# A Banach space with an infinite dimensional reflexive quotient algebra
$\mathscr{L}(X)/\mathscr{SS}(X)$

Anna Pelczar-Barwacz<EMAIL_ADDRESS>Jagiellonian University, Faculty
of Mathematics and Computer Science, Institute of Mathematics, Łojasiewicza 6,
30-348 Kraków, Poland

###### Abstract.

We construct a Banach space $X$ of Gowers-Maurey type such that the algebra of
bounded operators $\mathscr{L}(X)$ is a direct sum of an infinite dimensional
reflexive Banach space and the operator ideal of strictly singular operators
$\mathscr{SS}(X)$.

The research of the author was supported by the grant of the National Science
Centre (NCN), Poland, no. UMO-2020/39/B/ST1/01042

## Introduction

The study of the structure of the closed operator ideals in the Banach algebra
$\mathscr{L}(X)$ on a Banach space $X$, started in early 1940’s, developed
intensively in recent years. One of the main direction of research
concentrates on determining Banach algebras which can be represented as a
quotient algebra of $\mathscr{L}(X)$ for some Banach space $X$, with special
emphasis on the Calkin algebra of $X$, that is
$\mathscr{C}(X)=\mathscr{L}(X)/\mathscr{K}(X)$, where $\mathscr{K}(X)$ denotes
the ideal of compact operators. Concerning this direction, powerful examples
solving several long-open structural questions on Banach spaces were built
mainly within the framework of Gowers-Maurey and Argyros-Haydon type spaces
theory.

The Gowers-Maurey method, starting with techniques introduced by Th.
Schlumprecht [S] and developed widely, provides a list of Banach spaces
sharing many extreme properties, among others with an explicitly described
quotient algebra $\mathscr{L}(X)/\mathscr{SS}(X)$, where $\mathscr{SS}(X)$
denotes the closed operator ideal of strictly singular operators on $X$ (cf.
[GM1, GM2, AD]). Recall that an operator $S$ between Banach spaces $U,V$ is
called strictly singular if there is no infinite dimensional subspace
$W\subset U$ such that the restriction $S|_{W}:W\rightarrow S(W)$ is an
isomorphism. The Gowers-Maurey space $X_{GM}$ provided a negative answer to
the unconditional basic sequence problem and turned out to satisfy a much
stronger property than lack of unconditional basic sequences, i.e. hereditary
indecomposability (HI; no two closed infinite dimensional subspaces of
$X_{GM}$ form a direct sum) [GM1]. In close relation to this fact, the
quotient algebra $\mathscr{L}(X_{GM})/\mathscr{SS}(X_{GM})$ is one-dimensional
[GM1] (whereas $\mathscr{SS}(X_{GM})$ itself has rich structure, containing
$2^{\mathfrak{c}}$ many pairwise distinct closed operator ideals, [MP]). The
same result holds also in the case of $\ell_{1}$-asymptotic HI space of
Argyros-Deliyanni [AD]; in general for any HI space $X$ the quotient algebra
$\mathscr{L}(X)/\mathscr{SS}(X)$ is isomorphic either to $\mathbb{R}$,
$\mathbb{C}$ or $\mathbb{H}$ [F].

The general theory developed in [GM2], cf. also [Ma], proves in particular
that the convolution algebra $\ell_{1}(\mathbb{Z})$, among other examples, can
be represented as the quotient algebra $\mathscr{L}(X)/\mathscr{SS}(X)$ for
some Banach spaces $X$, with the left and right shifts generating
$\mathscr{L}(X)$ modulo strictly singular operators. Recently P.Motakis [Mo],
within the framework of Gowers-Maurey spaces, for any compact metric $K$ built
a space with the quotient algebra $\mathscr{L}(X)/\mathscr{SS}(X)$ isometric
to the space $C(K)$ of continuous functions on $K$, using diagonal operators
on $X$. Examples of non-separable spaces $X=C(K)$ with the quotient algebra
$\mathscr{L}(X)/\mathscr{SS}(X)$ isometric to $X$ were constructed in [K, P].

The results concerning Gowers-Maurey type spaces were lifted and augmented on
the level of Argyros-Haydon type spaces, with the ideal $\mathscr{SS}(X)$
replaced by $\mathscr{K}(X)$. The construction of the celebrated Argyros-
Haydon space $X_{AH}$, adapting the Gowers-Maurey method in the setting of
Bourgain-Delbaen $\mathscr{L}_{\infty}$-spaces, solves the fundamental
"scalar+compact" problem [AH], that is yields a one-dimensional Calkin algebra
$\mathscr{C}(X_{AH})$ [AH]. The space $X_{AH}$ served as the basis, in the
process of combining finitely many carefully chosen variants of $X_{AH}$,
certain iterative constructions of Argyros-Haydon sums of variants of $X_{AH}$
or similar procedures, for building other examples of Banach spaces with the
Calkin algebra of prescribed form, as any finite dimensional semi-simple
complex algebra [KL], any space $C(K)$ with countable compact $K$ [MPZ] or any
non-reflexive Banach spaces with an unconditional basis, as well as some HI
spaces [MPT].

On the other hand, direct lifting constructions of Gowers-Maurey type spaces
with specific quotient algebras (over the ideal of strictly singular
operators) to the setting of Bourgain-Delbaen spaces provides also powerful
examples, as a Banach space with Calkin algebra isomorphic to the convolution
algebra $\ell_{1}(\mathbb{N}_{0})$ [T], whose construction follows the
approach of [GM2], or, for any compact $K$, a Banach space with Calkin algebra
isometric to space $C(K)$ [Mo].

In the context of the results described above the natural question concerns
the existence of a Banach space with an infinite dimensional reflexive Calkin
algebra, cf. [Mo, MPT, MPZ]. As it was mentioned above, constructions of
Gowers-Maurey type spaces are often adaptable to the setting of Argyros-Haydon
spaces, which puts forward - as possibly the first step to solve the former
problem - the question on the existence of a Gowers-Maurey type space $X$ with
an infinite dimensional reflexive quotient algebra
$\mathscr{L}(X)/\mathscr{SS}(X)$. The aim of this paper is to present a
positive answer to this last question.

######  Main Theorem.

There is a Banach space $X$ with a basis so that $\mathscr{L}(X)$ is a direct
sum of $\mathscr{SS}(X)$ and an infinite dimensional reflexive space $V$ with
an unconditional basis $(I_{s})_{s=0}^{\infty}$. The space $V$ as a Banach
algebra is a unitisation of the space
$\overline{\operatorname{span}}\\{I_{s}:s=1,2,\dots\\}$ endowed with pointwise
multiplication.

More precisely each $I_{s}$ is a projection on an infinite dimensional
subspace $X_{s}\subset X$ spanned by some subsequence of the basis of $X$ and
the basic sequence $(I_{s})_{s=1}^{\infty}$ is equivalent to the canonical
basis of the dual space to a certain mixed Tsirelson space $Y$ defined in the
sequel. Notice that in the presented approach in the place of shifts or
diagonal operators used in the previous methods as operators generating
$\mathscr{L}(X)$ modulo strictly singular operators we use projections on
infinite dimensional subspaces of $X$.

The construction of $X$ is based on Gowers-Maurey method of constructing HI
(hereditary indecomposable) spaces, with two adaptations. The first one
guarantees the existence of a sequence $(X_{s})_{s=1}^{\infty}$ of totally
incomparable 1-complemented Gowers-Maurey subspaces of $X$, whose union is
linearly dense in $X$. This step without the second ingredient would produce
simply a Gowers-Maurey sum of spaces $(X_{s})_{s}$, thus with $(X_{s})_{s}$
forming an infinite dimensional Schauder decomposition of $X$ and the quotient
algebra $\mathscr{L}(X)/\mathscr{SS}(X)$ isomorpic to the space $c_{0}$.

The second ingredient added to the construction forces the equivalence of the
sequence $(I_{s})_{s=1}^{\infty}$ of projections on the subspaces
$(X_{s})_{s}$, to the canonical basis of the space $Y^{*}$, where $Y$ is a
certain mixed Tsirelson space, by adding to the norming set of $X$ a certain
"ground" set of functionals of $c_{0}$ type, that somewhat violate the
"horizontal" (i.e. with respect to $(X_{s})_{s}$) structure of $X$. This
"ground" set in the norming set of $X$ guarantees a rich and closely related
finite representation of $Y$ and $\ell_{1}$ in $X$. Precise choice of the
parameters defining $X,Y$ yields the desired equivalence.

Adding the second ingredient implies in particular that
$(X_{s})_{s=1}^{\infty}$ do not form a Schauder decomposition of $X$.
Therefore justifying Gowers-Maurey "horizontal" behaviour of $X$, i.e.
properties concerning sequences in $X$ that are block with respect to the
sequence $(X_{s})_{s=1}^{\infty}$, requires more technical work. The Gowers-
Maurey "horizontal" properties of $X$ combined with Gowers-Maurey properties
of all $X_{s}$, $s=1,2,\dots$, and bounded completeness of
$(I_{s})_{s=1}^{\infty}$ imply the desired representation of $\mathscr{L}(X)$
as the direct sum of $V$ and $\mathscr{SS}(X)$.

The author is very grateful to Pavlos Motakis for suggesting using $\ell_{1}$
instead of an originally introduced mixed Tsirelson space $Z$ as the second,
after $Y$, auxiliary space, which essentially simplifies the reasonings. The
author is also very grateful to the referee of the paper for thorough reading
of the manuscript and suggesting several improvements of its presentation.

The paper is organised as follows. In Section 1. we recall the notion of mixed
Tsirelson spaces and spaces of Gowers-Maurey type, as well as their basic
properties. Section 2. is devoted to the definition of $X$ and its basic
properties, including the key Lemma 2.5. Section 3. concentrates on the
properties of sequence of projections
$(I_{s})_{s=1}^{\infty}\subset\mathscr{L}(X)$. In Section 4. we prove the
Gowers-Maurey type properties of the space $X$ adapting the classical
reasoning with the aid of Lemma 4.2, and finally in Section 5. we examine the
structure of $\mathscr{L}(X)$. The main theorem follows by Corollaries 3.4 and
5.6.

We recall now some basic notation. By $c_{00}$ we denote the vector space of
sequences in $\mathbb{R}$ which are eventually zero, by $(e_{n})_{n}$ and
$(e_{n}^{*})_{n}$ resp., we denote the unit vector sequence in $c_{00}$,
treated resp. as vectors or as functionals on $c_{00}$. The support of a
$x=(a_{i})_{i}\in c_{00}$ is the set
$\operatorname{supp}(x)=\\{i\in\mathbb{N}:a_{i}\neq 0\\}$, the range
$\operatorname{rng}(x)$ \- the smallest interval in $\mathbb{N}$ containing
the support of $x$. For any $x=(a_{i})_{i}\in c_{00}$ and $E\subset\mathbb{N}$
let $Ex$ be the vector defined by $(Ex)_{i}=a_{i}$ if $i\in E$ and
$(Ex)_{i}=0$ otherwise. We write $n<x$ for $n\in\mathbb{N}$ and $x\in c_{00}$
if $n<\max\operatorname{supp}(x)$ and we write $x<y$ for vectors $x,y\in
c_{00}$, if $\max\operatorname{supp}(x)<\min\operatorname{supp}(y)$. A block
sequence is any sequence $(x_{n})_{n}\subset c_{00}$ satisfying
$x_{1}<x_{2}<\dots$, a block subspace of $c_{00}$ \- any closed subspace
spanned by an infinite block sequence. Given infinite $J\subset\mathbb{N}$ let
$c_{00}(J)=\\{x\in c_{00}:\operatorname{supp}(x)\subset J\\}$ and
$c_{00}^{\mathbb{Q}}$, resp. $c_{00}^{\mathbb{Q}}(J)$, be the
$\mathbb{Q}$-vector space of elements of $c_{00}$, resp. $c_{00}(J)$, with
coefficients in $\mathbb{Q}$. A subconvex combination of vectors
$(x_{i})_{i=1}^{n}\in c_{00}$ is any vector of the form
$\sum_{i=1}^{n}a_{i}x_{i}$ for some scalars $(a_{i})_{i=1}^{n}\subset[0,1]$
with $\sum_{i=1}^{n}a_{i}\leq 1$.

We say that a basic sequence $(e_{n})_{n}$ $C$-dominates a basic sequence
$(\tilde{e}_{n})_{n}$ if the mapping $T:e_{n}\to\tilde{e}_{n}$,
$n\in\mathbb{N}$, extends to a bounded operator on
$\operatorname{span}\\{e_{n}:n\in\mathbb{N}\\}$ with norm not exceeding $C$.
If additionally $(\tilde{e}_{n})_{n}$ $C$-dominates $(e_{n})_{n}$ we say that
these basic sequences are $C$-equivalent.

A tree is a non-empty partially ordered set $(\mathcal{T},\preceq)$ for which
the set $\\{y\in\mathcal{T}:y\preceq x\\}$ is linearly ordered and finite for
each $x\in\mathcal{T}$. The root is the smallest element of the tree (if it
exists). The terminal nodes are the maximal elements. A branch of
$\mathcal{T}$ is any maximal linearly ordered set in $\mathcal{T}$. The
immediate successors of $t\in\mathcal{T}$ are all the nodes $r\in\mathcal{T}$
such that $t\prec r$ but there is no $s\in\mathcal{T}$ with $t\prec s\prec r$,
the set of immediate successors of $t$ is denoted by $succ(t)$.

##  1\. Mixed Tsirelson spaces and spaces of Gowers-Maurey type

We recall the definition and basic properties of mixed Tsirelson spaces and
spaces built on their basis, following [AD, AT].

######  Notation 1.

For sequences $(n_{j})_{j},(m_{j})_{j}\subset\mathbb{N}$ we consider the
following properties:

  * $(*)$

$m_{1}=2$, $n_{1}\geq 4$, $m_{j+1}\geq m_{j}^{m_{j}}$,
$n_{j+1}\geq(10n_{j})^{m_{j+1}^{2}}$ for all $j\in\mathbb{N}$,

  * $(**)$

$m_{j},n_{j}\in\\{2^{k}:k\in\mathbb{N}\\}$ and for any $k\in\mathbb{N}$ there
is $j\in\mathbb{N}$ with $n_{2j}=2^{km}$ for some $m\in\mathbb{N}$.

Fix sequences $(\tilde{n}_{j})_{j},(\tilde{m}_{j})_{j}$ satisfying $(*)$ of
Notation 1.

