Operators preserving \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty $$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y$$\end{document} be a Banach space, let the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty $$\end{document} be real, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W$$\end{document} denote the Banach space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty /c_0,$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document} denote the quotient map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty \rightarrow W.$$\end{document} In 1981, Partington proved there is a topological embedding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty $$\end{document} into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W$$\end{document} such that the composition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$QJ$$\end{document} is an isometry; in particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document} preserves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty .$$\end{document} In this paper we prove that if the kernel of an operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T:\ell _\infty /c_0 \rightarrow Y$$\end{document}does not contain an isometric copy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_0$$\end{document} (in particular, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document} is injective), then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document}preserves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty ,$$\end{document}and hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document}is non-weakly compact. This, in turn, allows us to extend Partington’s theorem: we show that natural quotient mappings of some real function spaces preserve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty .$$\end{document} We also remark that our results apply to some quotients of both Orlicz and Marcinkiewicz spaces.


Introduction
In what follows we use notations from the abstract. For notions and notations undefined here we refer the reader to the monographs [2,3]. All operators are linear and continuous, spaces are of infinite dimension, and denotes an infinite set endowed with the discrete topology.
The present paper deals with operators preserving the real Banach space ∞ or, what comes to the same thing, real C(βN), where βN is theČech-Stone compactification of the discrete space of positive integers N.
Let X, Y, V be Banach spaces, and let K be a compact Hausdorff space. An operator resp.] to V, such that the restricted operator T |U is an isomorphism [isometry, resp.]. The first theorem on such operators is due to Pełczyński [5]. In 1962, he proved that (*) Every non-weakly compact operator T : C(K ) → Y preserves c 0 .
Here ∞ ( ) denotes the closed subspace of ∞ ( ) consisting of the elements with support included in .
Remark 1 It is easy to see that if, in the hypothesis of (R), the unit vector basis {e γ : γ ∈ } is replaced by any norm-bounded subset {u γ : γ ∈ } of ∞ ( ) whose elements are pairwise disjoint, then T is an isomorphism on a subspace U of ∞ ( ), isomorphic to ∞ ( ), spanned on the set {u γ : γ ∈ } as in (P) below.
In 1981, Partington added a new information to the Rosenthal result (**): he proved that every automorphism of the quotient space ∞ /c 0 preserves ∞ in a particular way [4, Theorem 1]: (P) For the real Banach space ∞ and every norm | | on W = ∞ /c 0 , equivalent to the usual (quotient) norm, there exist pairwise disjoint and norm-bounded elements {u n : n ∈ N} in ∞ such that for every (a n ) ∈ ∞ we have where ( p) ∞ n=1 a n u n denotes the formal pointwise sum of the a n u n in ∞ . The following result of the present author extends partially Partington's theorem (P) to the Banach lattice-case (see [11, (The examples of such pairs of E and M can be found both in the class of Orlicz and in the class of Marcinkiewicz spaces [6, Lemma 1 and remarks before Corollary 7]).
In the present paper, we extend and supplement the above-cited theorems on operators preserving ∞ . Our main results are included in Theorems 1 and 2, and their applications to concrete cases are given in Example 1, and Corollaries 4 and 5.

