)is
measurable w.r.t. Σ0.Thus
∀H
1⊂Hfinite dimensional rH1◦restH1◦qH(M)=0.(7)
But (7) implies
rH◦qH(M)=0.(8)
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2290 P. DOMA ´
NSKI, M. LINDSTR ¨
OM, AND G. SCHL ¨
UCHTERMANN
Hence we compute
0(8)
=h(id ⊗S)∗∗(g),r
H◦q
H(M)i=hg, (id ⊗S)∗(rH◦qH(M))i
=ZB(E∗)
hq∗
F◦r∗
F(g),S∗◦Midµ =ZB(E∗)
hS◦q∗
F◦r∗
F(g),Midµ
=ZB(E∗)
hg, Midµ.
Thus, this contradicts (3) and (5), and we are done.
We shall now apply Theorem 2.6 and an operator ideal approach to obtain the
announced result avoiding the assumption of the approximation property.
Corollary 2.7. Let Ebe a Schwartz space and Fa Banach space with the Grothen-
dieck property. Then E⊗εFis a Grothendieck space.
Proof. By a well-known representation of ε-tensor products as projective limits
E⊗εF=projU∈UEEU⊗εF,whereU
Eis a 0-basis in E. A locally convex space
Xis Grothendieck if and only if every continuous linear map from Xinto c0maps
bounded sets into relatively weakly compact ones. Now, each continuous linear
map T:E⊗εF→c0factorizes through EU⊗εFfor some U∈U
E
.SinceEis
a Schwartz space we can apply our main theorem so that for every U∈U
Ethere
exists a V∈U
Econtained in Usuch that the canonical map EV⊗εF→EU⊗εF
is a Grothendieck operator. The result follows immediately.
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Department of Mathematics, A. Mickiewicz University, 60-769 Pozna´
n, Poland
E-mail address:domanski@math.amu.edu.pl
De part me nt o f Mat he mat ic s, ˚
Abo Akademi University, FIN-20500 ˚
Abo, Finland
E-mail address:mikael.lindstrom@abo.fi
Mathematisches Institut der Universit¨
at M¨
unchen, Theresienstrasse 39,
D-80333 M¨
unchen, Germany
E-mail address:schluech@rz.mathematik.uni-muenchen.de
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