######  Definition 1.1 (Mixed Tsirelson space).

Let $\tilde{K}\subset c_{00}$ be the smallest set satisfying the following:

  1. (1)

$(\pm e_{n}^{*})_{n}\subset\tilde{K}$,

  2. (2)

for any $j\in\mathbb{N}$ and $f_{1}<\dots<f_{d}$ in $\tilde{K}$ with
$d\leq\tilde{n}_{j}$ we have
$\tilde{m}_{j}^{-1}(f_{1}+\dots+f_{d})\in\tilde{K}$.

  3. (3)

$\tilde{K}$ is closed under restriction to intervals and taking subconvex
combinations with rational coefficients.

Let $\|\cdot\|$ be the norm on $c_{00}$ with the norming set $\tilde{K}$, i.e.
$\|\cdot\|=\sup\\{|f(\cdot)|:f\in\tilde{K}\\}$. The mixed Tsirelson space
$T[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in\mathbb{N}}]$ is the
completion of $(c_{00},\|\cdot\|)$. The set $\tilde{K}$ we shall call the
canonical norming set of
$T[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in\mathbb{N}}]$.

It is well-known that under the above assumptions the space
$T[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in\mathbb{N}}]$ is a
reflexive space with an unconditional basis formed by the unit vectors
$(e_{n})_{n}$ [AT, Thm I.10]. Moreover, the set $\tilde{K}$ is closed under
restrictions to arbitrary subsets.

Any functional $f\in\tilde{K}$ produced as in (2) above, i.e. of the form
$f=\tilde{m}_{j}^{-1}(f_{1}+\dots+f_{d})$ is called weighted, with the weight
defined as $\operatorname{w}(f)=\tilde{m}_{j}^{-1}$. We note that the weight
of a functional in $\tilde{K}$ is not necessarily uniquely determined.

######  Definition 1.2 (The tree-analysis).

Let $f\in\tilde{K}$. By a tree-analysis of $f$ we mean a finite family
$(f_{t})_{t\in\mathcal{T}}\subset\tilde{K}$ indexed by a finite tree
$\mathcal{T}$ with a unique root $0\in\mathcal{T}$ such that

  1. (1)

$f_{0}=f$,

  2. (2)

$t\in\mathcal{T}$ is terminal iff $f_{t}\in(\pm e_{n}^{*})$,

  3. (3)

for every non-terminal $t\in\mathcal{T}$ one of the following holds true

    1. (a)

$f_{t}=\tilde{m}_{j}^{-1}E\sum_{r\in\operatorname{succ}(t)}f_{r}$, for some
$j\in\mathbb{N}$ with $\\#\operatorname{succ}(t)\leq\tilde{n}_{j}$, an
interval $E\subset\mathbb{N}$, and a sequence
$(f_{r})_{r\in\operatorname{succ}(t)}\subset\tilde{K}$ which is block with
respect to an appropriate linear ordering of $\operatorname{succ}(t)$,

    2. (b)

$f_{t}=\sum_{r\in\operatorname{succ}(t)}c_{r}f_{r}$ for some scalars
$(c_{r})_{r}\subset[0,1]\cap\mathbb{Q}$ with $\sum_{r}c_{r}\leq 1$ and
functionals $(f_{r})_{r\in\operatorname{succ}(t)}\subset\tilde{K}$.

We shall use also spaces of Gowers-Maurey type, introduced in [GM1]. We follow
the approach of [AT]. We fix first an injective function
$\tilde{\sigma}:\\{(f_{1},\dots,f_{d})\subset
c_{00}^{\mathbb{Q}}:f_{1}<\dots<f_{d}\\}\to 2\mathbb{N}$ such that
$\tilde{\sigma}((f_{1},\dots,f_{d}))\geq\max\operatorname{supp}(f_{d})\|f_{1}+\dots+f_{d}\|_{\infty}^{-1}$
for any non-zero $(f_{1},\dots,f_{d})$.

######  Definition 1.3 (Mixed Tsirelson space).

Let $\tilde{K}_{\tilde{\sigma}}\subset c_{00}$ be the smallest set satisfying
the following:

  1. (1)

$(\pm e_{n}^{*})_{n}\subset\tilde{K}_{\tilde{\sigma}}$,

  2. (2)

for any $j\in\mathbb{N}$ and $f_{1}<\dots<f_{d}$ in
$\tilde{K}_{\tilde{\sigma}}$ with $d\leq n_{2j}$ we have
$\tilde{m}_{2j}^{-1}(f_{1}+\dots+f_{d})\in\tilde{K}_{\tilde{\sigma}}$,

  3. (3)

for any $j\in\mathbb{N}$ and $j$-special sequence $f_{1}<\dots<f_{d}$ in
$\tilde{K}_{\tilde{\sigma}}$ with $d\leq n_{2j-1}$ we have
$\tilde{m}_{2j-1}^{-1}(f_{1}+\dots+f_{d})\in\tilde{K}_{\tilde{\sigma}}$,

  4. (4)

$\tilde{K}_{\tilde{\sigma}}$ is symmetric, closed under restrictions to
intervals and taking subconvex combinations with rational coefficients.

In order to complete the definition of $\tilde{K}_{\tilde{\sigma}}$ we define
special sequences as follows: $(f_{1},\dots,f_{d})\subset K_{\tilde{\sigma}}$
is $j$-special, $j\in\mathbb{N}$, if all $f_{i}$’s are weighted with
$\operatorname{w}(f_{1})=m_{2k}^{-1}$ for some $k\in\mathbb{N}$ with
$m_{2k}>n_{2j-1}^{2}$, and
$\operatorname{w}(f_{i+1})=m^{-1}_{\tilde{\sigma}((f_{1},\dots,f_{i}))}$ for
any $i=1,\dots,d-1$.

Let $\|\cdot\|$ be the norm on $c_{00}$ with the norming set
$\tilde{K}_{\tilde{\sigma}}$. The Gowers-Maurey space
$T_{\tilde{\sigma}}[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in\mathbb{N}}]$
is the completion of $(c_{00},\|\cdot\|)$. The set
$\tilde{K}_{\tilde{\sigma}}$ is called the canonical norming set of
$T_{\tilde{\sigma}}[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in\mathbb{N}}]$.

It follows that
$T_{\tilde{\sigma}}[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in\mathbb{N}}]$
is also a reflexive space with a basis formed by the unit vectors
$(e_{n})_{n}$. As in the case of $\tilde{K}$, we use the notions of a weighted
functional and a tree-analysis of a functional in
$\tilde{K}_{\tilde{\sigma}}$.

We state now some of the standard notions and results concerning Gowers-Maurey
spaces in the form we shall need later.

######  Definition 1.4.

A pair $(f,x)$, with $x\in
T_{\tilde{\sigma}}[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in\mathbb{N}}]$,
$f\in\tilde{K}_{\tilde{\sigma}}$, is called a $j$-exact pair,
$j\in\mathbb{N}$, provided

  1. (1)

$f$ is weighted with $\operatorname{w}(f)=\tilde{m}_{j}^{-1}$,

  2. (2)

$f(x)=1$, $\|x\|\leq 3$, $\operatorname{rng}(x)=\operatorname{rng}(f)$,

  3. (3)

$|g(x)|\leq 5\max\\{\tilde{m}_{j}^{-1},\operatorname{w}(g)\\}$ for any
weighted $g\in\tilde{K}_{\tilde{\sigma}}$ with weight
$\operatorname{w}(g)\neq\tilde{m}_{j}^{-1}$.

The next two lemmas follow by [AT, Prop. II.19 and Prop. II.25].

######  Lemma 1.5.

Any block subspace of
$T_{\tilde{\sigma}}[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in\mathbb{N}}]$
for any $j\in\mathbb{N}$ contains a vector $x$ so that $(x,f)$ is a $j$-exact
pair for some $f\in\tilde{K}_{\tilde{\sigma}}$.

######  Lemma 1.6.

For any infinite $L\subset\mathbb{N}$ any block subspace of
$T_{\tilde{\sigma}}[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in
L}]$ contains a normalized block sequence $(y_{n})_{n}$ so that for any
$j\in\mathbb{N}\setminus L$ and $A\subset\mathbb{N}$ with $j<A$ and
$\\#A\leq\tilde{n}_{j}$ we have $\|\sum_{n\in A}y_{n}\|\leq
5\tilde{n}_{j}\tilde{m}_{j}^{-2}$.

Finally we recall the crucial property of Gowers-Maurey spaces.

######  Theorem 1.7.

[AT, Thm IV.12] Let
$W=T_{\tilde{\sigma}}[(\mathcal{A}_{\tilde{n}_{j}},\tilde{m}_{j}^{-1})_{j\in\mathbb{N}}$
be a Gowers-Maurey space. Then any operator $T\in\mathscr{L}(W)$ is of the
form $T=aId_{W}+S$ for some $a\in\mathbb{R}$ and $S\in\mathscr{SS}(W)$.

##  2\. The definition and basic properties of the space $X$

In this section we define of the space $X$, and present its basic properties,
including Lemma 2.5, which provides one of the tools substituting for the lack
of "horizontal" Schauder decomposition of the space.

For the rest of the paper we fix sequences $(n_{j})_{j}$, $(m_{j})_{j}$
satisfying $(*)$ and $(**)$ of Notation 1.

######  Notation 2.

We fix a partition $\mathbb{N}=\bigcup_{s=0}^{\infty}L_{s}$ into pairwise
disjoint infinite sets with infinite $L_{s}\cap 2\mathbb{N}$ and
$L_{s}\cap(2\mathbb{N}-1)$ for any $s=0,1,\dots$.

Let $Y=T[(A_{n_{2j}},m_{2j}^{-1})_{j\in L_{0}}]$ with the canonical norming
set $K_{Y}$.

We fix also another partition $\mathbb{N}=\bigcup_{s=1}^{\infty}J_{s}$ of
integers into pairwise disjoint infinite sets. Given any finite
$A\subset\mathbb{N}$ we write $J_{A}=\bigcup_{s\in A}J_{s}$.

For any $x\in c_{00}$ let
$\operatorname{supp}_{h}(x)=\\{s\in\mathbb{N}:\operatorname{supp}(x)\cap
J_{s}\neq\emptyset\\}$ and let $\operatorname{rng}_{h}(x)$ be the minimal
interval in $\mathbb{N}$ containing $\operatorname{supp}_{h}(x)$. A vector
$x\in c_{00}$ is called simple, if $\operatorname{supp}_{h}(x)$ is a
singleton. A sequence $(x_{n})_{n}\subset c_{00}$ is called horizontally block
(h-block), provided it is block both with respect to the unit vector basis of
$c_{00}$ and with respect to $(J_{s})_{s\in\mathbb{N}}$, i.e.
$(\operatorname{supp}(x_{n}))_{n}$ and $(\operatorname{supp}_{h}(x_{n}))_{n}$
are block in $\mathbb{N}$. A subspace of $c_{00}$ is called h-block, if it is
generated by an infinite h-block sequence.

Now we set a family of mappings. For any $s=1,2,\dots$ set an injective
mapping

$\sigma_{s}:\\{(f_{1},\dots,f_{d})\subset
c_{00}^{\mathbb{Q}}(J_{s}):(f_{1},\dots,f_{d})\text{ block}\\}\to L_{s}\cap
2\mathbb{N},\ \ \ s=1,2\dots$

such that
$\sigma_{s}((f_{1},\dots,f_{d}))>\max\operatorname{supp}(f_{d})\|f_{1}+\dots+f_{d}\|_{\infty}^{-1}$
for any non-zero $(f_{1},\dots,f_{d})$.

Set a mapping

$\sigma:\\{(f_{1},\dots,f_{d})\subset
c_{00}^{\mathbb{Q}}:(f_{1},\dots,f_{d})\text{ h-block}\\}\to 2^{L_{0}\cap
2\mathbb{N}}$

such that $\sigma((f_{1},\dots,f_{d}))$ is infinite for any
$(f_{1},\dots,f_{d})$ and
$\sigma((f_{1},\dots,f_{d}))\cap\sigma((g_{1},\dots,g_{l}))=\emptyset$ for any
$(f_{1},\dots,f_{d})\neq(g_{1},\dots,g_{l})$.

Finally set a mapping

$\tau:\\{(f_{1},\dots,f_{d})\subset
c_{00}^{\mathbb{Q}}:(f_{1},\dots,f_{d})\text{ h-block},f_{1},\dots,f_{d}\text{
simple}\\}\to 2^{\mathbb{N}}$

such that for any $(f_{1},\dots,f_{d})$ and any $s\in\mathbb{N}$ we have
$\tau(f_{1},\dots,f_{d})\cap(L_{s}\cap 2\mathbb{N})\neq\emptyset$, and
$\tau(f_{1},\dots,f_{d})\cap\tau(g_{1},\dots,g_{l})=\emptyset$ for any
$(f_{1},\dots,f_{d})\neq(g_{1},\dots,g_{k})$.

######  Definition 2.1.

We define the set $K\subset c_{00}$ as the minimal set such that

  1. (1)

$\pm e_{i}^{*}\in K$ for any $i\in\mathbb{N}$,

  2. (2)

for any $s\in\mathbb{N}$, $2j\in L_{s}$ and any block sequence
$f_{1}<\dots<f_{d}$, of elements of $K\cap c_{00}(J_{s})$, $d\leq n_{2j}$, we
have $m_{2j}^{-1}(f_{1}+\dots+f_{d})\in K$,

  3. (3)

for any $s\in\mathbb{N}$, $2j-1\in L_{s}$ and any $j$-special (defined with
respect to $\sigma_{s}$) sequence $f_{1}<\dots<f_{d}$ of elements of $K\cap
c_{00}(J_{s})$, $d\leq n_{2j-1}$, we have $m_{2j-1}^{-1}(f_{1}+\dots+f_{d})\in
K$,

  4. (4)

for any $2j\in L_{0}$ and any h-block sequence $f_{1}<\dots<f_{d}$ of elements
of $K$, $d\leq n_{2j}$, we have $m_{2j}^{-1}(f_{1}+\dots+f_{d})\in K$,

  5. (5)

for any $2j-1\in L_{0}$ and any $j$-h-special (defined with respect to
$\sigma$) sequence $f_{1}<\dots<f_{d}$ of elements of $K$, $d\leq n_{2j-1}$,
we have $m_{2j-1}^{-1}(f_{1}+\dots+f_{d})\in K$,

  6. (6)

for any $N\in\mathbb{N}$ and any $N$-g-special sequence $f_{1}<\dots<f_{2N}$
of elements of $K$, and any $(\varepsilon_{i})_{i=1}^{N}\subset\\{-1,0,1\\}$
we have $\sum_{i=1}^{N}\varepsilon_{i}(f_{2i-1}+f_{2i})\in K$,

  7. (7)

$K$ is symmetric, closed under restrictions to intervals and taking subconvex
combination with rational coefficients.

