On Disjointly singular centralizers

TL;DR

This paper explores various notions of disjoint singularity in quasi-linear maps within K"othe function spaces, revealing their relationships and distinctions, especially in superreflexive and non-atomic contexts.

Contribution

It introduces and analyzes disjoint versions of triviality and singularity for centralizers, establishing key equivalences and non-existence results in specific Banach space settings.

Findings

Trivial and disjoint trivial notions coincide on reflexive spaces.

No singular centralizers exist on non-atomic superreflexive K"othe spaces.

Super disjointly singular centralizers include Kalton-Peck centralizers.

Abstract

We study ``disjoint" versions of the notions of trivial, locally trivial, strictly singular and super-strictly singular quasi-linear maps in the context of K\"othe function spaces. Among other results, we show: i) (locally) trivial and (locally) disjointly trivial notions coincide on reflexive spaces; ii) On non-atomic superreflexive K\"othe spaces, no centralizer is singular, although most are disjointly singular. iii) No super singular quasi-linear maps exist between superreflexive spaces although Kalton-Peck centralizers are super disjointly singular; iv) Disjoint singularity does not imply super disjoint singularity.

Equations

$$\begin{array}{ccccccccc}0& \to & Y& \to & Z& \to & X& \to & 0.\end{array}$$

$$\Vert \Omega (x_1+x_2)-\Omega x_1-\Omega x_2\Vert \le C(\Vert x_1\Vert +\Vert x_2\Vert ),$$

$$\Vert (y,x){\Vert }_{\Omega }=\Vert y-\Omega z\Vert +\Vert x\Vert .$$

$$\begin{CD}0@>{}>{}>Y@>{}>{}>Z_{1}@>{}>{}>X@>{}>{}>0\\ &&\Big{\|}@V{}V{T}V\Big{\|}\\ 0@>{}>{}>Y@>{}>{}>Z_{2}@>{}>{}>Z@>{}>{}>0.\end{CD}$$

$$\Vert \Omega (fx)-f\Omega (x){\Vert }_{\mathcal{K}}\le C\Vert f{\Vert }_{\infty }\Vert x{\Vert }_{\mathcal{K}}.$$

$$\begin{array}{ccccccccc}0& \to & \mathcal{K}& \stackrel{ȷ}{\to }& d_{\Omega }\mathcal{K}& \stackrel{Q}{\to }& \mathcal{K}& \to & 0\end{array}$$

$$\Vert (w,x){\Vert }_{d_{\Omega }\mathcal{K}}=\Vert x{\Vert }_{\mathcal{K}}+\Vert w-\Omega x{\Vert }_{\mathcal{K}}$$

$$\lim \underset{k}{\sup }\frac{\Vert \Omega ({\sum }_{n\in A_k}{u}_n)-{\sum }_{n\in A_k}\Omega (u_n)\Vert }{\Vert {\sum }_{n\in A_k}{u}_n\Vert }=+\infty .$$

$$\{u_n^1{\}}_{n\in F_1},\{u_n^2{\}}_{n\in F_2},\dots ,\{u_n^k{\}}_{n\in F_k},\dots$$

$$\mathcal{K}(x)=x\log \frac{|x|}{\Vert x\Vert }$$

$$\Omega (x)=q\left(\frac{1}{q_1}-\frac{1}{q_0}\right)\mathcal{K}(x)+\left(\frac{q}{p}\left(\frac{1}{q_0}-\frac{1}{q_1}\right)-\left(\frac{1}{p_0}-\frac{1}{p_1}\right)\right)\kappa (x)$$

$${\rm Y}(f)(\gamma )=\omega ({\alpha }_{\gamma }{\pi }_{\gamma }(f))$$

$${{\rm Y}}_{|\gamma }(f)(\gamma )=\omega ({\alpha }_{\gamma }{\pi }_{\gamma }(f))=\omega ({\alpha }_{\gamma }f)$$

$${\rm Y}(x)(\gamma )=\omega {\alpha }_{\gamma }(1_{\overline{\gamma }}x)$$

$$\Omega (x)(\gamma ,{\gamma }^{\prime })=\pi {{\rm Y}}^{\ast \ast }(1_{\overline{\gamma }x})({\gamma }^{\prime })$$

$$\Omega (x)(\gamma ,\gamma )=\pi {{\rm Y}}^{\ast \ast }(1_{\overline{\gamma }x})(\gamma )={\rm Y}(x)(\gamma )=\omega {\alpha }_{\gamma }(1_{\overline{\gamma }x})=\omega {\alpha }_{\gamma }(x)$$

$$\Vert \sum {\lambda }_n(\Lambda u_n,u_n)\Vert =\Vert \Omega \left(\sum {\lambda }_n{u}_n\right)-\sum \Lambda ({\lambda }_n{u}_n)\Vert +\Vert \sum {\lambda }_n{u}_n\Vert \le (C+1)\Vert \sum {\lambda }_n{u}_n\Vert .$$

$$\Vert \sum {\lambda }_n{v}_n-\Omega \left(\sum {\lambda }_n{u}_n\right)\Vert \le C\Vert \sum {\lambda }_n{u}_n\Vert$$

$$\Vert L\left(\sum {\lambda }_n{u}_n\right)-\Omega \left(\sum {\lambda }_n{u}_n\right)\Vert =\Vert \sum {\lambda }_n{v}_n-\Omega \left(\sum {\lambda }_n{u}_n\right)\Vert \le C\Vert \sum {\lambda }_n{u}_n\Vert$$

$${\nabla }_{[b]}\Omega ={\mathrm{Ave}}_{\pm }\Vert \Omega \left(\sum _{k=1}^n\pm b_k\right)-\sum _{k=1}^n\pm \Omega (b_k)\Vert ,$$

$${\nabla }_{[\lambda u]}\Omega \le C\Vert \lambda {\Vert }_p.$$

$${\nabla }_{[\lambda u]}\Omega \le C\Vert \lambda {\Vert }_p$$

$$\Vert \Omega \left(\sum _{i=1}^n{\lambda }_i{u}_i\right)-L\left(\sum _{i=1}^n{\lambda }_i{u}_i\right)\Vert \le C\Vert \sum _{i=1}^n{\lambda }_i{u}_i\Vert$$

$$\Vert \Omega \left(\sum _{i=1}^n{\lambda }_i{u}_i\right)-\sum _{i=1}^n\Omega ({\lambda }_i{u}_i)\Vert$$

$${\nabla }_{[\lambda u]}\Omega \le C^{\prime }\Vert \lambda {\Vert }_p$$

$$\Vert \Omega \left(\sum _{i=1}^n{v}_i\right)-\sum _{i=1}^n\Omega (v_i)\Vert$$

$$\Vert \Omega (ϵv)-ϵ\sum _{i=1}^n\Omega (v_i)\Vert$$

$$\Vert \Omega (v)-\sum _{i=1}^n\Omega (v_i)\Vert$$

Full text

The research of the first and fourth authors has been supported in part by project MTM2016-76958-C2-1-P de MINCIN and Project IB16056 de la Junta de Extremadura, Spain. The second author was supported by Fapesp, grants 2016/25574-8 and 2018/18593-1. The third author was supported by CNPq grant 303034/2015-7, and by Fapesp, grants 2015/17216-1 and 2016/25574-8.

1. Introduction

For all unexplained notation and terms, please keep reading. This paper has its roots in [10] where the authors introduced the notion of disjointly singular centralizer on Köthe function spaces, proved that disjoint singularity coincides with singularity on Banach spaces with unconditional basis and presented a technique to produce disjointly singular centralizers via complex interpolation.

A second equally important fact to consider is that the fundamental Kalton-Peck map [24] is disjointly singular on $L_{p}$ [10, Proposition 5.4], but it is not singular [33]. In fact, as the last stroke one could wish to foster the study of disjoint singularity is the argument of Cabello [2] that no centralizer on $L_{p}$ can be singular that we extend here by showing that no centralizer can be singular. It is thus obvious that while singularity is an important notion in the domain of Köthe sequence spaces, disjoint singularity is the core notion in Köthe function spaces. The purpose of this paper is then to study the disjointly supported versions of the basic (trivial, locally trivial, singular and supersingular) notions in the theory of centralizers and present several crucial examples.

