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Operator Ideals arising from Generating Sequences

In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks operators will be demonstrated.


Introduction
Let T : E → F be a linear operator between Banach spaces. Let U E , U F be the closed unit balls of E, F , respectively. Note that closed unit balls serve simultaneously the basic model for open sets and bounded sets in Banach spaces. The usual way to describe T is to state either the bornological property, via T U E , or the topological property, via T −1 U F , of T . An other way classifying T is through the sequential structures T preserves.
Here are some well-known examples. Examples 1.1.
(1) T is bounded (i.e., T U E is a bounded subset of F ) ⇔ T is continuous (i.e., T −1 U F is a 0-neighborhood of E in the norm topology) ⇔ T is sequentially bounded ( i.e., T sends bounded sequences to bounded sequences); (2) T is of finite rank (i.e., T U E spans a finite dimensional subspace of F ) ⇔ T is weak-norm continuous (i.e., T −1 U F is a 0-neighborhood of E in the weak topology) ⇔ T sends bounded sequences to sequences spanning finite dimensional subspaces of F ; (3) T is compact (i.e., T U E is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on norm compact subsets of E ′ (i.e., T −1 U F ⊇ K • , the polar of a norm compact subset K of the ⇔ T is sequentially compact (i.e., T sends bounded sequences to sequences with norm convergent subsequences); and (4) T is weakly compact (i.e., T U E is relatively weakly compact in F ) ⇔ T is continuous in the topology of uniform convergence on weakly compact subsets of E ′ (i.e., T −1 U F ⊇ K • , the polar of a weakly compact subset K of E ′ ) ⇔ T is sequentially weakly compact (i.e., T sends bounded sequences to sequences with weakly convergent subsequences).
In [18,17,15], we investigate the duality of the topological and bornological characters of operators demonstrated in the above examples, and applied them in the study of operator ideal theory in the sense of Pietsch [6]. In [16], we use these concepts in classifying locally convex spaces. They are also used by other authors in operator algebra theory in [13,2].
After giving a brief account of the equivalence among the notions of operator ideals, generating topologies, and generating bornologies developed in [18,17,15] in Section 2, we shall explore into the other option of using sequential structures in Section 3.
The theory is initialed by Stephani [11,12]. We provide a variance here. In particular, we will show that the notions of operator ideals, generating bornologies and generating sequences are equivalent in the context of Banach spaces. However, we note that the sequential methodology does not seem to be appropriate in the context of locally convex spaces, as successfully as in [18,16]. Some examples of popular operator ideals are treated in our new ways as demonstrations in Section 4.
The author would like to take this opportunity to thank Professor Kar-Ping Shum. He has learned a lot from Professor Shum since he was a student in the Chinese University of Hong Kong in 1980's. With about 300 publications, Professor Shum has served as a good model for the author and other fellows since then. The current work is based partially on the thesis of the author finished when he was studying with Professor Shum.

The triangle of operators, topologies and bornologies
Let X be a vector space over K = R or C. Following Hogbe-Nlend [4], by a vector bornology on X we mean a family B of subsets of X satisfies the following conditions: (VB 4 ) λB ∈ B whenever λ ∈ K and B ∈ B; (VB 5 ) the circle hull ch B of any B in B belongs to B. Let (X, T) be a locally convex space. For any subset B of X, the σ-disked hull of B is defined to be  (a) If B is σ-disked then B is infracomplete.

Elements in
(b) If B is infracomplete and closed then B is σ-disked.
Let X, Y be locally convex spaces. Denote by σ(X, X ′ ) the weak topology of X with respect to its dual space X ′ , while P ori (X) is the original topology of X. We employ the notion M fin (Y ) for the finite dimensional bornology of Y which has a basis consisting of all convex hulls of finite sets. On the other hand, M von (Y ) is used for the von Neumann bornology of Y which consists of all P ori (Y )-bounded subsets of Y . Ordering of topologies and bornologies are induced by set-theoretical inclusion, as usual. Moreover, we write briefly X P for a vector space X equipped with a locally convex topology P and Y M for a vector space Y equipped with a convex vector bornology M.
Let X, Y be locally convex spaces. We denote by L(X, Y ) and L × (X, Y ) the collection of all linear operators from X into Y which are continuous and (locally) bounded (i.e., sending bounded sets to bounded sets), respectively.
Definition 2.2 (see [16]). Let C be a subcategory of locally convex spaces.
(1) ("Operators") A family A = {A(X, Y ) : X, Y ∈ C} of algebras of operators associated to each pair of spaces X and Y in C is called an operator ideal if L(X 0 , X) for any X 0 , X, Y and Y 0 in C.
(2) ("Topologies") A family P = {P(X) : X ∈ C} of locally convex topologies associated to each space X in C is called a generating topology if GT 1 : σ(X, X ′ ) ⊆ P(X) ⊆ P ori (X) for all X in C; and GT 2 : L(X, Y ) ⊆ L(X P , Y P ) for all X and Y in C. (1) ("Operators" → "Topologies") For each X 0 in C, the A-topology of X 0 , denoted by T(A)(X 0 ), is the projective topology of X 0 with respect to the family In other words, a seminorm p of X 0 is T(A)(X 0 )-continuous if and only if there In this case, we call p an A-seminorm of X 0 . ( In other words, In this case, we call B an A-bounded subset of Y 0 . be the vector space of all continuous operators from X into Y which is still continuous with respect to the P(X)-topology.
be the vector space of all continuous operators from X into Y which send bounded sets to M(Y )-bounded sets.
is defined to be the bornology (resp. topology) polar to P(X) (resp. M(Y )). More precisely, Here are two examples: the ideals K of compact operators and P of absolutely summing operators (see e.g. [6]), the generating systems P pc of precompact topologies (see e.g. [7]) and P pn of prenuclear topologies (see e.g. [ The injective hull A inj and the surjective hull A sur , of A is the intersection of all injective and surjective operator ideals containing A, respectively. For example, K = K inj = K sur and P = P inj P sur (see, e.g., [6]).
In the following, we see that the notions of operator ideals, generating topologies, and generating bornologies are equivalent (see also [13] for the operator algebra version).
Theorem 2.5 ([9, 10, 18, 16]). Let A be an operator ideal, P a generating topology and M a generating bornology on Banach spaces. We have is a generating topology.  For generating topologies P and P 1 , and generating bornologies M and M 1 on Banach spaces, we can also associate to them the operator ideals with components These give rise to Here, the product A • B consists of operators of the form ST with S ∈ A and T ∈ B, Readers are referred to Pietsch's classic [6] for information regarding quotients and products of operator ideals.