In order to complete the definition we need to define special sequences. We
say that a block sequence $(f_{1},\dots,f_{d})\subset K$, $1<d$, is

  * •

$j$-special, $2j-1\in L_{s}$, $s\in\mathbb{N}$, provided $d\leq n_{2j-1}$,
$\operatorname{w}(f_{i+1})=m_{\sigma_{s}(f_{1},\dots,f_{i})}^{-1}$ for each
$i=1,\dots,d-1$ and $\operatorname{w}(f_{1})<n_{2j-1}^{-2}$,

  * •

$j$-h-special, $2j-1\in L_{0}$, provided $d\leq n_{2j-1}$, $(f_{i})_{i=1}^{d}$
is h-block, $\operatorname{w}_{h}(f_{i+1})=m_{k}^{-1}$ for some
$k\in\sigma(f_{1},\dots,f_{i})$ for each $i=1,\dots,d-1$ and
$\operatorname{w}_{h}(f_{1})<n_{2j-1}^{-2}$ (cf. Notation 3),

  * •

$N$-g-special, $N\in\mathbb{N}$, provided $d=2N$, all $f_{i}$’s are simple
with $\operatorname{supp}_{h}(f_{2i-1})=\\{i\\}$,
$\operatorname{supp}_{h}(f_{2i})=\\{N+i\\}$ for each $i\leq N$,
$\operatorname{w}(f_{i+1})=m_{k}^{-1}$ for some $k\in\tau(f_{1},\dots,f_{i})$
for each $i=1,\dots,N-1$, and $\operatorname{w}(f_{1})<N^{-1}$.

Let $\|\cdot\|$ be the norm on $c_{00}$ for which $K$ is the norming set, i.e.
$\|\cdot\|=\max\\{|f(\cdot)|:f\in K\\}$. Let $X$ be the completion of
$(c_{00},\|\cdot\|)$.

######  Notation 3.

A functional $f\in K$ (and its restrictions to an interval) obtained by (2) or
(3) is called weighted with weight $\operatorname{w}(f)=m_{j}^{-1}$ for
suitable $j$ (according to the standard terminology of mixed Tsirelson
spaces).

A functional $f$ obtained by (4) or (5) (or its restriction to an interval) is
called h-regular and h-special resp., and in both cases h-weighted with weight
$\operatorname{w}_{h}(f)=m_{j}^{-1}$ for suitable $j$.

Finally a functional obtained by (6) (or its restriction to an interval) is
called g-special (if we underline the maximal length of the sequence used to
generate the functional, we call it $N$-g-special for suitable
$N\in\mathbb{N}$).

Notice that by the definition of $K$ the unit vectors $(e_{i})_{i}$ form a
bimonotone a basis of $X$. We gather now basic facts concerning specific
subspaces of $X$ encoded in the definition of the set $K$.

######  Notation 4.

For any fixed $s\in\mathbb{N}$ let $X_{s}=\\{x\in
X:\operatorname{supp}_{h}(x)\subset J_{s}\\}$ and let $K_{s}$ be the the
smallest subset of $K$ satisfying (1), (2), (3) and (7) of the definition of
$K$ for the fixed $s$.

######  Remark 2.2.

Fix $s\in\mathbb{N}$. As $f|_{J_{s}}\in K_{s}$ for any $f\in K$, the space
$X_{s}\subset X$ is a Gowers-Maurey space
$T_{\sigma_{s}}[(\mathcal{A}_{n_{j}},m_{j}^{-1})_{j\in L_{s}}]$ with the
canonical norming set $K_{s}$.

Moreover, by the definition of $K$, the projection $I_{s}:X\to X_{s}$,
$s\in\mathbb{N}$, is bounded with norm 1.

######  Notation 5.

In addition to the projections $(I_{s})_{s=1}^{\infty}\subset\mathscr{L}(X)$
defined above, by $I_{0}\in\mathscr{L}(X)$ we denote the identity operator
$X\to X$.

For any finite $A\subset\mathbb{N}$ we let also $I_{A}=\sum_{s\in
A}I_{s}\in\mathscr{L}(X)$.

######  Notation 6.

By $\|\cdot\|_{h,\infty}$ we denote the supremum norm on $X$ with respect to
$(X_{s})_{s}$, namely for any $x\in X$ we let
$\|x\|_{h,\infty}=\sup\\{\|I_{s}x\|:s=1,2,\dots\\}$. This norm is well-defined
and satisfies $\|\cdot\|_{h,\infty}\leq\|\cdot\|$ by Remark 2.2.

As it was mentioned in the introduction, the definition of the norm of $X$
follows the scheme of mixed Tsirelson norm modelled on a certain "ground" set
(cf. [AT]), in our case formed by g-special functionals, in the "horizontal"
setting. The latter provides a countable family of bounded non-trivial
projections $(I_{s})_{s=1}^{\infty}$ (on the spaces $(X_{s})_{s=1}^{\infty}$),
whereas including the "ground" set in the norming set ensures a rich and
closely related finite representation of $Y$ and $\ell_{1}$ in $X$, in
consequence - control over the norm of linear combinations of the
aforementioned projections. Notice that the norming set without the ground set
of g-special functionals would define simply a Gowers-Maurey sum of Gowers-
Maurey spaces $(X_{s})_{s}$. The only "antihorizontal" construction appears in
g-special functionals and it imposes the fact that $(X_{s})_{s}$ do not form a
Schauder decomposition of $X$.

We introduce now the basic tool for future estimates, i.e. a variant of a
tree-analysis of a functional in $K$, here built up to simple functionals.

######  Definition 2.3.

An h-tree-analysis of $f\in K$ is a family $(f_{t})_{t\in\mathcal{T}}$ indexed
by a finite tree $\mathcal{T}$ with a unique root $0\in\mathcal{T}$ such that
the following hold

  1. (1)

$f_{0}=f$ and $f_{t}\in K$ for all $t\in\mathcal{T}$,

  2. (2)

$t\in\mathcal{T}$ is terminal iff $f_{t}$ is simple,

  3. (3)

if $t\in\mathcal{T}$ is non-terminal, then one of the following holds true:

    1. (a)

$f_{t}$ is an h-weighted functional of the form
$f_{t}=\pm\operatorname{w}_{h}(f_{t})E\sum_{r\in\operatorname{succ}(t)}f_{r}$
for some interval $E\subset\mathbb{N}$ and a sequence
$(f_{r})_{r\in\operatorname{succ}(t)}\subset K$ which is h-block with respect
to an appropriate linear ordering of $\operatorname{succ}(t)$,

    2. (b)

$f_{t}$ is a g-special functional of the form
$f_{t}=E\sum_{r\in\operatorname{succ}(t)}\varepsilon_{r}f_{r}$, for some
interval $E\subset\mathbb{N}$, signs
$(\varepsilon_{r})_{r}\subset\\{-1,0,1\\}$ and a sequence
$(f_{r})_{r\in\operatorname{succ}(t)}\subset K$ of simple functionals, which
is h-block with respect to an appropriate linear ordering of
$\operatorname{succ}(t)$,

    3. (c)

$f_{t}$ is a subconvex rational combination of its successors, i.e. of the
form $f_{t}=\sum_{r\in\operatorname{succ}(t)}c_{r}f_{r}$, for some scalars
$(c_{r})_{r}\subset[0,1]\cap\mathbb{Q}$ with $\sum_{r}c_{r}\leq 1$, and a
sequence $(f_{r})_{r\in\operatorname{succ}(t)}\subset K$.

Notice that every functional from a norming set $K$ admits an h-tree-analysis,
not necessarily unique.

The next two lemmas are crucial in the reasoning concerning horizontal
structure of $X$. The second one forms a tool compensating for the fact that
$(X_{s})_{s=1}^{\infty}$ do not form a Schauder decomposition of $X$.

######  Lemma 2.4.

Fix $N\in\mathbb{N}$ and $f\in K$. Then there are $f^{\prime},g\in K$ so that
$f|_{J_{[1,N]}}=(f^{\prime}+g)|_{J_{[1,N]}}$,
$\operatorname{supp}_{h}(f^{\prime})\subset[1,N]$ and $g$ is a subconvex
combination of g-special functionals.

###### Proof.

Take an h-tree-analysis $(f_{t})_{t\in\mathcal{T}}$ of $f$, assume that
$\operatorname{supp}_{h}(f)\cap[1,N]\neq\emptyset$. Let $A\subset\mathcal{T}$
be the family of (terminal) $r\in\mathcal{T}$ such that
$r\in\operatorname{succ}(t)$ for some $t\in\mathcal{T}$ with $f_{t}$ g-special
and $\min\operatorname{supp}_{h}(f_{t})\leq
N<\max\operatorname{supp}_{h}(f_{t})$. Let $B$ be the family of all terminal
$t\in\mathcal{T}\setminus A$ with $\min\operatorname{supp}_{h}(f_{t})\leq N$.
Let $\mathcal{T}_{A}\subset\mathcal{T}$ (resp. $\mathcal{T}_{B}$) be the
family of all nodes in $\mathcal{T}$ that are comparable with some node of $A$
(resp. $B$). Note that the root of $\mathcal{T}$ belongs to
$\mathcal{T}_{A}\cup\mathcal{T}_{B}$.

We define $f^{\prime}$ and $g$ simultaneously by defining their h-tree-
analyses $(g_{t})_{t\in\mathcal{T}_{A}}$ and
$(f^{\prime}_{t})_{t\in\mathcal{T}_{B}}$ inductively starting from terminal
nodes, so that

  1. (1)

$f_{t}|_{J_{[1,N]}}=(f^{\prime}_{t}+g_{t})|_{J_{[1,N]}}$ for any
$t\in\mathcal{T}_{B}\cup\mathcal{T}_{A}$ (we adopt the convention that
$g_{t}=0$ if $t\not\in\mathcal{T}_{A}$ and $f^{\prime}_{t}=0$ if
$t\not\in\mathcal{T}_{B}$),

  2. (2)

for any $t\in\mathcal{T}_{A}$, either $t\in\operatorname{succ}(s)$ for some
$s\in\mathcal{T}$ with $g_{s}$ g-special, or $g_{t}$ is a subconvex
combination of g-special functionals,

  3. (3)

$\operatorname{supp}_{h}(f^{\prime}_{t})\subset[1,N]$ for any
$t\in\mathcal{T}_{B}$,

  4. (4)

$f_{t}^{\prime}=f_{t}$ for any $t\in\mathcal{T}_{B}$ with
$\operatorname{supp}_{h}(f_{t})\leq N$,

  5. (5)

$f^{\prime}_{t}$ is h-weighted with
$\operatorname{w}_{h}(f^{\prime}_{t})=\operatorname{w}_{h}(f_{t})$ for any
$t\in\mathcal{T}_{A}\cup\mathcal{T}_{B}$ with $f_{t}$ h-weighted.

If $t\in\mathcal{T}_{A}\cup\mathcal{T}_{B}$ is terminal let $g_{t}=f_{t}$ for
$t\in\mathcal{T}_{A}$ and $f^{\prime}_{t}=f_{t}$ for $t\in\mathcal{T}_{B}$.

Pick non-terminal $t\in\mathcal{T}_{A}\cup\mathcal{T}_{B}$ and assume all
$f^{\prime}_{r}$ and $g_{r}$, $r\in\operatorname{succ}(t)$, are defined and
satisfy the above conditions.

If $f_{t}$ is g-special, then $t$ belongs to only one of the trees
$\mathcal{T}_{A},\mathcal{T}_{B}$. Let $f_{t}^{\prime}=f_{t}$ if
$t\in\mathcal{T}_{B}$ and $g_{t}=f_{t}$ if $t\in\mathcal{T}_{A}$. All desired
conditions are satisfied.

If $f_{t}$ is a subconvex combination of its successors, let
$\operatorname{succ}_{B}(t)=\operatorname{succ}(t)\cap\mathcal{T}_{B}$ and
$\operatorname{succ}_{A}(t)=\operatorname{succ}(t)\cap\mathcal{T}_{A}$. If
$f_{t}=\sum_{r\in\operatorname{succ}(t)}c_{r}f_{r}$ then let
$f^{\prime}_{t}=\sum_{r\in\operatorname{succ}_{B}(t)}c_{r}f^{\prime}_{r}$ and
$g_{t}=\sum_{r\in\operatorname{succ}_{A}(t)}c_{r}g_{r}$. Notice that
$f_{t}|_{J_{[1,N]}}=\sum_{r\in\operatorname{succ}_{A}(t)\cup\operatorname{succ}_{B}(t)}c_{r}f_{r}|_{J_{[1,N]}}$,
thus (1) is satisfied by the inductive assumption, (3) is satisfied by the
inductive assumption. For (4) notice that if
$\operatorname{supp}_{h}(f_{t})\leq N$, then also
$\operatorname{supp}_{h}(f_{r})\leq\mathbb{N}$ for any
$r\in\operatorname{succ}(t)$ and apply the inductive assumption. Notice also
that in the considered case, by the inductive assumption (2), for any
$r\in\operatorname{succ}_{A}(t)$ the functional $g_{r}$ is a subconvex
combination of g-special functionals. Therefore we obtain (2) for $t$. It is
clear also that $f^{\prime}_{t},g_{t}\in K$.

If $f_{t}=\pm\operatorname{w}_{h}(f_{t})E\sum_{r=1}^{d}f_{r}$, with
$f_{1}<\dots<f_{d}$, let
$r_{0}=\max\\{r=1,\dots,d:\operatorname{supp}_{h}(f_{r})\leq\mathbb{N}\\}$.
Note that by inductive assumption $f^{\prime}_{r}=f_{r}$ for any $1\leq r\leq
r_{0}$.