The results
Our basic theorem is given below. In what follows, for the set fixed, the letter Q denotes the natural quotient map from ∞ ( ) onto ∞ ( )/c 0 ( ), and W denotes the natural (quotient and lattice) norm on W := ∞ ( )/c 0 ( ).
If the kernel of an operator T : W → Y does not contain an isometric [isomorphic, resp.] copy of c 0 then T preserves ∞ . In particular, T is non-weakly compact.
Proof Since, by Partington's result (P), the isomorphic case follows from the isometric case, we shall assume without loss of generality that W is endowed with the norm W . Let us fix .
Let us consider first the case countable. Without loss of generality we set W = ∞ /c 0 . Our goal is to show that there is a pairwise disjoint and norm-bounded sequence (u n ) in W such that inf n≥1 T (Qu n ) Y > 0; then we shall apply Rosenthal's theorem (R).
We define a new norm | | on W by the formula where Y is a norm on Y. Now we apply (P) and choose a sequence (u n ) in ∞ as in (1), and we set x n := Qu n , n = 1, 2, . . . . Hence By (P), the sequence (u n ) consists of pairwise disjoint elements and "spans" a copy U of ∞ in ∞ in the same way as in (P); we thus have for every (a n ) ∈ ∞ . So that Q(U ) is an isometric copy of ∞ in W. Now we claim that Let us suppose the contrary, i.e., Let (N k ) be a sequence of infinite and pairwise disjoint subsets of N such that N = ∞ k=1 N k , and let us set By (4), the elements y k are well defined (in W ) and |y k | = 1 for all k. Since Q is a lattice homomorphism and the elements (u n ) are pairwise disjoint, we have for all n ∈ N k (the latter inequality ≥ follows from the inequality |a + b| ≥ |a| for the elements a and b disjoint). Hence, by (2) (and since W is a lattice norm), we obtain By (2), (3), (6) and (8), we obtain that 1 ≥ y k W ≥ lim n→∞ x n W = 1, i.e., whence, by (2) again, T y k = 0 for all k. But, by (7) and (9), the elements (y k ) are pairwise disjoint in W = C(βN\N) and are of norm one, thus they span an isometric copy of c 0 .
Hence the kernel of T contains an isometric copy of c 0 , but this contradicts the hypothesis of our theorem. Thus our claim (6) is false, so (5) must be true. Now we apply the general case of Rosenthal's result (R) (see Remark 1) to the space U 1 isomorphic to ∞ and spanned-as in (P)-by an infinite subsequence (u n j ) of (u n ) such that inf j→∞ T Q(u n j ) Y > 0; and without loss of generality we may assume that the operator T Q acts on U 1 as an isomorphism.
Set Q 1 := Q |U 1 , W 1 := Q 1 (U 1 ), T 1 := T |W 1 , and let S denote the composition T 1 • Q 1 . We thus have that U 1 and W 1 are isomorphic copies of ∞ , and that S is an isomorphism from U 1 onto Y 1 := T 1 (W 1 ). Moreover, by (4), Q 1 is an isomorphism from U 1 onto W 1 .
From all this follows that If the set is uncountable, then the space ∞ ( )/c 0 ( ) contains an isometric copy of ∞ /c 0 (see [13,Corollary 2]). Now we apply the just proved result for countable. Now let us consider the case when the operator T : ∞ ( )/c 0 ( ) → Y is injective. If is countable, Theorem 1 implies that T preserves ∞ . Moreover, if is uncountable, from the above-cited results (W ) and (R) we obtain immediately that T preserves ∞ ( ). More exactly: from the proof of Theorem 1 it follows that, in each of the either cases, there is a set {u γ : γ ∈ } of pairwise disjoint and norm bounded elements of ∞ ( ) such that inf γ ∈ T Q(u γ ) Y > 0.
Hence, by Theorem 1, we have the following result.

Corollary 1 Every injective operator T
where (u γ ) γ ∈ are pairwise disjoint and norm bounded elements of ∞ ( ); hence Rosenthal's result (R) applies to the operator T and the family (Qu γ ) γ ∈ . Moreover, if Y is a WCG-space, then the kernel of every operator T : ∞ /c 0 → Y contains an isometric copy of c 0 ; in particular, T is not injective.
(The second part of Corollary 1 follows from the well known fact that a weakly compactly generated (WCG) Banach space cannot contain an isomorphic copy of ∞ ).
In the next corollary we supplement Rosenthal's result (R) in a particular case of the condition inf γ ∈ T e γ = 0.