2. Background

Most of the action in this paper will take place in the ambient of Köthe functions spaces over a $\sigma$-finite measure space $(\Sigma,\mu)$ endowed with their $L_{\infty}$-module structure. A Köthe function space $K$ is a linear subspace of $L_{0}(\Sigma,\mu)$, the vector space of all measurable functions, endowed with a quasi-norm such that whenever $|f|\leq g$ and $g\in K$ then $f\in K$ and $\|f\|\leq\|g\|$ and so that for every finite measure subset $A\subset\Sigma$ the characteristic function $1_{A}$ belongs to $X$. A particular case of which is that of Banach spaces with a 1-unconditional basis (called Köthe sequence spaces in what follows) with their associated $\ell_{\infty}$-module structure.

2.1. Exact sequences, quasi-linear maps and centralizers

For a rather complete background on the theory of twisted sums see [11]. We recall that a twisted sum of two Banach spaces $Y$, $Z$ is a quasi-Banach space $X$ which has a closed subspace isomorphic to $Y$ such that the quotient $X/Y$ is isomorphic to $Z$. Equivalently, $X$ is a twisted sum of $Y$, $Z$ if there exists a short exact sequence

[TABLE]

According to Kalton and Peck [24], twisted sums can be identified with homogeneous maps $\Omega:X\to Y$ satisfying

[TABLE]

which are called quasi-linear maps, and induce an equivalent quasi-norm on $X$ (seen algebraically as $Y\times X$) by

[TABLE]

This space is usually denoted $Y\oplus_{\Omega}X$. When $Y$ and $X$ are, for example, Banach spaces of non-trivial type, the quasi-norm above is equivalent to a norm; therefore, the twisted sum obtained is a Banach space. Two exact sequences $0\to Y\to Z_{1}\to X\to 0$ and $0\to Y\to Z_{2}\to X\to 0$ are said to be equivalent if there exists an operator $T:Z_{1}\to Z_{2}$ such that the following diagram commutes:

[TABLE]

The classical 3-lemma (see [11, p. 3]) shows that $T$ must be an isomorphism.

Definition 2.1**.**

An $L_{\infty}$-centralizer (resp. an $\ell_{\infty}$-centralizer) on a Köthe function (resp. sequence) space $\mathcal{K}$ is a homogeneous map $\Omega:\mathcal{K}\to L_{0}$ such that there is a constant $C$ satisfying that, for every $f\in L_{\infty}$ (resp. $\ell_{\infty}$) and for every $x\in\mathcal{K}$, the difference $\Omega(fx)-f\Omega(x)$ belongs to $\mathcal{K}$ and

[TABLE]

The centralizer is called real when it sends real functions (sequences) to real functions (sequences).

When no confusion arises we will simply say: a centralizer. Observe that a centralizer $\Omega$ on $\mathcal{K}$ does not take values in $\mathcal{K}$, but in $L_{0}$, and still it induces an exact sequence

[TABLE]

as follows: $d_{\Omega}\mathcal{K}=\{(w,x):w\in L_{0},x\in\mathcal{K}:w-\Omega x\in\mathcal{K}\}$ endowed with the quasi-norm

[TABLE]

and with obvious inclusion $\jmath(x)=(x,0)$ and quotient map $Q(w,x)=x$. The reason is that a centralizer “is” quasi-linear, in the sense that for all $x,y\in\mathcal{K}$ one has $\Omega(x+y)-\Omega(x)-\Omega(y)\in\mathcal{K}$ and $\|\Omega(x+y)-\Omega(x)-\Omega(y)\|\leq C(\|x\|+\|y\|)$ for some $C>0$ and all $x,y\in\mathcal{K}$. To describe the fact that the centralizer acts $\Omega:\mathcal{K}\to L_{0}$ but defines a twisted sum of $\mathcal{K}$ with itself we will use sometimes the notation $\Omega:\mathcal{K}\curvearrowright\mathcal{K}$. Centralizers arise naturally by complex interpolation [1] as can be seen in [22].

2.2. Trivial maps

An exact sequence $0\to Y\to Z\to X\to 0$ is trivial if and only if it is equivalent to $0\to Y\to Y\oplus X\to X\to 0$, where $Y\oplus X$ is endowed with the product norm. In this case we say that the exact sequence splits. Two quasi-linear maps $\Omega,\Omega^{\prime}:X\to Y$ are said to be equivalent, denoted $\Omega\equiv\Omega^{\prime}$, if the difference $\Omega-\Omega^{\prime}$ can be written as $B+L$, where $B:X\to Y$ is a homogeneous bounded map (not necessarily linear) and $L:X\to Y$ is a linear map (not necessarily bounded). Two quasi-linear maps are equivalent if and only if the associated exact sequences are equivalent. A quasi-linear map is trivial if it is equivalent to the [math] map, which also means that the associated exact sequence is trivial. Given two Banach spaces $X,Y$ we will denote by $\ell(X,Y)$ the vector space of linear (not necessarily continuous) maps $X\to Y$. The distance between two homogeneous maps $T,S$ will be the usual operator norm (the supremum on the unit ball) of the difference; i.e., $\|T-S\|$, which can make sense even when $S$ and $T$ are unbounded. So a quasi-linear map $\Omega:X\longrightarrow Y$ is trivial if and only if $d(\Omega,\ell(X,Y))\leq C<+\infty$, in which case we will say that $\Omega$ is $C$-trivial. A centralizer $\mathcal{K}\curvearrowright\mathcal{K}$ is trivial if and only if there is a linear map $L:\mathcal{K}\to L_{0}$ so that $\Omega-L:\mathcal{K}\to\mathcal{K}$ is bounded.

2.3. Locally trivial maps

A quasi-linear map $\Omega:X\to Y$ is said to be locally trivial [20] if there exists $C>0$ such that or any finite dimensional subspace $F$ of $X$, there exists a linear map $L_{F}$ such that $\|\Omega_{|F}-L_{F}\|\leq C$. It is clear that a trivial map is locally trivial. The converse is not true, although locally trivial quasi-linear maps $F:X\to Y$ in which $Y$ is reflexive are trivial, by [5].

2.4. Singular maps

An operator between Banach spaces is said to be strictly singular if no restriction to an infinite dimensional closed subspace is an isomorphism. Analogously, a quasi-linear map (in particular, a centralizer) is said to be singular if its restriction to every infinite dimensional closed subspace is never trivial. An exact sequence induced by a singular quasi-linear map is called a singular sequence. A quasi-linear map is singular if and only if the associated exact sequence has strictly singular quotient map [13, Lemma 1]. Singular $\ell_{\infty}$-centralizers exist and the most natural example is the Kalton-Peck map $\mathscr{K}_{p}:\ell_{p}\curvearrowright\ell_{p}$, $0<p<+\infty$, defined by $\mathscr{K}_{p}(x)=x\log\frac{|x|}{\|x\|_{p}}.$ The proof that $\mathscr{K}_{p}$ is singular can be found in [24] for $1<p<+\infty$, [13] for $p=1$, and [8] for all $0<p<+\infty$. A simple characterization of singular $\ell_{\infty}$-centralizers on Banach sequence spaces can be presented

Proposition 2.2**.**

Let $X$ be a Banach space with an unconditional basis not containing $c_{0}$. Let $\Omega:X\curvearrowright X$ be an $\ell_{\infty}$-centralizer such that for every sequence $(A_{k})$ of consecutive subsets of $\mathbb{N}$ and every sequence $(u_{n})$ of consecutive normalized blocks of the basis, for which $\sup_{k}\|\sum_{n\in A_{k}}u_{n}\|\to+\infty$ one has

[TABLE]

Then $\Omega$ is singular.

Proof.