If $r_{0}=d$ or $r_{0}+1\not\in\mathcal{T}_{A}\cup\mathcal{T}_{B}$ then
$\operatorname{supp}_{h}(f_{t})\cap[1,N]\subset\bigcup_{r=1}^{r_{0}}\operatorname{supp}_{h}(f_{r})$
and we let $g_{t}=0$ and
$f^{\prime}_{t}=\pm\operatorname{w}_{h}(f_{t})E\sum_{r=1}^{r_{0}}f^{\prime}_{r}$.
Conditions (1) are (5) are satisfied by the definition, (3) and (4) are
satisfied by the inductive assumption. It follows easily that
$f^{\prime}_{t}\in K$.

Assume $r_{0}+1\in\mathcal{T}_{A}\cup\mathcal{T}_{B}$. Then
$\operatorname{supp}_{h}(f_{t})\cap[1,N]\subset\bigcup_{r=1}^{r_{0}+1}\operatorname{supp}_{h}(f_{r})$
and we let $g_{t}=\pm\operatorname{w}_{h}(f_{t})Eg_{r_{0}+1}\in K$ and
$f^{\prime}_{t}=\pm\operatorname{w}_{h}(t)E\sum_{r=1}^{r_{0}+1}f^{\prime}_{r}$.
Conditions (1), (2) and (5) are satisfied by the definition, (3) and (4) - by
the inductive assumption. Notice also that $f^{\prime}_{t}\in K$ both in the
case of $f_{t}$ h-regular and h-special by the inductive assumption (4) and
(5) for $r\in\\{1,\dots,r_{0}+1\\}$. This case ends the inductive construction
and thus the proof of the lemma. ∎

######  Lemma 2.5.

  1. (1)

For any normalized h-block sequence $(x_{n})_{n}$ there are $\delta>0$, an
infinite $J\subset\mathbb{N}$ and an h-block sequence $(f_{n})_{n\in J}\subset
K$ such that $f_{n}(x_{n})\geq\delta$ for any $n\in J$.

  2. (2)

For any normalized h-block sequences $(x_{n})_{n}$ and $(y_{n})_{n}$ with
$\inf_{n}\operatorname{dist}(x_{n},\mathbb{R}y_{n})>0$ and
$\|y_{n}\|_{h,\infty}\xrightarrow{n\to\infty}0$ there are $\delta>0$, infinite
$J\subset\mathbb{N}$ and an h-block sequence $(f_{n})_{n\in J}\subset K$ such
that $f_{n}(x_{n})\geq\delta$ for any $n\in J$ and
$f_{n}(y_{n})\xrightarrow{J\ni n\to\infty}0$.

###### Proof.

We shall prove (2), as the proof of (1) is a simpler version of (2). Take
$(x_{n})_{n}$ and $(y_{n})_{n}$ as above. Notice that by rational convexity
the set $K$ is $w^{*}$-dense in the unit ball of $X^{*}$. Therefore by Hahn-
Banach theorem we can pick $(\hat{f}_{n})_{n}\subset K$ with
$\hat{f}_{n}(y_{n})<(2n)^{-1}$ and $\hat{f}_{n}(x_{n})>\epsilon$ for each
$n\in\mathbb{N}$ and some universal $\epsilon>0$. Relabelling we can assume
that $\operatorname{supp}_{h}(x_{n})\cup\operatorname{supp}_{h}(y_{n})>n$ and
$\|y_{n}\|_{h,\infty}\leq(2n)^{-2}$ for all $n$.

We introduce some useful notation: for any $N$-g-special
$g=\sum_{i=1}^{N}\varepsilon_{i}(f_{2i-1}+f_{2i})$ and any $k\in\mathbb{N}$ we
write $g^{k}=\sum_{i=1}^{\min\\{k,N\\}}\varepsilon_{i}(f_{2i-1}+f_{2i})$.
Obviously $g^{k}\in K$ and $g-g^{k}\in K$. If $g\in K$ is a subconvex
combinations of g-special functionals, say $g=\sum_{r}c_{r}g_{r}$, then we
write $g^{k}=\sum_{r}c_{r}g_{r}^{k}$.

For any $n\in\mathbb{N}$ pick $f^{\prime}_{n},g_{n}\in K$ by Lemma 2.4 for
$N=n$ and $f=\hat{f}_{n}$. Let
$h_{n}^{k}=\hat{f}_{n}-f^{\prime}_{n}-g_{n}^{k}$ for each $k,n\in\mathbb{N}$.
Then for all $k,n\in\mathbb{N}$ we have the following

  1. (1)

$\frac{1}{3}h_{n}^{k}\in K$ by definition of $K$,

  2. (2)

$\operatorname{supp}_{h}(h_{n}^{k})>\min\\{k,n\\}$ as
$\hat{f}_{n}|_{[1,\min\\{k,n\\}]}=(f^{\prime}_{n}+g_{n}^{k})|_{[1,\min\\{k,n\\}]}$,

  3. (3)

$h^{k}_{n}(x_{n})=\hat{f}_{n}(x_{n})-g_{n}^{k}(x_{n})$ and
$h^{k}_{n}(y_{n})=\hat{f}_{n}(y_{n})-g_{n}^{k}(y_{n})$ as
$\operatorname{supp}_{h}(f^{\prime}_{n})\leq
n<\operatorname{supp}_{h}(x_{n})\cup\operatorname{supp}_{h}(y_{n})$,

  4. (4)

if additionally $k\leq n$, then we have
$|h_{n}^{k}(y_{n})|=|g_{n}^{k}(y_{n})|+(2n)^{-1}\leq
2k\|y_{n}\|_{h,\infty}+(2n)^{-1}<n^{-1}$ by assumption on $(y_{n})_{n}$, (3)
and definition of g-special functionals.

Assume that for any $k\in\mathbb{N}$ there is $n_{k}>k$, with
$|h^{k}_{n_{k}}(x_{n_{k}})|>\epsilon/2$. Then any h-block subsequence
$(\frac{1}{3}h^{k}_{n_{k}})_{k\in J}$ (which exists by (2) above) satisfies
the desired properties for $(y_{n_{k}})_{k\in J}$ and $(x_{n_{k}})_{k\in J}$
by (4).

Otherwise for some $k_{0}\in\mathbb{N}$ we have
$|g_{n}^{k_{0}}(x_{n})|>\epsilon/2$ for any $n>k_{0}$ by (3) above and the
choice of $(\hat{f}_{n})_{n}$.

Fix now $n>k_{0}$. We shall restrict $g_{n}^{k_{0}}$ to some
$g^{\prime}_{n}\in K$ with better properties.

First for any $N$-g-special $g=\sum_{i=1}^{N}\varepsilon_{i}(f_{2i-1}+f_{2i})$
let $g^{\prime}=g-g^{l}$, where $l=\max\\{n+1-N,0\\}$. Then in particular for
any g-special $g$ we have
$\operatorname{supp}_{h}((g^{k_{0}})^{\prime})\cap(k_{0},\infty)=\operatorname{supp}_{h}((g^{k_{0}})^{\prime})\cap(n,\infty)=\operatorname{supp}_{h}(g^{k_{0}})\cap(n,\infty)$.
If $g_{n}^{k_{0}}=\sum_{r}c_{r}g_{r,n}^{k_{0}}$ let
$g^{\prime}_{n}=\sum_{r}c_{r}(g_{r,n}^{k_{0}})^{\prime}\in K$ and notice that
also

  1. (5)

$\operatorname{supp}_{h}(g^{\prime}_{n})\cap(k_{0},\infty)=\operatorname{supp}_{h}(g^{\prime}_{n})\cap(n,\infty)=\operatorname{supp}_{h}(g^{k_{0}}_{n})\cap(n,\infty)$,

  2. (6)

$g^{\prime}_{n}(x_{n})=g^{k_{0}}_{n}(x_{n})$ by (5),

  3. (7)

moreover $\\#(\operatorname{supp}_{h}(g^{\prime}_{n})\cap(k_{0},\infty))\leq
k_{0}$ by definition of $(g_{n}^{k_{0}})$.

Let $f_{n}=k_{0}^{-1}g^{\prime}_{n}|_{J_{(k_{0},\infty)}}$ for all $n>k_{0}$.
Then $f_{n}\in K$ (as a convex combination of a family
$(g^{\prime}_{n}|_{J_{s}})_{s>n,s\in\operatorname{supp}_{h}(g^{\prime}_{n})}$,
by (7)), $\operatorname{supp}_{h}(f_{n})\geq n$ by (5),
$f_{n}(x_{n})>\epsilon/2k_{0}$ by (6) and
$|f_{n}(y_{n})|\leq\\#(\operatorname{supp}(g^{\prime}_{n})\cap(n,\infty))\|y_{n}\|_{h,\infty}\leq
1/n$ for all $n>k_{0}$ by (7), thus passing to a suitable subsequence ends the
proof. ∎

##  3\. The sequence $(I_{s})_{s=1}^{\infty}\subset\mathscr{L}(X)$

This section is devoted to the proof of Thm 3.3 which reduces the study of
$(I_{s})_{s}$ to the case of $(e^{*}_{s})_{s}\subset Y^{*}$. The reasoning
presented here relies on the properties of classical Tsirelson spaces and the
specific choice of the space $Y$.

We introduce first the notion of g-special pairs of sequences of vectors and
functionals, with the functional parts being a g-special sequences. By the
careful choice of g-special functionals included in the norming set $K$ of $X$
the vector parts of a g-special pair provide closely related finite
representation of $Y$ and $\ell_{1}$ in $X$ (cf. Prop. 3.2 (1) and (2)). This
representation is rich enough to guarantee that vectors from isomorphic copies
of finite dimensional subspaces of $Y$ are "seminormalizing" for any
$T\in\operatorname{span}\\{I_{s}:s=0,1,\dots\\}$ (cf. Prop. 3.2 (3)) which
provides the control on the norm of such $T$ described in Thm 3.3.

######  Definition 3.1.

A pair of sequences $((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$, with
$(x_{s})_{s}\subset X$, $(f_{s})_{s}\subset K$, is called a $N$-g-special pair
provided

  1. (1)

$(f_{1},\dots,f_{2N})$ is an $N$-g-special sequence,

  2. (2)

$\operatorname{supp}_{h}(x_{s})=\operatorname{supp}_{h}(f_{s})=\\{r_{s}\\}$
for suitable $r_{s}\in\mathbb{N}$, for any $s=1,\dots,2N$,

  3. (3)

$(f_{s},x_{s})$ is a $2j_{s}$-exact pair in $X_{r_{s}}$, where
$\operatorname{w}(f_{s})=m_{2j_{s}}^{-1}$ for any $s=1,\dots,2N$ (cf. Def.
1.4).

In particular any vector part $(x_{s})_{s=1}^{2N}$ of a g-special pair is a
block sequence with $1\leq\|x_{s}\|\leq 3$ for each $s$ (by definition of
exact pairs).

######  Proposition 3.2.

  1. (1)

For any $N$-g-special pair $((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$ the
sequence $(x_{2s-1})_{s=1}^{N}$ 1-dominates the unit vector basis
$(e_{s})_{s=1}^{N}\subset\ell_{1}$ and is 3-dominated by
$(e_{s})_{s=1}^{N}\subset\ell_{1}$.

  2. (2)

For any $N\in\mathbb{N}$ there is $M_{N}\in\mathbb{N}$ so that for any
$N$-g-special pair $((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$ with
$\operatorname{supp}(x_{1})>M_{N}$ the sequence $(x_{2s-1}-x_{2s})_{s=1}^{N}$
$1$-dominates the unit vector basis $(e_{s})_{s=1}^{N}\subset Y$ and is
$20$-dominated by $(e_{s})_{s=1}^{N}\subset Y$.

  3. (3)

For any scalars $(a_{s})_{s=1}^{M}$ and any $N$-g-special pair
$((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$ with $M\leq N$,
$\operatorname{supp}(x_{1})>M_{N}$, there are scalars $(b_{s})_{s=1}^{M}$ so
that the vector $x=\sum_{s=1}^{M}b_{s}(x_{2s-1}-x_{2s})\in X$ satisfies
$\|x\|\leq 1$ and $\|\sum_{s=1}^{M}a_{s}I_{s}\|\leq
40\|\sum_{s=1}^{M}a_{s}I_{s}x\|$.

###### Proof.

(1) As $\|x_{s}\|\leq 3$ for any $s$, the sequence $(x_{2s-1})_{s=1}^{N}$ is
3-dominated by $(e_{s})_{s=1}^{N}\subset\ell_{1}$. For the reverse domination
fix scalars $(b_{s})_{s=1}^{N}$. By definition
$f=\sum_{s=1}^{N}\operatorname{sgn}(b_{s})(f_{2s-1}+f_{2s})$ is a g-special
functional in $K$. Thus

$\|\sum_{s=1}^{N}b_{s}x_{2s-1}\|\geq
f(\sum_{s=1}b_{s}x_{2s-1})=\sum_{s=1}|b_{s}|$

(2) First notice that $(x_{2s-1}-x_{2s})_{s=1}^{N}$ 1-dominates
$(e_{s})_{s=1}^{N}\subset Y$. Indeed, pick scalars $(b_{s})_{s}$ and
$\tilde{f}\in K_{Y}$ with $\tilde{f}_{t}(\sum b_{s}e_{s})=\|\sum
b_{s}e_{s}\|_{Y}$. Write $\tilde{f}=\sum_{s}d_{s}e_{s}^{*}$, then, as $K$
contains all h-regular functionals, also $f=\sum_{s}d_{s}f_{2s-1}\in K$ and
thus

$\|\sum_{s}b_{s}(x_{2s-1}-x_{2s})\|\geq
f(\sum_{s}b_{s}(x_{2s-1}-x_{2s}))=\sum_{s}d_{s}b_{s}=\|\sum_{s}b_{s}e_{s}\|_{Y}$

For the reverse domination fix $N\in\mathbb{N}$ and let
$M_{N}=\sup\\{\operatorname{supp}(\mathbf{g}):\mathbf{g}\text{ an h-special
sequence},\sigma(\mathbf{g})\cap[1,12N]\neq\emptyset\\}$. Notice that $M_{N}$
is finite by injectivity of $\sigma$. It follows that for any h-special
sequence $(g_{1},\dots,g_{d})\subset K$, $1\leq n<d$ with
$\max\operatorname{supp}(g_{n})>M_{N}$ and an interval $E$ we have
$\|E(g_{n+1}+\dots+g_{d})\|_{h,\infty}\leq\operatorname{w}_{h}(g_{n+1})\leq
1/12N$.

