ISOMETRIC FACTORIZATION OF WEAKLY COMPACT
OPERATORS AND THE APPROXIMATION PR OPERTY
˚
ASVALD LIMA, OLAV NYGAARD, AND EVE OJA
Abstract. Using an isometric version of the Davis, Figiel, Johnson,
and Pe%lczy´nski factorization of weakly compact operators, we prove
that a Banach space X has the approximation property if and only
if, for every Banach space Y , the finite rank operators of norm ≤ 1
are dense in the unit ball of W(Y,X), the space of weakly compact
operators from Y to X, in the strong operator topology. We also show
that, for every finite dimensional subspace F of W(Y, X), there are a
reflexive space Z, a norm one operator J : Y → Z, and an isometry
Φ: F →W(Z, X) which preserves finite rank and compact operators
so that T =Φ(T ) ◦ J for all T ∈ F . This enables us to prove that X
has the approximation property if and only if the finite rank operators
form an ideal in W(Y, X ) for all Banach spaces Y .
Introduction
Let us recall that a linear subspace F of a Banach space E is an ideal
in E if F
⊥
is the kernel of a norm one projection in E
∗
. The notion of
an ideal was introduced and studied by Godefroy, Kalton, and Saphar in
[14].
J. Johnson [20] proved that if X is a Banach space with the metric
approximation property, then, for every Banach space Y , F(Y,X), the
space of finite rank operators from Y to X,isanidealinL(Y,X), the
space of bounded operators from Y to X. Lima [23] has shown that the
converse is true if X has the Radon-Nikod´ym property. It is not known
whether the converse is true in general.
Date: December 1998.
1991 Mathematics Subject Classification. Primary: 46B20, 46B28, 47D15.
Key words and phrases. Weakly compact operator, approximation property, Davis-
Figiel-Johnson-Pe%lczy´nski factorization.
The third-named author wishes to acknowledge the warm hospitality provided by
˚
Asvald Lima and his colleagues at Agder College, where a part of this work was done
in May-June 1998. Her visit was supported by the Norwegian Academy of Science and
Letters and by Estonian Science Foundation Grant 3055.
1
2
˚
ASVALD LIMA, OLAV NYGAARD, AND EVE OJA
In [25], Lima and Oja proved that X has the approximation property if
and only if F(Y,X)isanidealinK(Y,X), the space of compact operators
from Y to X, for all Banach spaces Y . In fact, they showed something
stronger: X has the approximation property if (and only if) F(Y,X)is
an ideal in K(Y,X) for all separable reflexive spaces Y , or, equivalently,
for all closed subspaces Y of c
0
.
It is natural to ask what happens if we look at F(Y,X) as a subspace of
W(Y,X), the space of weakly compact operators from Y to X, instead of
looking at F(Y,X) as a subspace of K(Y,X). The answer to this question
is the main result of this paper: X has the approximation property if and
only if F(Y,X) is an ideal in W(Y,X) for all Banach spaces Y ,whichin
turn, is equivalent to the condition that, for every Banach space Y and
every T ∈W(Y,X),thereisanet(T
α
) in F(Y,X) with sup
α
T
α
≤T
such that T
α
y → Ty for all y ∈ Y .
We depart from the remarkable factorization theorem due to Davis,
Figiel, Johnson, and Pelczy´nski [5] asserting that any weakly compact
operator factors through a reflexive Banach space. In Section 1 (cf.
Lemma 1.1), we make a quantitative change in the Davis-Figiel-Johnson-
Pelczy´nski construction which enables us to show, in Section 2, that one
can factorize weakly compact operators through reflexive Banach spaces
isometrically and even uniformly. In Theorem 1.2, we give a new char-
acterization of the approximation property in terms of the Davis-Figiel-
Johnson-Pelczy´nski factorization. We apply these results in Corollary 1.4
where we prove that X has the approximation property if and only if
every weakly compact operator into X can be approximated in the strong
operator topology by finite rank operators whose norms are at most equal
to the norm of the weakly compact operator.