Corollary 2 Let T be an operator from
Proof The operator T : ∞ ( )/c 0 ( ) → Y of the form T (Qx) := T x, is injective. By Corollary 1, from the form of T we obtain inf γ ∈ T u γ Y > 0, where {u γ : γ ∈ } is a norm-bounded subset of ∞ ( ) whose elements are pairwise disjoint. By Remark 1, the operator T preserves ∞ ( ).
The following example illustrates Corollary 2 (for another application of this corollary see the proof of Theorem 2).
Example 1 Let c denote the Banach space of all real convergent sequences, let y * be the "lim" functional on c, and let x * be any fixed continuous extension of y * to ∞ . Further, let F denote the set (with card(F) = 2 ℵ 0 ) of all strictly increasing functions f : N → N, and let ξ be any function F → (0, 1]. By x * f we denote the element of * ∞ defined by the formula is obviously continuous, and it is easy to check that kerT ξ = c 0 . From Corollary 2 we obtain that T ξ preserves ∞ .
The next two corollaries also follow from Corollary 1.
Let Y be a closed subspace of a Banach space X, and set W := ∞ ( )/c 0 ( ). Let S : ∞ ( ) be an operator such that S(c 0 ( )) = Y ∩ ImS, and let q and Q denote the natural quotient maps ∞ ( ) → W and X → X/Y, respectively. By [6, Theorem 2], the induced operator R : W → X/Y defined by the rule R • q = Q • S is injective. From Corollary 1 we thus obtain: Corollary 3 With the notations as above, and with the hypothesis S(c 0 ( )) = Y ∩ ImS, the quotient space X/Y contains an isomorphic copy of ∞ . Now let us consider the quotient space X * * /ι(X ), where ι denotes the canonical embedding of X into X * * . Assume there is an isomorphic embedding S 0 of c 0 into X. Then S := S * * 0 embeds ∞ into X * * with ι(X ) ∩ ImS = ι(S 0 (c 0 )) = S(c 0 ). By [6, Corollary 2], for every separable subspace V of ∞ containing a copy of c 0 , the quotient space X = ∞ /V contains a copy of c 0 ( ), where card( ) = 2 ℵ 0 . Hence, by Corollary 3, we obtain: Corollary 4 If X contains a copy of c 0 , then the quotient space X * * /ι(X ) contains an isomorphic copy of ∞ .
In particular, for the Banach space X = ∞ /C[0, 1], the quotient space X * * / X contains an isomorphic copy of ∞ .
The next theorem has an application to some quotient mappings.
Theorem 2 Let U be a closed subspace of a Banach space X. If U is isomorphic to ∞ and T : X → Y is an operator such that the subspace U 0 := U ∩ kerT is isomorphic to c 0 , then T preserves ∞ .
Proof Let R and S be two isomorphisms from ∞ onto U and from c 0 onto U 0 , respectively. Then the subspace X 0 := R −1 (U 0 ) of ∞ is isomorphic to c 0 , and the operator τ 0 := R −1 S maps c 0 onto X 0 . By [2, Theorem 2.f.12(i)], there is an automorphism τ of ∞ extending τ 0 . It follows that the operator S := Rτ is an extension of S, and S maps ∞ onto U. (10) Now let us consider the restricted operator T |U . By (10), the composition T |U S maps ∞ into Y, and ker(T |U S) = c 0 because ker(T |U ) = U 0 . By Corollary 2, the space ∞ contains an isomorphic copy V of ∞ such that T |U S acts on V as an isomorphism. But since S is an isomorphism and the subspace U 1 := S(V ) of U is also a copy of ∞ , the operator T acts on U 1 as an isomorphism; that is, T preserves ∞ , as claimed.
In the corollary below, we show how Theorem 2 works in concrete cases. Let X denote one of the real Banach spaces, endowed with the "sup"-norm and built on the interval Proof Let (x n ) be a sequence of positive, pairwise disjoint and norm-one elements of C[0, 1]. Then the closed linear span U 0 := [x n ] of (x n ) is an isometric copy of c 0 , and its pointwise "span" U, defined in X as in (P), is isometric to ∞ . Moreover, by Dini's theorem, the pointwise sum ( p) ∞ n=1 t n x n , where t n ≥ 0, n = 1, 2, . . . , lies in C[0, 1] if and only if t n → 0 as n →∞. It follows that U ∩ kerq =U 0 , and hence, by Theorem 2, q preserves ∞ .

Further applications of Theorem 1
Let E denote a real Banach lattice and let E a be the order continuous part of E (for unexplained in this section notions and fundamental facts concerning the theory of Banach lattices and Orlicz and Marcinkiewicz spaces we refer the reader to [1][2][3]10]; cf. [6,Section 4]). In [13, Theorem 1 and Corollary 3], it has been proved that if E is Dedekind complete with E = E a then the quotient Banach lattice E/E a contains an isomorphic (or isometric) copy of W = ∞ /c 0 (cf. [6,Corollary 6]).
For example, if φ denotes a sequence Orlicz space such that the Orlicz function φ does not fulfil the 2 -condition at 0, then the quotient space φ / h φ contains an isometric copy of W (see [6,Corollary 7]).
Similarly, if denotes a Marcinkiewicz function, then the quotient Marcinkiewicz space M( )/M 0 ( ) contains a copy of W whenever M( ) = M 0 ( ) (see [6,Corollary 9] Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.