If $\Omega:X\curvearrowright X$ is an $\ell_{\infty}$-centralizer verifying the condition above and, at the same time, trivial on some subspace $H$, by the blocking principle (see [8]), it must be trivial on the subspace $[u_{n}]$ spanned by some consecutive blocks of the basis. Standard manipulations (see [8, 10]) show that the linear map $\ell(u_{n})=\Omega(u_{n})$ is at finite distance from $\Omega$, which implies that $\lim\sup_{k}\|\sum_{n\in A_{k}}u_{n}\|<+\infty$ for all choices of $(A_{k})$, thus $(u_{n})$ is equivalent to the canonical basis of $c_{0}$ and consequently $H$ contains $c_{0}$. ∎

In sharp contrast, Cabello [2] proved that no $L_{\infty}$-centralizer is singular on $L_{p}[0,1]$. Let us observe that quite the same proof of Cabello provides:

Proposition 2.3**.**

No singular $L_{\infty}$-centralizers exist on (admissible) superreflexive Köthe funcion spaces. More precisely, every $L_{\infty}$-centralizer on an admissible superreflexive Köthe function space is bounded on some copy of $\ell_{2}$.

Sketch of proof: Recall that according to Kalton [22, p.482] a Köthe space is termed admissible when for some strictly positive functions $h,k\in L_{0}$ one has $\|hk\|_{1}\leq\|x\|_{\mathcal{K}}\leq\|kx\|_{\infty}$ for every $x\in\mathcal{K}$. By Kalton’s theorem [22, Thm. 7.6] plus the comments in [2, Section 1.3] there are two admissible Köthe spaces $A,B$ so that $\mathcal{K}=(A,B)_{1/2}$; these spaces can be assumed to be superreflexive by reiteration and [23, Thm. 7.8]. The admissibility hypothesis yields functions $h_{a},k_{a}$ such that $\|h_{a}f\|\leq\|f\|_{A}\leq\|k_{a}f\|_{\infty}$ for every $f\in A$; and functions $h_{b},k_{b}$ such that $\|h_{b}f\|\leq\|f\|_{B}\leq\|k_{b}f\|_{\infty}$ for every $f\in B$. Thus, one can find a positive measure set $S\subset\Sigma$ and a constant $M>0$ such that $k_{a},k_{b}\leq M$ and $h_{a},h_{b}\geq M^{-1}$. This provides continuous inclusions $L_{\infty}(S)\subset A(S)\subset L_{1}(S)$ and $L_{\infty}(S)\subset B(S)\subset L_{1}(S)$ .

By super-reflexivity, both spaces $A,B$ are $p$-convex and $q$-concave for some $1<p,q<+\infty$ ([27, Thm 1.f.12 and Thm 1.f.7.]) So, using the Johnson-Maurey-Schechtman-Tzafriri remark [18, p.14] then also $L_{q}(S)\subset A(S)\subset L_{p}(S)$ and $L_{q}(S)\subset B(S)\subset L_{p}(S)$ . Let $R(S)$ be the subspace generated by Rademacher functions supported in $S$. The $L_{p}$ and $L_{q}$-norms are equivalent on $R(S)$ by Khintchine’s inequality, and are also equivalent to $\|\cdot\|_{A}$ and to $\|\cdot\|_{B}$, and thus $R(S)\sim\ell_{2}$. The equivalence of norms $A$ and $B$ on $R(S)$ makes the differential $\Omega_{1/2}$ bounded on $R(S)$, and since $\Omega_{K}$ is boundedly equivalent to $\Omega_{1/2}$, it must be bounded too.

2.5. Super-singular maps

An operator $T:X\to Y$ between two Banach spaces is said to be super strictly singular (in short, super-SS) if there does not exist a number $c>0$ and a sequence of subspaces $E_{n}$ of $X$, with $\dim\,E_{n}=n$, such that $\|Tx\|\geq c\|x\|$ for every $x\in\bigcup_{n}E_{n}$. Equivalently [14, Lemma 1.1.], if every ultrapower of $T$ is strictly singular. Super strictly singular operators have also been called finitely strictly singular; they were first introduced in [28, 29], and form a closed ideal containing the ideal of compact and contained in the ideal of strictly singular operators. See also [14] for the study of such a notion in the context of twisted sums, as well as [9] where a few results are also mentioned in relation to complex structures on twisted sums.

It is a standard fact (see [11]) that given an exact sequence $0\to Y\to Z\to Z\to 0$ and an ultrafilter $\mathscr{U}$ the ultrapowers form an exact sequence $0\to Y_{\mathscr{U}}\to Z_{\mathscr{U}}\to X_{\mathscr{U}}\to 0$. If $\Omega$ is a quasi-linear map associated to the former sequence we will call $\Omega_{\mathscr{U}}$ any quasi-linear map associated to the later. We do not need for the moment to specify the construction of $\Omega_{\mathscr{U}}$. We will say, following [9] that a quasi-linear map $\Omega$ is super-singular if every ultrapower $\Omega_{\mathscr{U}}$ is singular. We need to state here two facts proved in [9]:

• •

$\Omega$ is super-singular if and only the quotient map $q_{\Omega}$ of the exact sequence it defines is super strictly singular.

• •

No super singular quasi-linear maps between $B$-convex Banach spaces exist. This follows from [30, Thm. 3], where it is proved that a super strictly singular operator on a $B$-convex space has super strictly singular adjoint. Since superreflexive spaces are $B$-convex, $B$-convexity is a $3$-space property (see [11]) and the adjoint of a quotient map is an into isomorphism, the result follows.

After these prolegomena, we tackle the study of the “disjoint” versions of the preceding properties. It is worth to observe that all our forthcoming “disjoint” notions admits an immediate translation to general quasi-linear maps on Banach lattices.

3. Disjoint local triviality

Definition 3.1**.**

A quasi-linear map $\Omega:\mathcal{K}\to Y$ defined on a Banach lattice is said to be disjointly trivial if it is trivial on any subspace generated by a sequence of disjointly supported elements. It is said to be locally disjointly trivial if there exists $C>0$ such that for any finite dimensional subspace $F$ of $\mathcal{K}$ generated by disjointly supported vectors, there exists a linear map $L_{F}$ such that $\|\Omega_{|F}-L_{F}\|\leq C$.

We can show:

Proposition 3.2**.**

Let $\Omega:\mathcal{K}\to Y$ be a quasi-linear map on a Köthe space $\mathcal{K}$. Consider the following assertions:

• (i)

$\Omega$* is trivial.*

• (ii)

$\Omega$* is disjointly trivial.*

• (iii)

$\Omega$* is locally disjointly trivial.*

• (iv)

$\Omega$* is locally trivial.*

Then $(i)\Rightarrow(ii)\Rightarrow(iii)\Leftrightarrow(iv)$. Moreover, if $Y$ is complemented in its bidual, then all assertions are equivalent.

Proof.

Assertions $(i)$ and $(iv)$ are well-known to be equivalent: trivial implies locally trivial while, see [5], a locally trivial quasi-linear map taking values in a space complemented in its bidual is trivial. That $(i)$ implies $(ii)$ is obvious. Let us show that $(iii)$ implies $(iv)$: Let $\Omega$ be a quasi-linear map verifying $(iii)$ and let $F$ be a finite dimensional subspace of $\mathcal{K}$. Approximating functions by characteristic functions we may find a nuclear operator $N$ on $\mathcal{K}$ of arbitrary small norm so that $(Id+N)(F)$ is contained in the linear span $[u_{n}]$ of a finite sequence of disjointly supported vectors. The restriction $\Omega_{|[u_{n}]}$ is trivial with constant $C$, thus using [9, Lemma 5.6], we get that $\Omega=\Omega(I+N)-\Omega N$ is trivial with constant $C+\epsilon$ on $F$. Therefore $(iv)$ holds.