Generating sequences
The following is based on a version in [11]. • every subsequence of a sequence in Φ(E) belongs to Φ(E).
Clearly, the biggest vector sequential structure on a Banach space E is the family   The condition (GB 1 ) follows directly from (GS 1 ). For (GB 2 ), let T ∈ L(E, F ) and M ∈ M Φ (E). Then for every sequence {T x n } in T M, we will have a subsequence of {x n } in Φ(E). By (GS 2 ), we have a further subsequence {T x n k } belonging to M Φ (F ).
Definition 3.6. A generating sequential structure Φ is said to be normal if for every {x n } in Φ(E) and every bounded scalar sequence (λ n ) in ℓ ∞ of norm not greater than 1, we have {λ n x n } ∈ Φ(E).

It is clear that S(A) is normal for any operator ideal A.
Lemma 3.7. For every normal generating sequential structure Φ, we have Proof. It suffices to show that every M in M Φ base (E) has its circle hull ch M in M Φ base (E) for each Banach space E, i.e., M Φ base is itself circled. To this end, let {x n } be from ch M. Then there is a (λ n ) from ℓ ∞ of norm not greater than 1 such that x n = λ n y n , n = 1, 2, . . . for some {y n } in M. Now, there is a subsequence {y n k } in Φ(E) ensuring that {x n k } = {λ n k y n k } ∈ Φ(E) for Φ being normal.    for any Banach space E.
Since {T x n } ∈ Ψ(F ) by (GS 2 ), we can verify the condition in Definition 3.12. Therefore, we might have a subsequence {T x n k } having no further subsequence in Φ(F ). However, by the facts that {x n k } ∈ Ψ(E) and T ∈ O(Φ/Ψ)(E, F ), we arrive at the asserted contradiction.
Corollary 3.14. Let Φ, Ψ be two generating sequential structures such that Φ ≪ Ψ and Ψ is normal. Then, the operator ideal O(M Φ , M Ψ ) consists of exactly those operators sending sequences in Ψ to sequences in Φ.
Examples 4.4 ( [11]). The following generating bornologies can be induced by the corresponding generating sequential structures as in Lemma 3.5.
(2) The compact bornology M c is defined by Φ c .
(3) The weakly compact bornology M wc is defined by Φ wc .
(4) The Rosenthal compact bornology M wCa is defined by Φ wCa . Conversely, if W is a bounded set in F such that every sequence {y n } in W has a limited subsequence {y n k }, we need to check that W ∈ M lim . Assume, on contrary, that there were some σ(F ′ , F )-null sequence {y ′ n } in F ′ and some sequence {a n } in W such that lim n→∞ | a n k , y ′ n k | = 0. This is a contradiction and thus W is a member of M lim (F ). Proof. For the case of M lim , see [1]. For all others, see [11].
We are now ready to give a number of examples.
Examples 4.7. Using Definition 2.3 and Theorem 3.9, we can obtain the following operator ideals on Banach spaces through the associated generating bornologies.
(1) The ideal N sur of co-nuclear operators for the nuclear bornology M ν .
The first four can be found in [11]. For the last one, we observe the fact that each limited sequence is weakly Cauchy (see [1]) and (c).
Examples 4.9. The following operator ideals have desirable representations.
(1) The ideal U of unconditionally summing operators  See [11] for a proof of (a), (b) and (c). See [3] for (d). For (e), we simply recall that a T in L(E, F ) is called weakly Banach-Saks if T sends each weakly convergent sequence to a sequence possessing a Banach-Saks subsequence, by definition. Proof. We present a proof here for completeness. Denote