In Section 2 (cf. Lemma 2.1), we show that on the absolutely convex
weakly compact set that is used in the factorization theorem of Davis,
Figiel, Johnson, and Pelczy´nski to construct the reflexive Banach space,
the two norm topologies coincide. (It was a part of the original construc-
tion that the two weak topologies coincide on the unit ball of the reflexive
Banach space.) This, together with the quantitative modification of the
Davis-Figiel-Johnson-Pelczy´nski construction made in Section 1, leads us
to an isometric version of the Davis-Figiel-Johnson-Pelczy´nski factoriza-
tion theorem (cf. Theorem 2.2). This also applies to show that the isomet-
ric factorization can even be uniform with respect to finite dimensional
subspaces in the space of weakly compact operators (cf. Theorem 2.3 and
Corollaries 2.4 and 2.5).
WEAKLY COMPACT OPERATORS 3
We apply the uniform isometric factorization from Section 2 in Sections
3 and 4. Our main results in Section 3 are Theorem 3.3 and Theorem 3.4.
They characterize the approximation property of X and X
∗
in terms of
ideals of finite rank operators. In particular, Theorem 3.3 shows that
X has the approximation property if and only if F(Y,X)isanidealin
W(Y,X) for all Banach spaces Y , and Theorem 3.4 shows that X
∗
has
the approximation property if and only if F(X, Y )isanidealinW(X, Y )
for all Banach spaces Y .
In Section 4, an easy example shows that it is not possible to character-
ize the compact approximation property of X by K(Y,X) being an ideal
in W(Y,X) for all Y (although this property characterizes the compact
approximation property for reflexive X). In Theorem 4.1, we give some
conditions equivalent to K(Y,X) being an ideal in W(Y,X) for all Y .We
also show, by using the description of duals of spaces of compact opera-
tors due to Feder and Saphar [12], that these conditions are implied by
the compact approximation property of X (cf. also Theorem 4.1).
In Theorems 5.1 and 5.2 of the final Section 5, we demonstrate how
the method of proof of Theorem 1.2 can be further developed to give
alternative proofs (through ideals of finite rank or compact operators) for
known results about cases when the (compact) approximation property
implies the metric (compact) approximation property. In particular, as an
immediate corollary, we obtain the result due to Godefroy and Saphar [15]
that X
∗
has the metric compact approximation property with conjugate
operators whenever X
∗
has the compact approximation property with
conjugate operators and X
∗
or X
∗∗
has the Radon-Nikod´ym property.
Let us fix some more notation. In a linear normed space X,wedenote
the closed unit ball by B
X
and the closed ball with center x and radius
r by B
X
(x, r). For a set A ⊂ X, its norm closure is denoted by A,its
linear span by span A, its convex hull by conv A, and the set of its strongly
exposed points by sexp A.
We shall write K
X
(resp. W
X
) for the family of all compact (resp.
weakly compact) absolutely convex subsets of B
X
.
1. Criteria of the approximation property in terms of the
Davis-Figiel-Johnson-Pelczy
´
nski factorization
In this section, we depart from the famous Davis, Figiel, Johnson, and
Pelczy´nski factorization construction (cf. Lemma 1 on p. 313 in [5], [6,
pp. 160-161], [7, p. 227], [33, p. 51] or Lemma 1.1 below) and apply
the Grothendieck-Feder-Saphar description of duals of spaces of compact
4
˚
ASVALD LIMA, OLAV NYGAARD, AND EVE OJA
operators (cf. [16] or [8] and [12]) to obtain several conditions equivalent
to the approximation property of Banach spaces, all of them expressed
in terms of the Davis-Figiel-Johnson-Pelczy´nski construction (cf. Theo-
rem 1.2 below). This leads us to an interesting “metric” characterization of
the approximation property (cf. Corollary 1.4) similar to the well-known
characterization of the metric approximation property as the denseness
of B
F(Y,X)
in B
L(Y,X)
in the topology of uniform convergence on compact
sets, for all Banach spaces Y .