It remains to show that $(ii)$ implies $(iii)$. Let $\Sigma$ be the $\sigma$-finite base space on which the Köthe function space $\mathcal{K}$ is defined. For a subset $A\subset\Sigma$ we will denote $\mathcal{K}(A)$ the subspace of $\mathcal{K}$ formed by those functions with support contained in $A$.

Claim 1. If $A$ and $B$ are disjoint and $\Omega$ is trivial on both $\mathcal{K}(A)$ and $\mathcal{K}(B)$ then it is trivial on $\mathcal{K}(A\cup B)$. Indeed, if $\|\Omega_{|\mathcal{K}(A)}-a\|\leq c$ and $\|\Omega_{|\mathcal{K}(B)}-b\|\leq d$, where $a$ and $b$ are linear, then $\|\Omega_{|\mathcal{K}(A\cup B)}-(a\oplus b)\|\leq 2(Z(\Omega)+c+d)$, where $a\oplus b$ is the obvious linear map on $\mathcal{K}(A\cup B)$.

Claim 2. If $\Omega$ is nontrivial on $X$ then $\Sigma$ can be split in two sets $\Sigma=A\cup B$ so that $\Omega_{|\mathcal{K}(A)}$ and $\Omega_{|\mathcal{K}(B)}$ are both nontrivial. We first assume that $\Sigma$ is a finite measure space. Assume the claim does not hold. Split $\Sigma=R_{1}\cup I_{1}$ in two sets of the same measure and assume $\Omega_{|\mathcal{K}(I_{1})}$ is trivial. Note that since the claim does not hold, given any $C\subset\Sigma$ and any splitting $C=A\cup B$ the map $\Omega$ is trivial on $\mathcal{K}(A)$ or $\mathcal{K}(B)$. So, split $R_{1}=R_{2}\cup I_{2}$ in two sets of equal measure and assume that $\Omega_{|\mathcal{K}(I_{2})}$ is trivial, and so on. If $\Omega$ is $\lambda$-trivial on $\mathcal{K}(\cup_{j\leq n}I_{j})$ for $\lambda<+\infty$ and for all $n$ then $\Omega$ is locally trivial on $\mathcal{K}$ and therefore is trivial, a contradiction. If $\lambda_{n}\to\infty$ is such that $\Omega_{\mathcal{K}(\cup_{j\leq n}I_{j})}$ is $\lambda_{n}+1$-trivial but not $\lambda_{n}$-trivial for all $n$, then by the Fact we note that for $m<n$, $\Omega$ cannot be trivial with constant less than $\lambda_{n}/2-Z(\Omega)-\lambda_{m}-1$ on $\mathcal{K}(\cup_{m<j\leq n}I_{j})$. From this we find a partition of $\mathbb{N}$ as $N_{1}\cup N_{2}$ so that if $A=\cup_{n\in N_{1}}I_{n}$ and $B=\cup_{n\in N_{2}}I_{n}$, then $\Omega$ is non trivial on $\mathcal{K}(A)$ and $\mathcal{K}(B)$, another contradiction.

If $\Sigma$ is $\sigma$-finite then the proof is essentially the same: either one can choose the sets $I_{n}$ all having measure, say, $1$ or at some step $R_{m}$ is of finite measure, and we are in the previous case. This concludes the proof of the claim.

We pass to complete the proof that $(ii)$ implies $(iii)$. Assume that $\Omega$ is not trivial on $\mathcal{K}$. By the claim, split $\Sigma=A_{1}\cup B_{1}$ so that $\Omega$ is trivial neither on $\mathcal{K}(A_{1})$ nor on $\mathcal{K}(B_{1})$. It cannot be locally trivial on them, so there is a finite number $\{u^{1}_{n}\}_{n\in F_{1}}$ of disjointly supported vectors on $\mathcal{K}(A_{1})$ on which $\Omega$ is not $2$-trivial. By the claim applied to $\mathcal{K}(B_{1})$ split $B_{1}=A_{2}\cup B_{2}$ so that $\Omega$ is trivial neither on $\mathcal{K}(A_{2})$ nor in $\mathcal{K}(B_{2})$. It cannot be locally trivial on them, so there is a finite number of disjointly supported vectors $\{u^{2}_{n}\}_{n\in F_{2}}$ on $X(A_{2})$ on which $\Omega$ is not $4$-trivial. Iterate the argument to produce a subspace $Y$ generated by an infinite sequence

[TABLE]

of disjointly supported vectors, where $\Omega$ cannot be trivial.∎

An immediate corollary of (the proof of) Proposition 3.2 is:

Corollary 3.3**.**

Given a Köthe space $\mathcal{K}$ with base space $(\Sigma,\mu)$ and a non-locally trivial quasi-linear map $\Omega$ defined on $\mathcal{K}$ then there is a sequence $(A_{n})$ of finite measure mutually disjoint subsets of $\Sigma$ so that the restriction $\Omega_{|[1_{A_{n}}]}$ is not locally trivial.

In $L_{p}(\Sigma,\mu)$ a subspace $[1_{A_{n}}]$ is isomorphic to $\ell_{p}$ when $0<p<+\infty$ and to $c_{0}$ in $L_{\infty}(\Sigma,\mu)$. It is not clear whether Proposition 3.2 can be translated to the domain of Banach lattices. $C(K)$-spaces are not, as a rule, Köthe spaces; however, the following essential part of Proposition 3.2 still survives [6, Theorem 2.1] and since a disjointly supported sequence in a $C(K)$-space generates $c_{0}$, one has:

Corollary 3.4**.**

Let $0<p<\infty$. Given a non-locally trivial map $\Omega$ defined on $L_{p}(\Sigma,\mu)$ (resp. $L_{\infty}(\Sigma,\mu)$, $C(K)$) there is a copy of $\ell_{p}$ (resp. $c_{0}$) spanned by disjointly supported vectors on which the restriction of $\Omega$ is not locally trivial.

4. Disjoint singularity

Theorem 3.2 shows that (local) triviality and disjoint (local) triviality are essentially equivalent. We shall now see that the situation is much more complex regarding singularity notions.

Definition 4.1**.**

A quasi-linear map on a Banach lattice is called disjointly singular if its restriction to every infinite dimensional subspace generated by a disjointly supported sequence is never trivial.

Of course that a singular quasi-linear map is disjointly singular and a disjointly singular quasi-linear map on a Köthe sequence space is singular. An open question, to the best of our knowledge due to Félix Cabello, is about the existence of singular quasi-linear maps on Köthe function spaces; recall that no singular $L_{\infty}$-centralizers exist on any reasonable Köthe space [2] (cf. Proposition 2.3); see also [33]).

4.1. Examples

1. (1)

As we mentioned at the introduction, the methods in [10] actually produce disjointly singular centralizers. In particular, it is shown [10, Proposition 5.4] that the Kalton-Peck centralizer

[TABLE]

is disjointly singular on any reflexive, $p$-convex Köthe function space, $p>1$ 2. (2)

Given two Lorentz spaces $L_{p_{0},q_{0}},L_{p_{1},q_{1}}$, it was proved in [4] that $(L_{p_{0},q_{0}},L_{p_{1},q_{1}})_{\theta}=L_{p,q}$ for $p^{-1}=(1-\theta){p_{0}}^{-1}+\theta{p_{1}}^{-1}$ and $q^{-1}=(1-\theta){q_{0}}^{-1}+\theta{q_{1}}^{-1}$ with associated derivation

[TABLE]

Here $\mathscr{K}(\cdot)$ is the Kalton-Peck map earlier defined and $\kappa(\cdot)$ is the so-called Kalton map [19]; see also [4], given by $\kappa(x)=x\;r_{x}$ where $r_{x}$ is the rank function $r_{x}(t)=m\{s:|x(s)|>|x(t)|$ or$|x(s)|=|x(t)|$ and $s\leq t\}$ (see [32]).

