Representing the Banach operator ideal of completely continuous operators

Let V ,W∞, and W be the operator ideals of completely continuous, weakly ∞-compact, and weakly compact operators, respectively. In a recent paper, William B. Johnson, Eve Oja, and the author proved that V = W∞ ◦W −1 (Johnson, W. B., Lillemets, R., and Oja, E. Representing completely continuous operators through weakly ∞-compact operators. Bull. London Math. Soc., 2016, 48, 452–456). We show that this equality also holds in the context of Banach operator ideals.


INTRODUCTION
Let L , K , W , and V denote the operator ideals of bounded linear, compact, weakly compact, and completely continuous operators.Let X and Y be Banach spaces.Recall that a linear map T : X → Y is completely continuous, i.e.T ∈ V (X,Y ), if T takes weakly null sequences in X to null sequences in Y .It is well known that operator ideals K , V , and W are Banach operator ideals with the usual operator norm.
Let (x n ) ⊂ X be a bounded sequence.It is well known and easy to see that (x n ) defines an operator Φ (x n ) ∈ L (ℓ 1 , X) through the equality Denote the classes of all null sequences and weakly null sequences in X by c 0 (X) and c w 0 (X), respectively.Both of them are Banach spaces with the supremum norm.According to the Grothendieck compactness principle (see [3] or, e.g.[4, Proposition 1.e.2]), a subset K ⊂ X is relatively compact if and only if for every ε > 0 there exists Weakly ∞-compact (more generally, weakly p-compact) operators were considered by Sinha and Karn [7] in 2002 (for an even more general version of weakly (p, r)-compact operators, see [2]).Denote by W ∞ the class of all weakly ∞-compact operators acting between arbitrary Banach spaces.An easy straightforward verification (as in [1, Proposition 2.1]) shows that W ∞ is an operator ideal.
Recall that the right-hand quotient A •B −1 of two operator ideals A and B is the operator ideal that consists of all operators T ∈ L (X,Y ) such that T S ∈ A (X 0 ,Y ) whenever S ∈ B(X 0 , X) for some Banach space X 0 (see [6, 3.1.1]).
Let (A , ∥•∥ A ) and (B, ∥•∥ B ) be quasi-Banach operator ideals.The quotient A • B −1 becomes a quasi-Banach operator ideal if for every operator where the supremum is taken over all Banach spaces X 0 (see [6, 7.2.1]).
In [5] Johnson, Oja, and the author proved that V = W ∞ • W −1 as operator ideals.Now, we will show that this equality holds in the context of Banach operator ideals.For this, we introduce a norm on the operator ideal As Proposition 1 below shows, W ∞ is a Banach operator ideal with this norm.The main result of this paper (Theorem 3) is that the equality V = W ∞ • W −1 indeed holds in the context of Banach operator ideals.Throughout this paper, let K denote the scalar field R or C.

BANACH OPERATOR IDEAL W ∞
In this section we verify that W ∞ is indeed a Banach operator ideal endowed with the norm and we have shown that For this, take ε > 0 and sequences Assume that sup n∈N ∥x n ∥ ̸ = 0 and that sup n∈N ∥y n ∥ ̸ = 0 (otherwise, either S = 0 or T = 0, and the proof is trivial).Put We check that sup For this purpose, we use the fact that We have that sup It remains to show that because the operator R (as every bounded linear operator) is weaklyweakly continuous.Therefore . This gives us that To prove that it is a Banach operator ideal, we need to verify that . Furthermore, for every m ≥ 2 there exists a sequence (y m k ) k∈N ∈ c w 0 (Y ) so that sup k∈N y m k ≤ 1 4 m and S m (B X ) ⊂ Φ (y m k ) (B ℓ 1 ).We define the sequence (z n ) as any permutation of the following elements: 2y 1  1 , 2y We complete the proof by observing that Proof.Clearly, T ∈ W ∞ (X,Y ).The Grothendieck compactness principle allows us to write Therefore ∥T ∥ W ∞ ≤ ∥T ∥, since infimum in the definition of ∥T ∥ W ∞ is taken over a larger set than in the previous formula.On the other hand, ∥T ∥ ≤ ∥T ∥ W ∞ because W ∞ is a Banach operator ideal.
where the supremum is taken over all Banach spaces X 0 .

Theorem 3 .
The equality V = W ∞ • W −1 holds in the context of Banach operator ideals.Proof.Fix an operatorT ∈ V (X,Y ) = W ∞ • W −1 (X,Y ).By definition, where z n = 2 j n y j n i n .To prove that (z n ) ∈ c w 0 (Y ), we take any f ∈ Y * , let ε > 0, and show that the set{n ∈ N | | f (z n )| > ε} is finite.It is so because 2 m sup k∈N y m