We shall need a quantitative version of the classical Davis, Figiel, John-
son, Pelczy´nski factorization construction, which in fact consists in re-
placing the number 2 in the original construction by
√
a for any a>1.
We now fix the notation to describe the Davis-Figiel-Johnson-Pelczy´nski
construction, and we shall also use this notation in the following sections.
Let a>1. Let X be a Banach space and let K be a closed absolutely
convex subset of its unit ball B
X
.Foreachn ∈ N = {1, 2,...}, put
B
n
= a
n/2
K + a
−n/2
B
X
and denote by
n
the equivalent norm on X
defined by the gauge of B
n
.Letx
K
=(
∞
n=1
x
2
n
)
1/2
, X
K
= {x ∈
X : x
K
< ∞} and C
K
= {x ∈ X : x
K
≤ 1}. Further, let J
K
denote
the identity embedding of X
K
into X. Finally, we put
f(a)=
∞
n=1
a
n
(a
n
+1)
2
1/2
and note that f :(1, ∞) → R is a continuous, strictly decreasing function
with lim
a→1
+
f(a)=∞ and lim
a→∞
f(a) = 0. Hence, there is a unique
point ˜a ∈ (1, ∞) such that f(˜a) = 1. (A “good” estimate of this ˜a is
exp(4/9) = 1.55962349761....) For this ˜a, one has K ⊂ C
K
⊂ B
X
(this is
clear from Lemma 1.1 below).
The following is the classical Davis-Figiel-Johnson-Pelczy´nski factoriza-
tion lemma with some “cosmetic” changes.
Lemma 1.1 (cf. p. 313 in [5]).
(i) K ⊂ f(a)C
K
.
(ii) X
K
is a Banach space with the closed unit ball C
K
,andJ
K
∈
L(X
K
,X),andJ
K
≤1/f(a).
(iii) J
∗∗
K
is injective.
(iv) X
K
is reflexive if and only if K is weakly compact.
Proof. Only (i) and J
K
≤1/f(a) in (ii) need to be verified.
Suppose x ∈ K.Sincex ∈ B
X
,weget
a
n/2
x + a
−n/2
x ∈ B
n
,
WEAKLY COMPACT OPERATORS 5
so that
x
n
≤
1
a
n/2
+ a
−n/2
=
a
n/2
a
n
+1
for all n. Hence x
K
≤ f (a). This proves (i).
Since B
X
is convex and K ⊂ B
X
,wehave
1
a
n/2
+ a
−n/2
(a
n/2
K + a
−n/2
B
X
) ⊂ B
X
,
that is
a
n/2
a
n
+1
B
n
⊂ B
X
.
Hence
x
n
≥
a
n/2
a
n
+1
x
and therefore x
K
≥ f(a)x for all x ∈ X
K
, meaning that J
K
≤
1/f(a).
Theorem 1.2. For a Banach space X, the following assertions are equiv-
alent.
(i) X has the approximation property.
(ii) F(X
K
,X) is an ideal in L(X
K
,X) for every K ∈W
X
.
(iii) For every K ∈W
X
, there exists a net (A
α
) in F(X
K
,X) with
sup
α
A
α
≤J
K
such that A
α
x −→
α
J
K
x for all x ∈ X
K
.
(iv) For every K ∈W
X
, there exists a bounded net (A
α
) in F(X
K
,X)
such that A
α
x −→
α
J
K
x for all x ∈ X
K
.
(v) For every K ∈K
X
, there exists a net (A
α
) in F(X
K
,X) such
that A
α
− J
K
−→
α
0.
Remark 1.1. Condition (v) means that J
K
belongs to the norm closure of
F(X
K
,X)inL(X
K
,X) and (iii) can be viewed as its “metric” version:
J
K
belongs to the closure of the ball F(X
K
,X) ∩B(0, J
K
) in the strong
operator topology of L(X
K
,X).
The proof of Theorem 1.2, as well as some other proofs of this paper,
will use the following result.
Lemma 1.3. Let X and Y be Banach spaces. Let A be a subspace of
L(Y,X) containing F(Y,X) and let T ∈L(Y,X).IfA is an ideal in