The map $\mathscr{K}$ is disjointly singular while $\kappa$ has the property that every infinite dimensional subspace contains a further infinite dimensional subspace where it is trivial [4], so it is clear that $\Omega$ is disjointly singular. 3. (3)

A different set of examples will be presented now in $C(K)$ or $L_{\infty}$ spaces. These examples are relevant because no singular quasi-linear map is possible on a space containing $\ell_{1}$, say $C[0,1]$ or $\ell_{\infty}$. It is necessary to remark that $C[0,1]$ is not a Köthe space and thus the example lives in the domain of Banach lattices; see the comments after the examples.

Proposition 4.2**.**

There exist disjointly singular quasi-linear maps on $C[0,1]$ and $\ell_{\infty}$.

Proof.

Let us consider first the case of $C[0,1]$. As we have already remarked, one just needs to construct a $c_{0}$-singular map. Let $\omega:c_{0}\to C[0,1]$ be a nontrivial quasi-linear map (see [7] for explicit examples). Let $\Gamma$ the set of all $2$-isomorphic copies $\gamma$ of $c_{0}$ inside $C[0,1]$, which are necessarily $4$-complemented in $C[0,1]$ via some projection $\pi_{\gamma}$ and let $\alpha_{\gamma}:\gamma c_{0}$ be a 2-isomorphism. Define a quasi-linear map $\Upsilon:C[0,1]\to\ell_{\infty}(\Gamma,C[0,1])$ by means of

[TABLE]

This map is $c_{0}$-singular because if there is a copy of $c_{0}$ in which $\Upsilon$ is trivial, that copy must contains some $\gamma\in\Gamma$, on which $\Upsilon$ must be trivial too. But if $f\in\gamma$ one has

[TABLE]

thus, if $\delta_{\gamma}:\ell_{\infty}(\Gamma,C[0,1])\to C[0,1]$ is the canonical evaluation at the coordinate $\gamma$ we have obtained $\delta_{\gamma}\Upsilon_{|\gamma}=\omega\alpha_{\gamma}$. This map cannot be trivial since, otherwise, so it would be $\omega=\delta_{\gamma}\Upsilon_{|\gamma}\alpha_{\gamma}^{-1}$, which is not. But that means that $\Upsilon_{|}\gamma$ cannot be trivial because $\delta_{\gamma}\Upsilon_{|\gamma}$ is not trivial.

A standard reduction (see [13]) allows one to find an equivalent quasi-linear map $\Omega:C[0,1]\to\ell_{\infty}(\Gamma,C[0,1])$ having separable range. Since $\ell_{\infty}(\Gamma,C[0,1])$ is a Banach algebra, $\Omega$ can also be considered taking values in the closed subalgebra generated by $[\Omega(C[0,1])]$, which, being separable, is contained in $C[0,1]$. Thus, $\Omega:C[0,1]\to C[0,1]$ is a $c_{0}$-singular quasi-linear map, as desired.

The case of $\ell_{\infty}$ has to be treated differently because the projections $\pi_{\gamma}$ do not exist now. Pick to start a nontrivial quasi-linear map $\omega:c_{0}\to\ell_{2}$, which can be constructed as follows: pick the Kalton-Peck map $\mathscr{K}:\ell_{2}\to\ell_{2}$ and a quotient map $Q:C[0,1]\to\ell_{2}$. The map $\mathscr{K}Q$ is not trivial (see [5, 7]). It cannot be locally trivial either since $\ell_{2}$ is reflexive and Proposition 3.2 would make it trivial. Thus, using Corollary 3.4 (cf. [6, Theorem 2.1]) there is a copy of $c_{0}$ inside $C[0,1]$ via some isomorphic embedding $j$ so that the restriction $\mathscr{K}Qj$ is not trivial. Let us simplify and call this map $\omega$. Let $\Gamma$ be the set of infinite sequences of finite subsets $\mathbb{N}$. Given such a sequence $\gamma=(A_{n})$ we will call $\overline{\gamma}=\cup_{A_{n}\in\gamma}A$. Let also $\alpha_{\gamma}:[1_{A_{n}}]\to c_{0}$ be an isometry. Define a quasi-linear map $\Upsilon:c_{0}\to\ell_{\infty}(\Gamma,\ell_{2})$ as

[TABLE]

The bidual map $\Upsilon^{**}:\ell_{\infty}\to\ell_{\infty}(\Gamma,\ell_{2})^{**}$ cannot be trivial either since $\ell_{\infty}(\Gamma,\ell_{2})$ is complemented in its bidual. If $\pi$ denotes a projection, the map $\pi\Upsilon^{**}:\ell_{\infty}\to\ell_{\infty}(\Gamma,\ell_{2})$ cannot be trivial either. We define a new map $\Omega:\ell_{\infty}\to\ell_{\infty}(\Gamma\times\Gamma,\ell_{2})$ in the form

[TABLE]

This map $\Omega$ cannot be disjointly singular: if it becomes trivial on some $\gamma$ then for $x\in\gamma$ one has

[TABLE]

This map cannot be trivial since $\alpha_{\gamma}$ is an isomorphism and $\omega$ is not trivial.∎

It is an open problem posed in [2] whether there exists a singular quasi-linear map $\Omega:L_{p}\to L_{p}$ for $0<p<2$. Singular quasi-linear maps (not centralizers) $\Omega:L_{p}\to L_{p}$ exist for $2\leq p<+\infty$ (see [8, Theorem 2(c)]); observe that in this case the Kadec-Pełczyński alternative immediately yields that a quasilinear map $\Omega$ on $L_{p}$ that is both disjointly singular and $\ell_{2}$-singular must be singular. Thus, we could use a construction similar to that in Proposition 4.2 to obtain singular maps in $L_{p}$, $2\leq p<+\infty$. None of these can be $L_{\infty}$-centralizers, nonetheless.

The papers [15, 16, 17] study the behaviour of strictly singular operators in Banach lattices by considering the more general notion of lattice singular operator (one for which no restriction to an infinite dimensional sublattice is an isomorphism). Obviously, strict singularity implies lattice singularity and this implies disjoint singularity. The authors obtain an interesting result [15]: Let $X,Y$ be Banach lattices such that $X$ has finite cotype and $Y$ admits a lower $2$-estimate. Then an operator $T:X\to Y$ is strictly singular if and only if it is disjointly singular and $\ell_{2}$-singular. A non-vacuous centralizer version for this result is not possible since Proposition 2.3 establishes that no $L_{\infty}$-centralizer can be $\ell_{2}$-singular. It makes however sense the question of which conditions ensure that a quasi-linear map on a Köthe space that is simultaneously disjointly singular and $\ell_{2}$-singular is necessarily singular.

4.2. Characterizations

Regarding characterizations, given a quasi-linear map $\Omega:\mathcal{K}\curvearrowright\mathcal{K}$ the fact that the twisted sum space $d_{\Omega}\mathcal{K}$ is not necessarily a Köthe space complicates the characterization of disjointly singular maps in terms of the quotient operator. This difficulty can be overcome for centralizers, which always admit a version satisfying that $\mathop{\mathrm{supp}}\nolimits\Omega x\subset\mathop{\mathrm{supp}}\nolimits x$ for all $x\in\mathcal{K}$. Although, as we have just said, the space $d_{\Omega}\mathcal{K}$ is not a Köthe space, its elements are couples of functions of $L_{0}$; i.e., functions $S\to\mathbb{C}\times\mathbb{C}$. The following definition makes sense:

Definition 4.3**.**

A pair of nonzero elements $f=(w_{0},x_{0}),g=(w_{1},x_{1})$ of $d_{\Omega}\mathcal{K}$ are said to be disjoint if the functions $f,g:S\to\mathbb{C}\times\mathbb{C}$ are disjointly supported. An operator $\tau\,:\,d_{\Omega}\mathcal{K}\to\mathcal{K}$ is said to be disjointly singular if the restriction of $\tau$ to any infinite dimensional subspace generated by a disjoint sequence of vectors is not an isomorphism.

One has:

Lemma 4.4**.**

A centralizer $\Omega$ on a maximal Köthe space $\mathcal{K}$ is disjointly singular if and only if $q_{\Omega}$ is disjointly singular.

Proof.