We shall show that for any g-special pair
$((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$ with
$\operatorname{supp}(x_{1})>M_{N}$ and any scalars $(b_{s})_{s=1}^{N}$ with
$\|(b_{s})_{s}\|_{\infty}\leq 1$ we have
$\|\sum_{s=1}^{N}b_{s}(x_{2s-1}-x_{2s})\|\leq
10\|(b_{s})_{s=1}^{N}\|_{Y}+\frac{1}{2}$. Having this for any
$(b_{s})_{s=1}^{N}$ with $\|\sum_{s=1}^{N}b_{s}(x_{2s-1}-x_{2s})\|=1$ we have
$\|(b_{s})_{s=1}^{N}\|_{Y}\geq\frac{1}{20}$, which ends the proof.

Pick $(b_{s})_{s=1}^{N}$ with $\|(b_{s})_{s}\|_{\infty}\leq 1$, and a
g-special pair $((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$ with
$\operatorname{supp}(x_{1})>M_{N}$. Let
$x=\sum_{s=1}^{N}b_{s}(x_{2s-1}-x_{2s})$, then
$\|x\|_{h,\infty}\leq\max\\{\|x_{s}\|:s=1,\dots,2N\\}\leq 3$ by def. of
g-special pairs and exact pairs.

We notice first the following two estimates.

By the definition of $M_{N}$ for any h-special functional
$g=\pm\operatorname{w}_{h}(g)E\sum_{n=1}^{d}g_{n}$ and
$n_{0}=\min\\{n:\max\operatorname{supp}(g_{n})>M_{N}\\}$ we have the following
estimate

$\displaystyle|g(x)|$
$\displaystyle\leq\operatorname{w}_{h}(g)|Eg_{n_{0}}(x)|+\sum_{n>n_{0}}\sum_{s=1}^{2N}|Eg_{n}(x_{s})|$
$\displaystyle\leq\operatorname{w}_{h}(g)|Eg_{n_{0}}(x)|+\Bigl{\|}E\sum_{n>n_{0}}g_{n}\Bigr{\|}_{h,\infty}\cdot\max_{s}\|x_{s}\|\cdot\\#\Bigl{\\{}s=1,\dots,2N:\Bigl{(}E\sum_{n>n_{0}}g_{n}\Bigr{)}(x_{s})\neq
0\Bigr{\\}}$ (3.1)
$\displaystyle\leq\operatorname{w}_{h}(g)|Eg_{n_{0}}(x)|+(4N)^{-1}\\#([1,2N]\cap\operatorname{supp}_{h}(E\sum_{n>n_{0}}g_{n}))$

Take any g-special functional
$g=E\sum_{s=1}^{L}\varepsilon_{s}(g_{2s-1}+g_{2s})$. For any $s=1,\dots,2L$
put $s^{\prime}=\lceil\frac{1}{2}s\rceil$. Let
$s_{0}=\max\\{s=1,\dots,2L:g_{s}=f_{s}\\}$ if the latter set is non-empty,
otherwise let $s_{0}=0$. Pick also $s_{1}$ with $\min
E\in\operatorname{supp}(g_{s_{1}})$. We shall consider only the case of odd
$s_{0}\geq s_{1}>0$ with $s_{0},s_{1}\in\operatorname{supp}_{h}(g)$, as in
other cases the computation is even simpler. By the definition of g-special
pairs and exact pairs we have

$\displaystyle|g(x)|$
$\displaystyle=|E\sum_{s=1}^{L}\varepsilon_{s}b_{s}(g_{2s-1}(x_{2s-1})-g_{2s}(x_{2s}))|$
$\displaystyle\leq|Eg_{s_{1}}(b_{s^{\prime}_{1}}x_{s_{1}})|+|Eg_{s_{0}}(b_{s_{0}^{\prime}}x_{s_{0}})|+|Eg_{s_{0}+1}(b_{(s_{0}+1)^{\prime}}x_{s_{0}+1})|+\sum_{s_{0}+1<s\in\operatorname{supp}_{h}(g)}|Eg_{s}(b_{s^{\prime}}x_{s})|$
$\displaystyle\leq|b_{s^{\prime}_{1}}|\|x_{s_{1}}\|+|b_{s_{0}^{\prime}}|\|x_{s_{0}}\|+|b_{(s_{0}+1)^{\prime}}|\|x_{s_{0}+1}\|+\sum_{s_{0}+1<s\in\operatorname{supp}_{h}(g)}|b_{s^{\prime}}|\max\\{\operatorname{w}(g_{s}),\operatorname{w}(f_{s})\\}$
$\displaystyle\leq
9\|(b_{s^{\prime}})_{s\in\operatorname{supp}_{h}(g)}\|_{\infty}+5/4^{s_{0}+1}\|(b_{s^{\prime}})_{s\in\operatorname{supp}_{h}(g)}\|_{\infty}$

Therefore we have the following

(3.2) $|g(x)|\leq
10\|(b_{s^{\prime}})_{s\in\operatorname{supp}_{h}(g)}\|_{\infty}$

Take any functional $g\in K$ and its h-tree-analysis
$(g_{t})_{t\in\mathcal{T}}$. We shall produce a functional $\tilde{g}\in
K_{Y}$ so that $|g(x)|\leq 10\tilde{g}((b_{s})_{s=1}^{N})+\frac{1}{2}$ by
constructing its tree-analysis $(\tilde{g}_{t})_{t\in\tilde{\mathcal{T}}}$.

We prove inductively, starting from the terminal nodes, that for any node
$t\in\mathcal{T}$ there is a functional $\tilde{g}_{t}\in K_{Y}$ with
$\operatorname{supp}(\tilde{g}_{t})\subset\operatorname{supp}_{h}(g_{t})$ so
that $|g_{t}(x)|\leq 10\tilde{g}_{t}((b_{s})_{s=1}^{N})+\delta_{t}$, where
$\delta_{t}=(4N)^{-1}\\#([1,2N]\cap\operatorname{supp}_{h}(g_{t}))$.

If $t\in\mathcal{T}$ is terminal, then we let
$\tilde{g}_{t}=\operatorname{sgn}(b_{s_{t}})e^{*}_{s_{t}}$ where
$\operatorname{supp}_{h}(g_{t})=\\{s_{t}\\}$ obtaining $|g_{t}(x)|\leq
3|b_{s_{t}}|=3\tilde{g}_{t}((b_{s})_{s})$.

Take non-terminal $t\in\mathcal{T}$ and assume that all
$(\tilde{g}_{r})_{r\in\operatorname{succ}(t)}$ are defined and satisfy the
desired condition.

If $g_{t}$ is g-special, then all of successors of $t$ are terminal, we pick
$s_{0}\in\operatorname{supp}_{h}(g_{t})$ with
$|b_{s^{\prime}_{0}}|=\|(b_{s^{\prime}})_{s\in\operatorname{supp}_{h}(g_{t})}\|_{\infty}$,
let $\tilde{g}_{t}=\operatorname{sgn}(b_{s^{\prime}_{0}})e^{*}_{s_{0}}$ and
obtain $|g_{t}(x)|\leq 10\tilde{g}_{t}((b_{s})_{s})$ by (3.2).

If $g_{t}$ is a subconvex combination of
$(g_{r})_{r\in\operatorname{succ}(t)}$, then we pick
$r_{0}\in\operatorname{succ}(t)$ with the biggest
$10|\tilde{g}_{r_{0}}((b_{s})_{s})|+\delta_{r_{0}}$ and let
$\tilde{g}_{t}=\tilde{g}_{r_{0}}$. Then
$\operatorname{supp}(\tilde{g}_{t})\subset\operatorname{supp}_{h}(g_{t})$ and
$|g_{t}(x)|\leq 10\tilde{g}_{t}((b_{s})_{s})+\delta_{t}$ by the inductive
assumption.

If $g_{t}$ is h-regular,
$g_{t}=\pm\operatorname{w}_{h}(g_{t})E\sum_{r\in\operatorname{succ}(t)}g_{r}$,
then let
$\tilde{g}_{t}=\pm\operatorname{w}_{h}(g_{t})\operatorname{supp}_{h}(g_{t})\sum_{r\in\operatorname{succ}(t)}\tilde{g}_{r}$
and notice $\tilde{g}_{t}\in K_{Y}$ by the inductive assumption and the
definition of h-weighted functionals in $K$. As
$\sum_{r\in\operatorname{succ}(t)}\delta_{r}\leq\delta_{t}$ by inductive
assumption we have the desired estimate.

If $g_{t}$ is h-special we pick $n_{0}$ as in (3) and let
$\tilde{g}_{t}=\operatorname{supp}_{h}(g_{t})\tilde{g}_{n_{0}}$. The desired
estimate follows by the inductive assumption and (3), which finishes the
inductive construction.

Therefore we have $g(x)\leq 10\|(b_{s})_{s}\|_{Y}+\delta_{g}\leq
10\|(b_{s})_{s}\|_{Y}+\frac{1}{2}$, which ends the proof.

(3) Let $T=\sum_{s=1}^{M}a_{s}I_{s}$, pick a block $w\in X$ with $\|w\|=1$ and
$\|T\|\leq 2\|Tw\|$. Take $f\in K$ with its h-tree-analysis
$(f_{t})_{t\in\mathcal{T}}$ such that $f(Tw)=\|Tw\|$. For any
$s\in\operatorname{supp}_{h}(f)$ let $b_{s}=f_{t_{s}}(w)$, where
$t_{s}\in\mathcal{T}$ with $f_{t_{s}}\in K_{s}$ and let $b_{s}=0$ otherwise.

We claim that $\|(b_{s})_{s}\|_{Y}\leq 2\|w\|=2$. First for any
$t\in\mathcal{T}$ with $f_{t}$ g-special of the form
$f_{t}=E\sum_{r=1}^{L}\varepsilon_{r}(f_{2r-1}+f_{2r})$ define
$A_{t}=\operatorname{supp}_{h}(E\sum_{r=1}^{L}\varepsilon_{r}f_{2r-1})$ and
$B_{t}=\operatorname{supp}_{h}(E\sum_{r=1}^{L}\varepsilon_{r}f_{2r})$. Let
$A=\bigcup_{t\in\mathcal{T}:f_{t}\text{ g-special}}A_{t}$,
$B=\operatorname{supp}_{h}(f)\setminus A$ and notice that for any
$s,\bar{s}\in\mathbb{N}$, if $s,\bar{s}\in A$ or $s,\bar{s}\in B$, then we
have $s<\bar{s}$ iff $f_{t_{s}}<f_{t_{\bar{s}}}$.

Take $\tilde{f}\in K_{Y}$ with $\tilde{f}((b_{s})_{s})=\|(b_{s})\|_{Y}$ of the
form $\tilde{f}=\sum_{s\in\operatorname{supp}_{h}(f)}d_{s}e_{s}^{*}$ (by
unconditionality of the basis of $Y$). By the above property of $A$ and $B$,
as $K$ contains all h-regular functionals, we have $f_{A}=\sum_{s\in
A}d_{s}f_{t_{s}}\in K$ and $f_{B}=\sum_{s\in B}d_{s}f_{t_{s}}\in K$. Therefore

$\tilde{f}((b_{s})_{s})=\sum_{s\in\operatorname{supp}_{h}(f)}d_{s}f_{t_{s}}(w)=f_{A}(w)+f_{B}(w)\leq
2\|w\|$

Consider the vector $x=\sum_{s=1}^{M}b_{s}(x_{2s-1}-x_{2s})$. Then by (2)
$\|x\|\leq 20\|(b_{s})_{s=1}^{M}\|_{Y}\leq 40$. On the other hand by (1)

$\|Tw\|=\sum_{s=1}^{M}a_{s}f_{t_{s}}(w)\leq\|\sum_{s=1}^{M}a_{s}b_{s}x_{2s-1}\|=\|Tx\|$

which proves that $x/40$ is the desired vector. ∎

######  Theorem 3.3.

  1. (1)

For any $T\in\operatorname{span}\\{I_{s}:s=1,2,\dots\\}$ we have $\|T\|\leq
2^{7}\operatorname{dist}(T,\mathscr{SS}(X))$.

  2. (2)

The sequences $(I_{s})_{s=1}^{\infty}\subset\mathscr{L}(X)$ and
$(e^{*}_{s})_{s}\subset Y^{*}$ are $2^{7}$-equivalent.

###### Proof.

(1) Fix scalars $(a_{s})_{s=1}^{N}$, let $T=\sum_{s=1}^{N}a_{s}I_{s}$ and fix
$S\in\mathscr{SS}(X)$. By the strict singularity of $S$, Remark 2.2 and Lemma
1.5 pick inductively on $s=1,\dots,2N$ a g-special pair
$((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$ with
$\operatorname{supp}(x_{1})>M_{N}$ and $\|Sx_{s}\|\leq\|T\|/2^{s+6}$ for each
$s$. By Prop. 3.2 (3) for some scalars $(b_{s})_{s=1}^{N}\subset[-1,1]$ the
vector $x=\sum_{s=1}^{N}b_{s}(x_{2s-1}-x_{2s})$ has norm at most 1 and
$\|T\|\leq 40\|Tx\|$. Moreover, by the choice of $(x_{s})$ we have
$\|Sx\|\leq\|T\|/2^{6}$, thus $\|T-S\|\geq\|(T-S)x\|\geq\|T\|/2^{7}$ and we
finish the proof of (1).

(2) We shall prove that for any $(a_{s})_{s=1}^{N}$ we have

$\frac{1}{120}\|\sum_{s=1}^{N}a_{s}I_{s}\|\leq\|\sum_{s=1}^{N}a_{s}e^{*}_{s}\|_{Y^{*}}\leq
20\|\sum_{s=1}^{N}a_{s}I_{s}\|$

Let $T=\sum_{s=1}^{N}a_{s}I_{s}$.

In order to show the first inequality pick by Remark 2.2 and Lemma 1.5 any
g-exact pair $((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$ with
$\operatorname{supp}(x_{1})>M_{N}$. By Prop. 3.2 (3) and (2) for some scalars
$(b_{s})_{s=1}^{N}$ with $\|(b_{s})_{s=1}^{N}\|_{Y}\leq 1$ the vector
$x=\sum_{s=1}^{N}b_{s}(x_{2s-1}-x_{2s})$ satisfies $\|T\|\leq 40\|Tx\|$. Thus

$\|T\|\leq 40\|Tx\|=40\|\sum_{s=1}^{N}a_{s}b_{s}x_{2s-1}\|\leq
120\sum_{s=1}^{N}|a_{s}b_{s}|\leq 120\|\sum_{s=1}a_{s}e^{*}_{s}\|_{Y^{*}}$

as $\|(b_{s})_{s=1}^{N}\|_{Y}\leq 1$ and the basis $(e_{n})_{n}\subset Y$ is
1-unconditional.