We choose the form $q_{\Omega}:\mathcal{K}\oplus_{\Omega}\mathcal{K}\to\mathcal{K}$ given by $q_{\Omega}(v,u)=u$. If $\Omega$ is trivial on the span $[(u_{n})]$ of a disjointly supported sequence there is a linear map $L:[u_{n}]\to\mathcal{K}$ so that $\|\Omega-L\|\leq C$. From [10, Lemma 3.17] we get that there is a linear map $\Lambda:[u_{n}]\to\mathcal{K}$ such that $\mathop{\mathrm{supp}}\nolimits\Lambda x\subset\mathop{\mathrm{supp}}\nolimits x$ and $\|\Omega-\Lambda\|\leq C$ and thus $(\Lambda u_{n},u_{n})$ is a disjointly supported sequence on which $q_{\Omega}$ is trivial since

[TABLE]

In this way, $q_{\Omega}$ disjointly singular implies $\Omega$ disjointly singular. To get the converse, assume that $q_{\Omega}$ is not disjointly singular, so there is a disjointly supported sequence $(v_{n},u_{n})$ in $\mathcal{K}\oplus_{\Omega}\mathcal{K}$ such that $q_{\Omega}$ is an isomorphism on $[(v_{n},u_{n})]$. This means that

[TABLE]

The linear map $L(u_{n})=v_{n}$ verifies

[TABLE]

∎

Now we want to mimicry Proposition 2.2. Let $\mathcal{K}$ be a Köthe space and let $\Omega:\mathcal{K}\to L_{0}$ be a centralizer. Given a finite sequence $b=(b_{k})_{k=1}^{n}\subset\mathcal{K}$ we will follow [9] and define

[TABLE]

where the average is taken over all the signs $\pm 1$. The triangle inequality holds for $\nabla_{[b]}\Omega$: if $\Omega$ and $\Psi$ are centralizers then $\nabla_{[b]}(\Omega+\Psi)\leq\nabla_{[b]}\Omega+\nabla_{[b]}\Psi.$ If $\lambda=(\lambda_{k})_{k}$ is a finite sequence of scalars and $x=(x_{k})_{k}$ a sequence of vectors of $\mathcal{K}$, we write $\lambda x$ to denote the finite sequence obtained by the non-zero vectors of $(\lambda_{1}x_{1},\lambda_{2}x_{2},\ldots)$.

Recall from [10, Definiton 3.10] that a centralizer $\Omega$ on a Köthe function space $\mathcal{K}$ is contractive if $\mathop{\mathrm{supp}}\nolimits\Omega(x)\subseteq\mathop{\mathrm{supp}}\nolimits x$ for every $x\in\mathcal{K}$. Our next result provides a characterization of disjointly singularity for contractive centralizers in the $L_{p}$ spaces. The contractive restriction is not so severe, since every centralizer $\Omega$ on a Köthe function space $\mathcal{K}$ admits a contractive centralizer $\omega$ such that $\Omega-\omega$ is bounded ([21, Proposition 4.1]). Also it is easy to see that the canonical centralizer induced by interpolation of Köthe spaces is contractive, see [10].

Proposition 4.5**.**

A contractive centralizer $\Omega$ defined on $L_{p}$ is not disjointly singular if and only if there is a disjointly supported normalized sequence $u=(u_{n})_{n}$ and a constant $C>0$ such that for every $\lambda=(\lambda_{k})_{k}\in c_{00}$ one has

[TABLE]

The proof follows from the following three lemmas.

Lemma 4.6**.**

Let $\Omega$ be a contractive centralizer on a Köthe space $\mathcal{K}$ satisfying an upper $p$-estimate, and let $u=(u_{n})_{n}$ be a disjointly supported normalized sequence of vectors. Suppose that the restriction of $\Omega$ to the closed linear span of the $u_{n}$’s is trivial. Then there is a constant $C>0$ such that

[TABLE]

for every $\lambda=(\lambda_{k})_{k}\in c_{00}$.

Proof.

If the restriction of $\Omega$ to $[u_{n}]$ is trivial then there is a linear map $L:[u_{n}]\to\mathcal{K}$ so that $\|\Omega-L\|\leq C<+\infty$. From [10, Lemma 3.17] we can take such $L$ so that $\mathop{\mathrm{supp}}\nolimits L(x)\subset\mathop{\mathrm{supp}}\nolimits x$. Then for every $\lambda\in c_{00}$

[TABLE]

which implies that

[TABLE]

From where

[TABLE]

∎

Lemma 4.7**.**

Let $\Omega$ be a contractive centralizer on a Köthe space $\mathcal{K}$. Then there exists a constant $c>0$ such that for every disjointly supported normalized sequence $(v_{i})$ of $\mathcal{K}$ and every $n\in\mathbb{N}$

[TABLE]

Proof.

Let $(\epsilon_{i})$ be a sequence of signs. Let $v=\sum_{i=1}^{n}v_{i}$ and $\sum_{i=1}^{n}\epsilon_{i}v_{i}=\epsilon v$ for some function $\epsilon$ taking values $\pm 1$.

[TABLE]

Thus the centralizer $\Omega$ verifies for some constant $c>0$,

[TABLE]

Since $\Omega$ is contractive, then also the $\Omega(v_{i})$ are disjointly supported. Applying to $\epsilon_{i}v_{i}$ instead of $v_{i}$,

[TABLE]

By taking the average

[TABLE]

∎

Lemma 4.8**.**

Let $\Omega$ be a contractive centralizer on a Köthe space $\mathcal{K}$ satisfying a lower $q$-estimate, and let $u=(u_{n})_{n}$ be a disjointly supported normalized sequence of vectors. Suppose that there is a constant $C>0$ such that

[TABLE]

for every $\lambda=(\lambda_{k})_{k}\in c_{00}$. Then the restriction of $\Omega$ to the closed linear span of the $u_{n}$’s is trivial.

Proof.

Let $\lambda=(\lambda_{k})_{k}\in c_{00}$ and $(\epsilon_{i})$ be a sequence of signs. It follows from the previous lemma that for some constant $c>0$ and every $n\in\mathbb{N}$

[TABLE]

Since $\mathcal{K}$ satisfies a lower $q$-estimate

[TABLE]

Then $\|\Omega-L\|\leq C^{\prime}$, where $L$ is a linear map such that $L(u_{i})=\Omega(u_{i})$. ∎

5. Disjoint super singularity

It is part of the folklore that ultrapowers of Köthe spaces are again Köthe spaces. Thus, it makes sense to define an operator $T:\mathcal{K}\to Y$ to be super-disjointly singular if every ultrapower of $T$ is disjointly singular; this means that for every sequence of subspaces $E_{n}\subseteq\mathcal{K}$ that are generated by disjointly supported elements and so that $\dim E_{n}=n$ there is a sequence $(F_{n})$ of subspaces, $F_{n}\subset E_{n}$ generated by disjointly supported elements such that $\dim F_{n}\to\infty$ and $\lim\|T_{|F_{n}}\|\to 0$. To transplant these ideas to the domain of quasi-linear maps $\Omega$ on Köthe function spaces it will be useful to define the modulus of superdisjoint singularity of a quasi-linear map $\Omega$ as

[TABLE]

where the infimum is taken over all $n$-dimensional subspaces $E_{n}$ of $\mathcal{K}$ generated by disjointly supported vectors. One has:

Lemma 5.1**.**

Let $\Omega:\mathcal{K}\to Y$ be a quasi-linear map defined on a Köthe space. The following are equivalent

1. (1)

All ultrapowers of $\Omega$ are disjointly singular. 2. (2)

$\lim\psi_{\Omega}(n)=+\infty$.

If $\Omega$ is a centralizer, the conditions above are equivalent to 3. (3)

The quotient map $q_{\Omega}$ is super-disjointly singular.

Proof.