For the second inequality pick $y=(b_{s})_{s=1}^{N}\in Y$ of norm 1 on which
$\sum_{s=1}^{N}a_{s}e_{s}^{*}$ attains its norm, by Remark 2.2 and Lemma 1.5
pick any g-special pair $((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$ with
$\operatorname{supp}(x_{1})>M_{N}$ and let
$x=\sum_{s=1}^{N}b_{s}(x_{2s-1}-x_{2s})$. By Prop. 3.2 (2) $\|x\|\leq 20$,
whereas by Prop. 3.2 (1) we have

$\|\sum_{s=1}^{N}a_{s}e^{*}_{s}\|_{Y^{*}}=\sum_{s=1}^{N}a_{s}b_{s}\leq\|\sum_{s=1}^{N}a_{s}b_{s}x_{2s-1}\|=\|Tx\|$

∎

As $dist(I_{0},\overline{\operatorname{span}}\\{I_{s}:s=1,2,\dots\\})=1$ and
the space $Y$ is reflexive, we have the following

######  Corollary 3.4.

The sequence $(I_{s})_{s=0}^{\infty}\subset\mathscr{L}(X)$ is an unconditional
shrinking and boundedly complete basic sequence.

##  4\. The Gowers-Maurey type notions and properties in the space $X$

We repeat in this section the standard scheme of Gowers-Maurey spaces theory
as presented in [AT] in the setting the horizontal structure of $X$,
presenting in detail the proofs requiring some adjustment due to
"antihorizontal" behaviour of g-special functionals and quoting the rest. The
adjustments rely on Lemma 2.5 and the following notion, essential in the proof
of the Basic Inequality (Lemma 4.8) for certain special h-block sequences.

######  Definition 4.1 (g-separated sequences).

An h-block sequence $(x_{n})_{n=1}^{N}\subset X$ is called
$(C,\epsilon)$-g-separated, $C>1$ and $\epsilon>0$, if there is some
increasing sequence $(K_{n})_{n=1}^{N}\subset\mathbb{N}$ such that

  1. (1)

$\|x_{n}\|\leq C$, for any $n=1,\dots,N$

  2. (2)

$f(x_{n})=0$ for any $K$-g-special $f$ with $K\leq K_{n}$, for any
$n=1,\dots,N$

  3. (3)

$|f(x_{n})|<\epsilon$ for any $K$-g-special $f$ with $K\geq K_{n+1}$, for any
$n=1,\dots,N$.

######  Lemma 4.2.

For any $C>1$, $\epsilon>0$ and $(C,\epsilon)$-g-separated
$(x_{n})_{n=1}^{N}\subset X$ there is $K_{N+1}\in\mathbb{N}$ so that for any
$x\in X$ with $\|x\|\leq C$ and $\operatorname{supp}_{h}(x)>2K_{N+1}$ the
sequence $(x_{1},\dots,x_{N},x)$ is $(C,\epsilon)$-g-separated.

In particular any bounded h-block sequence $(x_{n})_{n}\subset X$ for any
$\epsilon>0$ has a $(C,\epsilon)$-g-separated subsequence, where
$C=\sup_{n}\|x_{n}\|$.

###### Proof.

Consider $(x_{n})_{n=1}^{N}$ as above, with associated $(K_{n})_{n=1}^{N}$.
Pick $K_{N+1}>K_{N}$ with
$C\max\operatorname{supp}(x_{N})/K_{N+1}<\epsilon$. Notice that for any
$K$-g-special $f$ with $K\geq K_{N+1}$ we have $|f(x_{N})|\leq
C\max\operatorname{supp}(x_{N})\|f\|_{\infty}\leq
C\max\operatorname{supp}(x_{N})/K_{N+1}<\epsilon$ and we obtain (3) for
$x_{N}$. Pick $x\in X$ with $\operatorname{supp}_{h}(x)>2K_{N+1}$ and notice
that for any $K$-g-special $f$ with $K\leq K_{N+1}$ we have
$\operatorname{supp}_{h}(f)\leq 2K_{N+1}<\operatorname{supp}_{h}(x)$ and hence
(2) is satisfied for $x$, which ends the proof. ∎

######  Definition 4.3 ($\ell_{1}$-averages).

A normalized vector $x\in X$ is called a $C$-$\ell_{1}$-average of length
$N\in\mathbb{N}$, $C>1$, if there is an h-block sequence
$(x_{n})_{n=1}^{N}\subset X$ with $x=N^{-1}(x_{1}+\dots+x_{N})$ and
$\|x_{n}\|\leq C$ for each $n=1,\dots,N$.

If $C=2$ we will omit it in the notation and say that $x$ is an
$\ell_{1}$-average of suitable length.

######  Lemma 4.4.

Any h-block subspace of $X$ contains an $\ell_{1}$-average of arbitrary
length.

###### Proof.

We follow here the proof of [GM1, Lemma 1]. Fix $k\in\mathbb{N}$ and a
normalized h-block sequence $(x_{n})_{n}\subset X$. Passing to a subsequence
if necessary by Lemma 2.5 (1) we can assume that there is an h-block sequence
$(f_{n})_{n}\subset K$ and $\delta>0$ such that $f_{n}(x_{n})>\delta$ and
$f_{n}(x_{m})=0$ for any $n\neq m$. By property $(**)$ of sequences
$(n_{j})_{j}$, $(m_{j})_{j}$ (cf. Notation 1) pick $j\in\mathbb{N}$ so that
$n_{2j}=2^{km}$, for $m$ big enough to ensure
$m_{2j}<\delta\sqrt[k]{n_{2j}}=\delta 2^{m}$ (the latter is possible by $(*)$,
as $m_{j}/\sqrt[k]{n_{j}}\to 0$). For every $0\leq i\leq m$ and $1\leq j\leq
2^{k(m-i)}$ let $x(i,j)=\sum_{n=(j-1)2^{ki}+1}^{j2^{ki}}x_{n}$. Assume towards
contradiction that for $1\leq i\leq m$ and $1\leq j\leq 2^{k(m-i)}$,
$2^{-ki}2^{-(i-1)}x(i,j)$ is not an $\ell_{1}$-average of length $2^{k}$.
Notice that for any $0\leq i<m$, $x(i+1,j)$ is a sum of $2^{k}$ many terms of
the form $x(i,l)$. It follows by induction that $\|x(i,j)\|\leq 2^{-i}2^{ki}$,
in particular $\|x(m,1)\|\leq 2^{-m}2^{km}$. On the other hand

$\|x(m,1)\|=\|\sum_{n=1}^{2^{km}}x_{n}\|\geq
m_{2j}^{-1}\sum_{n=1}^{2^{km}}f_{n}(x_{n})\geq
n_{2j}m_{2j}^{-1}\delta=2^{km}m_{2j}^{-1}\delta$

which contradicts the choice of $m$. Therefore we proved that there exist
$\ell_{1}$-averages of length $2^{k}$ for any $k\in\mathbb{N}$ in the subspace
spanned by $(x_{n})_{n}$, thus also $\ell_{1}$-averages of arbitrary length.
Indeed, for $l\in\mathbb{N}$ pick $k\in\mathbb{N}$ with $l\leq 2^{k}$ and an
$\ell_{1}$-average $x=2^{-k}\sum_{i=1}^{2^{k}}y_{i}$. Let
$\mathcal{F}=\\{F\subset\\{1,\dots,2^{k}\\},\\#F=l\\}$ and write

$x=l2^{-k}\binom{2^{k}-1}{l-1}^{-1}\sum_{F\in\mathcal{F}}y_{F}$

where $y_{F}=l^{-1}\sum_{i\in F}y_{i}$ for each $F\subset\\{1,\dots,2^{k}\\}$.
Thus $x=(\\#\mathcal{F})^{-1}\sum_{F\in\mathcal{F}}y_{F}$ and so there is at
least one $F\in\mathcal{F}$ with $\|y_{F}\|=1$, i.e. $y_{F}$ is an
$\ell_{1}$-average of length $l$. ∎

The next Lemma is standard, cf. [AT, Lemma II.23]

######  Lemma 4.5.

Let $x\in X$ be an $\ell_{1}$-average of length $N$. Then for any intervals
$E_{1}<\dots<E_{d}$ with $d\leq N$ we have $\sum_{i=1}^{d}\|I_{E_{i}}x\|\leq
6$.

We modify below the classical definition of RIS by adding the assumption on
g-separation in (4).

######  Definition 4.6 (Rapidly Increasing Sequences).

An h-block sequence $(x_{k})\subset X$ is called a $(C,\epsilon)$\- rapidly
increasing sequence (RIS), for $C>1$, $\epsilon>0$, if there is a strictly
increasing sequence $(j_{k})_{k}\subset L_{0}$ such that

  1. (1)

$\|x_{k}\|\leq C$, for any $k$,

  2. (2)

$2m_{j_{1}}^{-1}<\epsilon$ and
$2m_{j_{k+1}}^{-1}\max\operatorname{supp}_{h}(x_{k})<\epsilon$ for any $k$,

  3. (3)

$|f(x_{k})|\leq C\operatorname{w}_{h}(f)$ for any h-weighted $f\in K$ with
$\operatorname{w}_{h}(f)>m_{j_{k}}^{-1}$ for any $k$,

  4. (4)

$(x_{k})_{k}$ is $(C,\epsilon)$-g-separated.

######  Remark 4.7.

Let $(x_{k})$ be an h-block sequence of $\ell_{1}$-averages of strictly
increasing lengths $(n_{j_{k}})\subset\mathbb{N}$. Then by Lemmas 4.2 and 4.5
there is infinite $J\subset\mathbb{N}$ so that $(x_{k})_{k\in J}$ is a
$(6,\epsilon)$-RIS.

The next Lemma is the key tool, classical by now, for estimating the norms of
averages of RIS, reducing it to computing norms of averages of the unit basic
vectors in the auxiliary mixed Tsirelson space
$W=T[(\mathcal{A}_{2n_{j}},\frac{1}{m_{j}})_{j\in L_{0}}]$ with the canonical
norming set $K_{W}$.

######  Lemma 4.8 (Basic Inequality).

Let $(x_{k})_{k=1}^{N}\subset X$ be a $(C,\epsilon)$-RIS for some $C>1$ and
$\epsilon>0$.

Then for any norming functional $f\in K$ there is a norming functional $g\in
K_{W}$ satisfying

$|f(\sum_{k=1}^{N}x_{k})|\leq Cg(\sum_{k=1}^{N}e_{k})+C\epsilon N$

and such that if $f$ is h-weighted, then either $g$ is also weighted with
$\operatorname{w}(g)=\operatorname{w}_{h}(f)$ or $g$ is some unit vector
$e_{t}^{*}$.

Moreover, if additionally for a fixed $j_{0}\in\mathbb{N}$ and for any
h-weighted $h\in K$ with $\operatorname{w}_{h}(h)=m_{j_{0}}^{-1}$ and any
interval $E$ we have $|h(\sum_{k\in E}x_{k})|\leq C$, then the functional $g$
can be chosen to have a tree-analysis $(g_{t})_{t\in\mathcal{T}}$ with
$\operatorname{w}(g_{t})\neq m_{j_{0}}$ for any $t\in\mathcal{T}$ with $g_{t}$
weighted.

###### Proof.

Let $(m_{j_{k}})_{k}$ be the sequence of weights associated to the RIS
$(x_{k})_{k}$. We repeat the standard reasoning, with the only difference in
CASE 5 below.

Take $f\in K$ with a tree-analysis $(f_{t})_{t\in\mathcal{T}}$. We will prove
the Lemma under the additional assumption for some fixed $j_{0}$. The proof in
general case follows along the same lines, just without referring to the case
of weight $m_{j_{0}}$. We prove by induction, starting from the terminal
nodes, that for any $t\in\mathcal{T}$ we have the following: for any interval
$E\subset\mathbb{N}$ there is functional $g_{t}\in K_{W}$ with
$\operatorname{supp}g_{t}\subset E$ such that

$|f_{t}(\sum_{k\in E}x_{k})|\leq Cg_{t}(\sum_{k\in E}e_{k})+C\epsilon\\#E$

and if $f_{t}$ is h-weighted, then either $g_{t}$ is weighted with
$\operatorname{w}(g_{t})=\operatorname{w}_{h}(f_{t})$ or $g_{t}$ is a unit
vector $e_{n_{t}}^{*}$. Moreover, if $g_{t}$ is weighted, then
$\operatorname{w}(g)\neq m_{j_{0}}^{-1}$.

If $t\in\mathcal{T}$ be terminal, then $f_{t}$ is simple with
$\operatorname{supp}_{h}(f_{t})=\\{s_{t}\\}$ for some $s_{t}\in\mathbb{N}$,
thus letting $g_{t}=e_{\min E}^{*}$ we finish the case.

Take a non-terminal node $t\in\mathcal{T}$ and interval $E$ and assume we
proved the assertion for all its successors. Consider the following cases.

CASE 1. $f_{t}$ is h-weighted with
$\operatorname{w}_{h}(f_{t})=m_{j_{0}}^{-1}$, then by the additional
assumption

$|f_{t}(\sum_{k\in E}x_{k})|\leq C\leq Ce^{*}_{\min E}(\sum_{k\in E}e_{k})$

and we let $g_{t}=e^{*}_{\min E}$.

CASE 2. $f_{t}$ is h-weighted with $\operatorname{w}_{h}(f_{t})\neq
m_{j_{0}}^{-1}$ and there is some $k\in E$ such that
$m_{j_{k}}\leq\operatorname{w}_{h}(f_{t})^{-1}$. Let $k_{t}$ be maximal in $E$
with this property. By the condition (2) of RIS we have

$|f_{t}(\sum_{k\in
E,k<k_{t}}x_{k})|\leq\operatorname{w}_{h}(f_{t})\max\operatorname{supp}_{h}(x_{k_{t}-1})\leq\frac{\epsilon}{2}$

and by conditions (2) and (3) of RIS

$|f_{t}(\sum_{k\in E,k>k_{t}}x_{k})|\leq
C\operatorname{w}_{h}(f_{t})\\#E\leq\frac{C\epsilon}{2}\\#E$

Hence for $g_{t}=e^{*}_{k_{t}}$ we have the desired estimate.