Condition (1) says that it does not exist $c>0$ and a sequence of finite dimensional subspaces $F_{n}$ of $Y\oplus_{\Omega}\mathcal{K}$ such that $E_{n}=q_{\Omega}(F_{n})$ is generated by $n$ disjointly supported vectors so that $\|q_{\Omega}(x)\|\geq c\|x\|$ for every $x\in\cup F_{n}$. But if $\Omega$ is $C$-trivial on $E_{n}$, which is generated by the disjointly supported vectors $[u_{i}]_{i=1}^{n}$ then we claim that there is a linear map $L_{n}:E_{n}\to\mathcal{K}$ such that $\mathop{\mathrm{supp}}\nolimits\,L_{n}(x)\subseteq\,\mathop{\mathrm{supp}}\nolimits\,x$ for every $x\in E_{n}$ and $\|\Omega|_{E_{n}}-L_{n}\|\leq C$: Indeed assume $L$ is linear such that $\|(\Omega-L)_{|E_{n}}\|\leq C$, and let $G$ be the finite group of units generated by the vectors $v_{i}$ that take value $1$ on the support of $u_{i}$ and $-1$ elsewhere); then it is enough to pick $L_{n}(x)={\rm Ave}_{v\in G}\;\;vL(vx)$. The rest of the argument goes as in Lemma 4.4. Done that, $\Omega_{\mathscr{U}}$ is trivial on $(E_{n})_{\mathscr{U}}$, which yields the equivalence between (1) and (2). The equivalence with (3) follows from Lemma 4.4. ∎

Definition 5.2**.**

A quasi-linear map (rep. a centralizer) $\Omega:\mathcal{K}\to Y$ on a Köthe space is said to be super disjointly singular if it satisfies the two (resp. three) equivalent conditions in the Lemma.

It is clear that either singularity or super disjoint singularity imply disjoint singularity. We will present two proofs for the following fact.

Proposition 5.3**.**

The Kalton-Peck map on $L_{p}$ is super disjointly singular for $1<p<\infty$.

Proof.

Assume there exist a linear map $L:E_{n}=[u]\to L_{p}$ and a constant $C>0$ so that $||\mathscr{K}_{|E_{n}}-L||\leq C$. By the proof of the previous lemma we may assume that $\mathop{\mathrm{supp}}\nolimits\,L(u_{i})\subseteq\,\mathop{\mathrm{supp}}\nolimits\,u_{i}$ for all $i$. Put $\Omega^{\prime}=\mathscr{K}_{|E_{n}}-L$ to get

[TABLE]

which is impossible.∎

Two functions $f,g:\mathbb{N}\to\mathbb{R}^{+}$ are called equivalent, and denoted $f\sim g$, if $0<\lim\inf f(n)/g(n)\}\leq\limsup f(n)/g(n)<+\infty$. We recall from [10] the parameter

[TABLE]

The interest of this parameter lies in [10, Proposition 5.3]:

Proposition 5.4**.**

Let $(X_{0},X_{1})$ be an interpolation couple of two Köthe function spaces so that $M_{X_{0}}$ and $M_{X_{1}}$ are not equivalent. Let $0<\theta<1$. Assume that $X_{\theta}$ is reflexive, that $M_{W}\sim M_{X_{\theta}}$ for every infinite-dimensional subspace generated by a disjoint sequence $W\subset X_{\theta}$, and $M_{X_{\theta}}\sim M_{X_{0}}^{1-\theta}M_{X_{1}}^{\theta}$. Then $\Omega_{\theta}$ is disjointly singular.

We observe that:

Lemma 5.5**.**

Let $X$ be a Köthe space. Then $M_{X}\sim M_{X_{\mathscr{U}}}$

Proof.

Since $X\subset X_{\mathscr{U}}$ it is clear that $M_{X}\leq M_{X_{\mathscr{U}}}$. Given $n$, pick $u^{k}=[u^{k}_{i}]\in X_{\mathscr{U}}$ for $1\leq k\leq n$, disjointly supported so that $M_{X_{\mathscr{U}}}(n)\sim\|u^{1}+\dots+u^{n}\|$ (we can freely assume that all $u^{k}_{i}$ are norm one elements). Since $u^{t}$ and $u^{s}$ are disjointly supported this means that the set of all $i$ so that $u^{s}_{i}$ and $u^{t}_{i}$ are disjointly supported belongs to $\mathscr{U}$. And the same for the set $A$ of all $i$ so that all $\{u^{k}_{i},1\leq i\leq n\}$ are disjointly supported. Since $B=\{j:\|u^{1}(j)+\dots+u^{n}(j)\|\geq M_{X_{\mathscr{U}}}(n)-\varepsilon\}\in\mathscr{U}$, also $A\cap B\in\mathscr{U}$. Thus, for any $i\in A\cap B$ we have $M_{X_{\mathscr{U}}}(n)-\varepsilon\ \leq\|u^{1}(i)+\dots+u^{n}(i)\|\leq M_{X}(n)$. ∎

In the case of Köthe spaces, complex interpolation is actually simple factorization. Recall that given two Köthe function spaces $Y,Z$ we define the space

[TABLE]

endowed with the quasi-norm $\|x\|=\inf\|y\|_{Y}\|z\|_{Z}$ where the infimum is taken on all factorizations as above. Now, assuming that one of the spaces $X_{0}$, $X_{1}$ has the Radon-Nikodym property, the Lozanovskii decomposition formula allows us to show (see [23, Theorem 4.6]) that the complex interpolation space $X_{\theta}$ is isometric to the space $X_{0}^{1-\theta}X_{1}^{\theta}$, with

[TABLE]

If $a_{0}(x),a_{1}(x)$ is an $(1+\epsilon)$-optimal Lozanovskii decomposition for $x$ then it is standard (see [10]) that

[TABLE]

Lemma 5.6**.**

Let $(X_{0},X_{1})$ be a couple of Kothe function spaces with non trivial concavity. Let $\mathscr{U}$ be an ultrafilter on $\mathbb{N}$ then $(X_{\theta})_{\mathscr{U}}=((X_{0})_{\mathscr{U}},(X_{1})_{\mathscr{U}})_{\theta}$ and $(\Omega_{\theta})_{\mathscr{U}}=(\Omega_{\mathscr{U}})_{\theta}$.

Proof.

According to [31], given an interpolation couple $(A,B)$ of Köthe spaces with non-trivial concavity their ultrapowers $(A_{\mathscr{U}},B_{\mathscr{U}})$ form an interpolation couple. The point now is to show that $(X_{0}^{1-\theta}X_{1}^{\theta})_{\mathscr{U}}=(X_{0})_{\mathscr{U}}^{1-\theta}(X_{1})_{\mathscr{U}}^{\theta}$. Indeed, given $[x_{i}]\in(X_{0}^{1-\theta}X_{1}^{\theta})_{\mathscr{U}}$ then pick an almost optimal factorization $x_{i}=y_{i}z_{i}$ and then $[x_{i}]=[y_{i}][z_{i}]$ is an almost optimal factorization. Conversely, if $x\in(X_{0})_{\mathscr{U}}^{1-\theta}(X_{1})_{\mathscr{U}}^{\theta}$ and set $x=[y_{i}][z_{i}]$ an almost optimal factorization then of course that $x_{i}=y_{i}z_{i}$ is not an almost optimal factorization for $x_{i}$, but it is so when the indices $i$ belong to a certain element of $\mathscr{U}$, and thus $[x_{i}]\in(X_{\theta})_{\mathscr{U}}$. The assertion about the induced centralizer follows from this. ∎

To apply the general criteria proved in [10] (see below) we need to analyze the estimate $M_{W}$ associated to any subspace $W$ generated by a sequence of disjoint vectors of $(X_{\theta})_{\mathscr{U}}$. As a rule, it is false that the ultrapower or the interpolated space is the interpolated between ultrapowers. To overcome this we concentrate first on the test case in which $X_{\theta}$ is an $L_{p}(\mu)$-space. In this situation $M_{W}(n)\sim n^{1/p}$ for all subspaces $W$ of $(X_{\theta})_{\mathscr{U}}$ generated by disjointly supported vectors, and from Proposition 5.4 we deduce:

Proposition 5.7**.**

Let $(X_{0},X_{1})$ be an interpolation couple of two Köthe function spaces and let $0<\theta<1$ so that $X_{\theta}$ is an $L_{p}(\mu)$-space. If

[TABLE]

Then the induced centralizer $\Omega_{\theta}$ on $X_{\theta}$ is super disjointly singular

Proof.