CASE 3. $f_{t}$ is h-weighted with $\operatorname{w}_{h}(f_{t})\neq
m_{j_{0}}^{-1}$ and for all $k\in E$ we have
$m_{j_{k}}>\operatorname{w}_{h}(f_{t})^{-1}$. Let

$E_{0}=\\{k\in E:\ \operatorname{rng}_{h}(x_{k})\text{ intersects horizontal
ranges of at least two different }f_{s},s\in\operatorname{succ}(t)\\}$
$E_{s}=\\{k\in E\setminus E_{0}:\ \operatorname{rng}_{h}(x_{k})\text{
intersects }\operatorname{rng}_{h}(f_{s})\\},\ \ \ \
s\in\operatorname{succ}(t)$

Since each $E_{s}$ is an interval, by inductive assumption we pick for each
$f_{s}$ and $E_{s}$ a suitable $g_{s}\in K_{W}$ with
$\operatorname{supp}(g_{s})\subset E_{s}$. Notice
$\\#E_{0}\leq\\#\operatorname{succ}(t)$ and thus $\\{E_{s}:\
s\in\operatorname{succ}(t)\\},\\{\\{k\\}:\ k\in E_{0}\\}$ is a family of
pairwise disjoint intervals of cardinality at most
$2\\#\operatorname{succ}(t)$. Therefore $g_{t}\in K_{W}$ and
$\operatorname{supp}(g_{t})\subset E$, where

$g_{t}=\operatorname{w}_{h}(f_{t})(\sum_{s\in\operatorname{succ}(t)}g_{s}+\sum_{k\in
E_{0}}e^{*}_{k})$

Estimate by the condition (2) of RIS and the inductive assumption

$\displaystyle|f_{t}(\sum_{k\in E}x_{k})|$
$\displaystyle\leq\operatorname{w}_{h}(f_{t})\sum_{s\in\operatorname{succ}(t)}|f_{s}(\sum_{k\in
E_{s}}x_{k})|+\sum_{k\in E_{0}}|f_{t}(x_{k})|$ $\displaystyle\leq
C\operatorname{w}_{h}(f_{t})\sum_{s\in\operatorname{succ}(t)}g_{s}(\sum_{k\in
E_{s}}e_{k})+C\epsilon\sum_{s\in\operatorname{succ}(t)}\\#E_{s}+C\operatorname{w}_{h}(f_{t})\\#E_{0}$
$\displaystyle\leq Cg_{t}(\sum_{k\in E}e_{k})+C\epsilon\\#E$

CASE 4. $f_{t}$ is a subconvex rational combination of functionals:
$f_{t}=\sum_{s\in\operatorname{succ}(t)}c_{s}f_{s}$. By the inductive
assumption we pick for any $s\in\operatorname{succ}(t)$ and $E$ a suitable
$g_{s}\in W$, let $g_{t}=\sum_{s\in\operatorname{succ}(t)}c_{s}g_{s}$ and
compute

$\displaystyle|f_{t}(\sum_{k\in E}x_{k})|$
$\displaystyle\leq\sum_{s\in\operatorname{succ}(t)}c_{s}|f_{s}(\sum_{k\in
E}x_{k})|\leq C\sum_{s\in\operatorname{succ}(t)}c_{s}g_{s}(\sum_{k\in
E}e_{k})+C\epsilon\\#E\sum_{s\in\operatorname{succ}(t)}c_{s}$
$\displaystyle=Cg_{t}(\sum_{k\in E}e_{k})+C\epsilon\\#E$

CASE 5. $f_{t}$ is $K$-g-special for some $K\in\mathbb{N}$. Let
$(K_{n})_{n}\subset\mathbb{N}$ be the sequence associated to $(x_{n})$ as a
g-separated sequence. Pick $n_{0}$ with $K_{n_{0}}<K\leq K_{n_{0}+1}$. Then
$f_{t}(x_{n})=0$ for any $n>n_{0}$ and $|f_{t}(x_{n})|\leq\epsilon$ for any
$n<n_{0}$. Let $g_{t}=e^{*}_{\min E}$, then

$|f_{t}(\sum_{k\in E}x_{k})|\leq Cg_{t}(\sum_{k\in E}e_{k})+\sum_{k<n_{0},k\in
E}\epsilon\leq Cg_{t}(\sum_{k\in E}e_{k})+\epsilon\\#E$

which ends the proof. ∎

######  Proposition 4.9.

Fix $j\in\mathbb{N}$. Let $(x_{k})_{k=1}^{n_{j}}\subset X$ be a
$(C,\frac{1}{m_{j}^{2}})$-RIS. Then for any h-weighted functional $f\in K$ we
have

$|f(\frac{1}{n_{j}}\sum_{k=1}^{n_{j}}x_{k})|\leq\begin{cases}\frac{2C}{m_{j}^{2}}\
\ \ \ \ \ \ \ \text{if}\ \ \ \operatorname{w}_{h}(f)<m_{j}^{-1}\\\
\frac{2C}{m_{j}}\ \ \ \ \ \ \ \ \text{if}\ \ \
\operatorname{w}_{h}(f)=m_{j}^{-1}\\\ \frac{3C\operatorname{w}_{h}(f)}{m_{j}}\
\ \ \text{if}\ \ \ \operatorname{w}_{h}(f)>m_{j}^{-1}\end{cases}$

In particular

$\|\frac{1}{n_{j}}\sum_{k=1}^{n_{j}}x_{k}\|\leq\frac{2C}{m_{j}}$

Moreover, if the additional assumption for $j_{0}=j$ in the Basic Inequality
(Lemma 4.8) is satisfied, then

$\|\frac{1}{n_{j}}\sum_{k=1}^{n_{j}}x_{k}\|\leq\frac{4C}{m_{j}^{2}}$

###### Proof.

The proof is a straightforward application of estimates on the basis of $W$
(cf. [AT, Lemma II.9]) and Lemma 4.8. ∎

The next definition follows along the lines of the classical one with slight
changes required in our setting, which do not affect the results.

######  Definition 4.10 (Dependent sequences).

A double sequence $(x_{k},f_{k})_{k=1}^{n_{2j-1}}\subset X\times K$ is called
a $(\eta,D,j)$-dependent sequence, for $\eta\in\\{0,1\\}$, $D\geq 1$, $2j-1\in
L_{0}$, if

  1. (1)

$(f_{k})_{k}$ is a $j$-h-special sequence,

  2. (2)

$f_{k}(x_{k})=1$ if $\eta=1$ and $|f_{k}(x_{k})|\leq 2^{-k}$ if $\eta=0$ and
$\operatorname{rng}_{h}(x_{k-1})<\operatorname{rng}_{h}(f_{k})<\operatorname{rng}_{h}(x_{k+1})$
for any $k$,

  3. (3)

$2m_{j_{1}}^{-1}<m_{2j-1}^{-2}$ and
$2m_{j_{k+1}}^{-1}\max\operatorname{supp}_{h}(x_{k})<m_{2j-1}^{-2}$ for any
$k$.

  4. (4)

$x_{k}=d_{k}\frac{m_{j_{k}}}{n_{j_{k}}}\sum_{i=1}^{n_{j_{k}}}y_{i}^{k}$ for
some $(6,\frac{1}{m_{j_{k}}^{2}})$-RIS $(y^{k}_{i})_{i=1}^{n_{j_{k}}}$ where
$m_{j_{k}}^{-1}=\operatorname{w}_{h}(f_{k})$ and $d_{k}\in[1,D]$ for any $k$,

  5. (5)

$(x_{k})_{k}$ is a $(12D,\frac{1}{m_{2j-1}^{2}})$-g-separated sequence,

######  Remark 4.11.

Notice that the condition (3) of dependent sequences, redundant in the
classical situation, is required in order to prove that the vector part of a
dependent sequence is itself a suitable RIS, as we cannot manage
$\operatorname{rng}_{h}(x_{k})=\operatorname{rng}_{h}(f_{k})$ (since
projections $(I_{[1,N]})_{N}$ are not uniformly bounded). This condition is
also the reason for defining $\sigma$ as a multivalued function, which
guarantees existence of dependent sequences.

######  Proposition 4.12.

Let $(x_{k},f_{k})_{k=1}^{n_{2j-1}}$ be a $(\eta,D,j)$-dependent sequence,
$D\geq 1$, $2j-1\in L_{0}$. Then

$\displaystyle\|\frac{1}{n_{2j-1}}\sum_{k=1}^{n_{2j-1}}(-1)^{k}x_{k}\|\leq\frac{180D}{m_{2j-1}^{2}}$
$\displaystyle\ \ \ \ \ \text{if}\ \ \ \eta=1$
$\displaystyle\|\frac{1}{n_{2j-1}}\sum_{k=1}^{n_{2j-1}}x_{k}\|\leq\frac{180D}{m_{2j-1}^{2}}$
$\displaystyle\ \ \ \ \ \text{if}\ \ \ \eta=0$

###### Proof.

Notice that the sequence $(x_{k})_{k=1}^{n_{2j-1}}$ is itself a
$(18D,\frac{1}{m_{2j-1}^{2}})$-RIS with $\|x_{k}\|\leq 12D$ for any $k$ by
Prop. 4.9 applied to the $(6,\frac{1}{m_{j_{k}}^{2}})$-RIS
$(y_{i}^{k})_{i=1}^{n_{j_{k}}}$ for each $k$ and conditions (3) and (5). Thus
the same hold for $((-1)^{k}x_{k})_{k=1}^{n_{2j-1}}$ also.

Let $\eta=1$. By Prop. 4.9 it is enough to show that for any $j$-h-special
$h\in K$ and any interval $E$ we have $|h(\sum_{k\in E}(-1)^{k}x_{k})|\leq
45D$. Take $j$-h-special $h\in K$ and an interval $E\subset\mathbb{N}$. Then
$h$ is of the form
$h=\frac{1}{m_{2j-1}}(Af_{l}+\dots+f_{r}+h_{r+1}+\dots+h_{d})$ for some
$j$-h-special sequence $(f_{1},\dots,f_{r},h_{r+1},\dots,h_{n_{2j-1}})$,
$1\leq l\leq r\leq d\leq n_{2j-1}$, $A\subset\mathbb{N}$ interval, where $r$
is minimal with $h_{r+1}\neq f_{r+1}$.

For any $k,i$ with $\operatorname{w}_{h}(h_{i})\neq m_{j_{k}}$ by Prop. 4.9
and the definition of dependent sequences we have
$|h_{i}(x_{k})|\leq\frac{18D}{n_{2j-1}^{2}}$. Now assume that $\min E\leq r$
and compute

$\displaystyle|h(\sum_{k\in E}(-1)^{k}x_{k})|$
$\displaystyle\leq|(Af_{l}+\dots+f_{r})((-1)^{\min E}x_{\min
E}+\dots+(-1)^{r}x_{r})|$ $\displaystyle+|h_{r+1}(x_{r+1})|+\sum_{w(h_{i})\neq
m_{j_{k}}}|h_{i}(x_{k})|$ $\displaystyle\leq
12D+1+12D+n_{2j-1}^{2}\frac{18D}{n_{2j-1}^{2}}\leq 43D$

In other cases of $E$ the computation is even simpler. The case $\eta=0$
follows analogously. ∎

######  Lemma 4.13.

For any h-block subspaces $U,V$ of $X$ there is some $D\geq 1$ so that for any
$2j-1\in L_{0}$ there is a $(1,D,j)$-dependent sequence
$(x_{k},f_{k})_{k=1}^{n_{2j-1}}$ with $(x_{2k})_{k=1}^{n_{2j-1}/2}\subset U$
and $(x_{2k-1})_{k=1}^{n_{2j-1}/2}\subset V$.

###### Proof.

By Lemma 4.2 pick h-block sequences $(y_{2i})_{i}\subset U$,
$(y_{2i-1})_{i}\subset V$ of $\ell_{1}$-averages of increasing length. By
Lemma 2.5, passing to subsequences and allowing some perturbation, we can
assume that for some h-block sequences $(f_{2i})_{i},(f_{2i-1})_{i}\subset K$
and $0<\delta_{1},\delta_{2}\leq 1$ we have $f_{2i-1}(y_{2i-1})=\delta_{1}$
and $f_{2i}(y_{2i})=\delta_{2}$.

Now fix $j\in L_{0}$ and pick inductively on $k$, using Remark 4.7, Lemma 4.2,
the definition of $\sigma$ and the fact that $(y_{2i-1})_{i}$, $(y_{2i})_{i}$,
$(f_{2i-1})_{i}$, $(f_{2i})_{i}$ are h-block,
$(6,\frac{1}{m_{j_{k}}^{2}})$-RIS
$(y_{i}^{k})_{i=1}^{n_{j_{k}}}\subset(y_{2i})_{i}$ and
$(f_{i}^{k})_{i=1}^{n_{j_{k}}}\subset(f_{2i})_{i}$ for $k\in 2\mathbb{N}$ and
$(6,\frac{1}{m_{j_{k}}^{2}})$-RIS
$(y_{i}^{k})_{i=1}^{n_{j_{k}}}\subset(y_{2i-1})_{i}$ and
$(f_{i}^{k})_{i=1}^{n_{j_{k}}}\subset(f_{2i-1})_{i}$ for $k\in 2\mathbb{N}-1$
such that for $f_{k}=m_{j_{k}}^{-1}\sum_{i=1}^{n_{j_{k}}}f_{i}^{k}$,
$k=1,\dots,n_{2j-1}$, $d_{2k}=\delta_{2}^{-1}$, $d_{2k-1}=\delta_{1}^{-1}$ and
$x_{k}=d_{k}\frac{m_{j_{k}}}{n_{j_{k}}}\sum_{i=1}^{n_{j_{k}}}y_{i}^{k}$,
$k=1,\dots,n_{2j-1}$ conditions (1)-(4) of dependent sequences are satisfied.
In particular with multivalued $\sigma$ we can guarantee (3). Passing to a
further sequence by Lemma 4.2 we obtain the condition (5).

Then the double sequence $(x_{k},f_{k})_{k=1}^{n_{2j-1}}$, is a
$(1,D,j)$-dependent sequence, with
$D=\max\\{\frac{1}{\delta_{1}},\frac{1}{\delta_{2}}\\}$, with desired
properties. ∎

Now we are ready to prove the horizontal HI property of the space $X$.

######  Theorem 4.14.

No two h-block subspaces of $X$ form a direct sum.