Thanks to Lemma 5.5 we observe that the hypotheses and Proposition 5.4 imply that the centralizer $(\Omega_{\mathscr{U}})_{\theta}$ is disjointly singular. By Lemma 5.6 we conclude that $\Omega_{\theta}$ is super disjointly singular. ∎

This provides the second proof that the Kalton-Peck centralizer is super disjointly singular on $L_{p}$-spaces. Let us present more examples

• •

If $\mathcal{S}$ denotes the Schreier space then $(\mathcal{S},\mathcal{S}^{*})_{1/2}=\ell_{2}$ then the associated centralizer is super disjointly singular. Since these are Köthe sequence spaces, it is also singular. All this follows from the estimates $M_{\mathcal{S}}(n)=n$ and $M_{\mathcal{S}^{*}}(n)\sim\log_{2}(n)$.

In the case of Köthe spaces on a discrete measure space (i.e. the unconditional basis case), as a consequence of the fact that (disjoint) singularity and ”block” singularity are equivalent, the conclusion of Proposition 5.7 still holds if one replaces the parameter $M_{X}$ by the parameter $M_{X}^{s}$, where the supremum is over sucessive vectors instead of disjointly supported. Therefore:

• •

If $\mathscr{S}$ denotes the Schlumprecht space then $(\mathscr{S},\mathscr{S}^{*})_{1/2}=\ell_{2}$ then the associated centralizer is super disjointly singular. Since these are Köthe sequence spaces, it is also singular. This follows from the estimates $M_{\mathscr{S}}^{s}(n)=n$ and $M_{\mathscr{S}^{*}}^{s}(n)\sim\log_{2}(n)$.

• •

Let $L_{p_{0},p_{1}},L_{p_{1},q_{1}}$ be Lorentz function spaces. Then, $(L_{p_{0},p_{1}},L_{p_{1},q_{1}})_{\theta}=L_{p,q}$ as in Section 4.1 and the associated centralizer is singular when $q_{0}\neq q_{1}$ and super disjointly singular when $\min\{p_{0},p_{1}\}\neq\min\{p_{1},q_{1}\}$ as it follows from the estimate $M_{L_{p,q}}(n)=n^{\frac{1}{\min\{p,q\}}}$. Observe that in this case we require a variation of Proposition 5.7: it is not true now that “$X_{\theta}$ is an $L_{p}(\mu)$-space”; rather “$X_{\theta}$ is an $L_{p,q}(\mu)$-space” and thus their subspaces generated by disjointly supported vectors are $\ell_{p,q}$, whose parameters are the same as those of $L_{p,q}$.

Although it es easy to believe that disjoint singularity implies super disjoint singularity, it is not so:

Proposition 5.8**.**

Let $1\leq p_{1}<p_{0}\leq\infty$ and $0<\theta<1$. For $p^{-1}=(1-\theta)p_{1}^{-1}+\theta p_{0}^{-1}$ one has $(\ell_{p_{0}}(\bigoplus\ell_{2}^{n}),\ell_{p_{1}}(\bigoplus\ell_{2}^{n}))_{\theta}=\ell_{p}(\bigoplus\ell_{2}^{k})$ with associated centralizer

[TABLE]

Thus, if $p_{1}<p_{0}<2$ then $\Omega$ is disjointly singular but not super disjointly singular

Proof.

The map $\Omega$ has been obtained [12, Theorem 3.4]. It is disjointly singular by Proposition 5.4 and is not super disjointly singular since $\Omega(x)=0$ for every $x\in\ell_{2}^{k}$ and every $k\in\mathbb{N}$.∎

A few more precise estimates can be presented. Observe first that given a centralizer $\Omega:\mathcal{K}\curvearrowright\mathcal{K}$ on a Köthe function space then for $n$ disjoint vectors $(u_{i})_{1}^{n}$ in the unit ball one has

[TABLE]

for some linear map $L:[u_{n}]\to\mathcal{K}$. Let us invoke now the estimate [10, Proposition 5.1]: given two Köthe spaces $(X_{0},X_{1})$ (on the same base space), fixing $0<\theta<1$ and considering $\Omega_{\theta}$ the induced centralizer on $X_{\theta}=X_{0}^{1-\theta}X_{1}^{\theta}$ then

[TABLE]

Therefore, in the same conditions as above, we get

[TABLE]

Let us define a new parameter

[TABLE]

Observe that $m_{\mathcal{K}}(n)M_{\mathcal{K}^{*}}(n)\geq n$ for every Köthe function space. We obtain

[TABLE]

which yields

[TABLE]

In particular:

• •

For $\Omega_{1/2}$ the centralizer obtained on $\ell_{2}=(\mathcal{S},\mathcal{S}^{*})_{1/2}$ when $\mathcal{S}$ is the Schreier space, one gets

[TABLE]

which shows that also the centralizers $\Omega_{1/2}$ are super-disjointly singular.

• •

In general, under a few minimal conditions [10, Proposition 6.2], on a Köthe function space $\mathcal{K}$ with base space $S$ one has $(X,X^{*})_{1/2}=L_{2}(S)$; and thus, if $M_{\mathcal{K}}$ and $M_{\mathcal{K}^{*}}$ are not equivalent then the induced centralizer on $L_{2}(S)$ is super disjointly singular.

• •

For $\Omega_{\theta}$ the Kalton-Peck map on $L_{p}$ obtained from $X_{0}=\L_{1},X_{1}=L_{\infty}$ and $\theta=1/p^{*}$ one gets

[TABLE]

which, as promised, shows again that $\mathscr{K}_{p}$ is super disjointly singular.

• •

Assume more generally that $p>1$ and $X$ is a $p$-convex Köthe space with base space $S$. Then $X=(L_{\infty}(S),X^{p})_{1/p}$, where $X^{p}$ denotes the $p$-concavification of $X$, and [10] Proposition 3.7, this induces as centralizer the map $p{\mathcal{K}}$, where ${\mathcal{K}}(f)=f\log(|f|/\|f\|)$ is the Kalton-Peck map on $X$. Since $M_{X^{p}}(n)=M_{X}(n)^{p}$, we obtain the following criteria for the super DSS property of Kalton-Peck map:

[TABLE]

• •

If for example ${\mathcal{S}}^{(p)}$ is the $p$-convexification of Schreier space then since $M_{{\mathcal{S}}^{(p)}}(n)=M_{\mathcal{S}}(n)^{1/p}=n^{1/p}$ and $m_{{\mathcal{S}}^{(p)}}^{s}(n)=m_{\mathcal{S}}(n)^{1/p}=(n/\log n)^{1/p}$ we obtain

[TABLE]

and deduce that Kalton-Peck map is super disjointly singular on ${\mathscr{S}}^{(p)}$.

• •

The same estimates hold, in the case of Köthe sequence spaces, using the successive vectors versions $M_{X}^{s}(n)$, $m_{X}^{s}(n)$ and $\psi^{s}_{X}(n)$ of the parameters and of the modulus. So for example, if ${\mathscr{S}}^{(p)}$ is the $p$-convexification of Schlumprecht space then since $M^{s}_{{\mathscr{S}}^{(p)}}(n)=n^{1/p}$ and $m_{{\mathscr{S}}^{(p)}}^{s}(n)=(n/\log n)^{1/p}$ we also obtain

[TABLE]

and deduce that Kalton-Peck map is ”super successively singular”, therefore disjointly singular, hence singular on ${\mathscr{S}}^{(p)}$.

Acknowledgements: The authors thank Félix Cabello and Pedro Tradacete for fruitful discussions about the topics in the paper.

.