###### Proof.

Fix h-block subspaces $U,V$ of $X$. Pick $D\geq 1$ ensured by Lemma 4.13 and
for any $2j-1\in L_{0}$ choose a $(1,D,j)$-dependent sequence
$(x_{k},f_{k})_{k=1}^{n_{2j-1}}$ with $x_{2k}\in U$ and $x_{2k-1}\in V$ for
any $k=1,\dots,n_{2j-1}/2$. Let

$y=\frac{m_{2j-1}}{n_{2j-1}}\sum_{k=1}^{n_{2j-1}/2}x_{2k}\in U,\ \ \ \
z=\frac{m_{2j-1}}{n_{2j-1}}\sum_{k=1}^{n_{2j-1}/2}x_{2k-1}\in V$

Then by Proposition 4.12 we have $\|y-z\|\leq\frac{180D}{m_{2j-1}}$, whereas

$\|y+z\|\geq\frac{1}{m_{2j-1}}\sum_{k=1}^{n_{2j-1}}f_{k}(y+z)\geq 1$

which, as $j$ is arbitrary big, ends the proof. ∎

##  5\. Bounded operators on $X$

In this section we prove the "scalar+horizontal strictly singular" property of
$X$ continuing the Gowers-Maurey scheme from the previous section and describe
the behaviour of bounded operators on $X$ with respect to subspaces $X_{s}$,
$s\in\mathbb{N}$. The reasoning, based on classical mixed Tsirelson spaces
tools including the results from the previous section and on already shown
complete boundedness of $(I_{s})_{s}$, leads to Cor. 5.6. The latter combined
with Cor. 3.4 proves the main theorem.

######  Proposition 5.1.

Fix $s\in\mathbb{N}$.

  1. (1)

The spaces $X_{s}$ and $(I-I_{s})(X)$ are totally incomparable (i.e. no
infinite dimensional subspaces $U\subset X_{s}$ and $V\subset(I-I_{s})(X)$ are
isomorphic).

  2. (2)

For any $T\in\mathscr{L}(X)$ there is $a_{s}\in\mathbb{R}$ with $T\circ
I_{s}-a_{s}I_{s}\in\mathscr{SS}(X)$.

###### Proof.

(1) Assume the contrary. Let $T:U\to V$, for some infinite dimensional
subspaces $U\subset X_{s}$, $V\subset(I-I_{s})(X)$, be an isomorphism.
Allowing some perturbation we can pick a normalized block sequence
$(u_{n})_{n}\subset U$ with $(Tu_{n})_{n}\subset V$ also block. Let
$U^{\prime}=\overline{\operatorname{span}}\\{u_{n}:n\in\mathbb{N}\\}$.

Assume first that for some $s\neq t\in\mathbb{N}$ the operator $I_{t}\circ
T:U^{\prime}\to X_{t}$ is not strictly singular. Passing to a further block
sequence we can assume that $I_{t}\circ T:U^{\prime}\to X_{t}$ is an
isomorphism onto its image. Then by Remark 2.2 and Lemma 1.6 for any $2j\in
L_{t}$ pick a normalized block sequence $(z_{i})_{i=1}^{n_{2j}}$ of
$(u_{n})_{n}$ with $\|z_{1}+\dots+z_{n_{2j}}\|\leq 5n_{2j}m_{2j}^{-2}$. On the
other hand, by the lower estimate in $X_{t}$, as $(Tz_{i})_{i}$ is also a
block sequence, we have $\|I_{t}Tz_{1}+\dots+I_{t}Tz_{n_{2j}}\|\geq
n_{2j}m_{2j}^{-1}\|T\|^{-1}$, which for large $2j\in L_{t}$ yields a
contradiction.

As all operators $I_{t}\circ T$, $t\neq s$, are strictly singular, allowing
some perturbation pick a normalized block sequence
$(u^{\prime}_{n})_{n}\subset U^{\prime}$ with $(Tu^{\prime}_{n})_{n}$
(seminormalized) h-block. By Remark 2.2 and Lemma 1.6 for any $2j\in L_{0}$
pick a normalized block sequence $(w_{i})_{i}$ of $(u^{\prime}_{n})_{n}$ with
$\|\sum_{i\in A}w_{i}\|\leq 5n_{2j}m_{2j}^{-2}$ provided $\\#A\leq n_{2j}$. By
Lemma 2.5 (1) we obtain h-block $(f_{i})_{i\in J}\subset K$ with
$f_{i}(Tw_{i})\geq\delta$ for some universal $\delta>0$ and all $i\in J$ and
$f_{i}(Tw_{k})=0$ for any $i\neq k\in J$. Therefore for any $2j\in L_{0}$ and
$A\subset J$, $\\#A\leq n_{2j}$, we have

$\|\sum_{i\in A}Tw_{i}\|\geq m_{2j}^{-1}\sum_{i=1}^{n_{2j}}f_{i}(Tw_{i})\geq
n_{2j}m_{2j}^{-1}\delta$

which by the choice of $(w_{i})_{i}$ contradicts the boundedness of $T$.

(2) For any bounded operator $T\in\mathscr{L}(X)$ write $T\circ
I_{s}=(I-I_{s})\circ T\circ I_{s}+I_{s}\circ T\circ I_{s}$. Then apply (1) and
the "scalar+strictly singular" property of $X_{s}$ (Remark 2.2 and Thm 1.7). ∎

We shall use the horizontal version of the strictly singular property.

######  Definition 5.2.

An operator $Q\in\mathscr{L}(X)$ is horizontally strictly singular if for no
h-block subspace $U$ of $X$ the operator $Q|_{U}:U\to X$ is an isomorphism
onto its image.

######  Proposition 5.3.

Let $S\in\mathscr{L}(X)$ satisfy $S\circ I_{s}\in\mathscr{SS}(X)$ for any
$s\in\mathbb{N}$. Then $S\in\mathscr{SS}(X)$.

###### Proof.

By the assumption on $S$ pick a normalized h-block sequence
$(x_{s})_{s}\subset X$, so that $\operatorname{supp}_{h}(x_{s})=\\{s\\}$ and
$\|Sx_{s}\|<4^{-s}$ for each $s\in\mathbb{N}$. It follows that for any
normalized h-block sequence $(y_{n})_{n}$ of $(x_{s})_{s}$ we have $Sy_{n}\to
0$. By Thm 4.14 the operator $S$ is horizontally strictly singular.

Assume that there is a block subspace $U$ of $X$ such that $S|_{U}$ is an
isomorphism onto its image.

Fix $s\in\mathbb{N}$. Notice there is $\varepsilon_{0}>0$ and
$k_{0}\in\mathbb{N}$ such that for any normalized $u\in U$ with
$\operatorname{supp}(u)>k_{0}$ we have $\|(I-I_{s})(u)\|\geq\varepsilon_{0}$.
Otherwise we could pick a block sequence $(u_{n})_{n}\subset U$ with
$u_{n}-I_{s}u_{n}\to 0$, which contradicts the strict singularity of $S\circ
I_{s}$.

It follows that for any $\epsilon>0$ and $k\in\mathbb{N}$ there is a
normalized vector $u\in U$ with $\operatorname{supp}(u)>k$ and
$\|I_{s}u\|<\epsilon$. Otherwise there would be a finite-codimensional
subspace $U^{\prime}$ of $U$ with spaces $I_{s}(U^{\prime})$,
$(I-I_{s})(U^{\prime})$ isomorphic to $U^{\prime}$, which contradicts (1) of
Prop. 5.1.

As $s\in\mathbb{N}$ is arbitrary, with to some perturbation we can pick a
normalized h-block sequence $(u_{n})_{n}\subset U$, which contradicts the
horizontal strict singularity of $S$. ∎

Continuing the reasoning from the previous section we prove the
"scalar+horizontal strictly singular" property of $X$. Recall that $I_{0}$
denotes the identity on $X$.

######  Theorem 5.4.

Any bounded operator $T\in\mathscr{L}(X)$ is of the form $a_{0}I_{0}+Q$, where
$Q$ is horizontally strictly singular.

###### Proof.

By Prop. 5.1 (1) for any $s\leq r\in\mathbb{N}$ the operator
$(I_{1}+\dots+I_{s})\circ T\circ(I-(I_{1}+\dots+I_{r}))$ is strictly singular,
thus for any $s\leq r\in\mathbb{N}$ and any $\epsilon>0$ there is a block
$z\in X$ with $\|z\|=1$, $\operatorname{supp}_{h}(z)>r$ and
$\|(I_{1}+\dots+I_{s})(Tz)\|<\epsilon$. Applying this observation inductively
and allowing some perturbation we pick a normalized h-block sequence
$(z_{i})_{i}$ such that $(Tz_{i})_{i}$ is also h-block.

By Lemma 4.4 pick an h-block sequence $(y_{i})_{i}$ of $\ell_{1}$-averages of
increasing length in the block subspace spanned by $(z_{i})_{i}$. We claim
that $\operatorname{dist}(Ty_{i},\mathbb{R}y_{i})\xrightarrow{i\to\infty}0$.

Assume the contrary. Then $\inf_{i\in
J}\operatorname{dist}(Ty_{i},\mathbb{R}y_{i})>0$ for some infinite
$J\subset\mathbb{N}$. By Lemma 2.5 (2), passing to a subsequence if necessary
and allowing some perturbation we can assume that there is h-block
$(f_{n})_{n}\subset K$ and $\delta>0$ with $f_{n}(Ty_{n})>\delta$ and
$|f_{n}(y_{n})|\leq 2^{-n}$ for all $n$.

As in the proof of Lemma 4.13, using Remark 4.7, Lemma 4.2 and the definition
of $\sigma$, produce for any $j\in\mathbb{N}$ a $(0,1,j)$-dependent sequence
$(x_{k},f_{k})_{k=1}^{n_{2j-1}}$, picking for each $k=1,\dots,n_{2j-1}$
suitable $(y_{i}^{k})_{i=1}^{n_{j_{k}}}$ and $(f_{i}^{k})_{i=1}^{n_{j_{k}}}$
from $(y_{i})$ and $(f_{i})$ and letting
$x_{k}=\frac{m_{j_{k}}}{n_{j_{k}}}\sum_{i=1}^{n_{j_{k}}}y_{i}^{k}$ and
$f_{k}=\frac{1}{m_{j_{k}}}\sum_{i=1}^{n_{j_{k}}}f_{i}^{k}$ (notice the choice
here is simpler, as the condition $|f_{k}(x_{k})|\leq 2^{-k}$ is automatic,
therefore $d_{k}=1$ for all $k$ and thus $D=1$). For such $x_{k},f_{k}$ we
have $f_{k}(Tx_{k})>\delta$ for each $k$. Let
$x=\frac{1}{n_{2j-1}}\sum_{k=1}^{n_{2j-1}}x_{k}$ and
$f=\frac{1}{m_{2j-1}}\sum_{k=1}^{n_{2j-1}}f_{k}$. Then
$\|x\|\leq\frac{180}{m_{2j-1}^{2}}$ by Prop. 4.12. On the other hand, since
$f\in K$, we have $\|Tx\|\geq f(Tx)>\frac{\delta}{m_{2j-1}}$, which for
sufficiently large $j$ yields contradiction and thus proves that
$\operatorname{dist}(\mathbb{R}y_{i},Ty_{i})\to 0$.

Passing to a subsequence of $(y_{i})$, find $a_{0}\in\mathbb{R}$ with
$\|Ty_{i}-a_{0}y_{i}\|\to 0$. Thus for some infinite $J\subset\mathbb{N}$ the
operator $(T-a_{0}I_{0})|_{\operatorname{span}\\{y_{i}:i\in J\\}}$ is compact.
By Thm 4.14 the operator $T-a_{0}I_{0}$ is horizontally strictly singular. ∎

We shall need next the following strong property of horizontally strictly
singular operators, not obvious as $(X_{s})_{s}$ do not form the Schauder
decomposition of $X$. We profit here from the already shown bounded
completeness of the basic sequence $(I_{s})_{s=1}^{\infty}$.

######  Proposition 5.5.

For any horizontally strictly singular $Q\in\mathscr{L}(X)$ let
$(a_{s})_{s=1}^{\infty}\subset\mathbb{R}$ be the scalars picked by Prop. 5.1
(3). Then the series $\sum_{s=1}^{\infty}a_{s}I_{s}$ is convergent in
$\mathscr{L}(X)$.

###### Proof.

We show first that $a_{s}\to 0$ as $s\to\infty$. Assume that $|a_{s}|>\delta$
for some $\delta>0$ and $s\in J$, $J\subset\mathbb{N}$ infinite. Let $a\neq 0$
be a cluster point of $(a_{s})_{s\in J}$. Choose $(s_{n})_{n}\subset J$ with
$|a_{s_{n}}-a|<a/8^{n}$ for all $n\in\mathbb{N}$. Pick a normalized block
sequence $(x_{n})_{n}$ with $x_{n}\in X_{s_{n}}$ (thus $(x_{n})_{n}$ is
h-block) and $\|Qx_{n}-a_{s_{n}}x_{n}\|<a/8^{n}$, thus
$\|Qx_{n}-ax_{n}\|<a/4^{n}$ for each $n\in\mathbb{N}$. As $Q$ is horizontally
strictly singular, there is a normalized h-block sequence $(z_{i})_{i}$ of
$(x_{n})_{n}$ with $Qz_{i}\to 0$. On the other hand $\|Qz_{i}-az_{i}\|<a/2$,
which yields the contradiction.

Now we prove that $(\sum_{s=1}^{M}a_{s}I_{s})_{M}$ is bounded in
$\mathscr{L}(X)$. This ends the proof by Cor. 3.4.

Fix $M\in\mathbb{N}$. Pick $M\leq N\in\mathbb{N}$ big enough to ensure
$M|a_{s}|\leq 1$ for $s\geq N$. By Remark 2.2 and Lemma 1.5 pick an
$N$-g-special pair $((x_{1},\dots,x_{2N}),(f_{1},\dots,f_{2N}))$ with
$\operatorname{supp}(x_{1})>M_{N}$ and $\|(Q-a_{s}I_{s})x_{s}\|\leq 1/2^{s}$
for each $s=1,\dots,M,N+1,\dots,N+M$. By Prop. 3.2 (3) for some
$(b_{s})_{s=1}^{M}\subset[-1,1]$ the vector
$x=\sum_{s=1}^{M}b_{s}(x_{2s-1}-x_{2s})$ satisfies $\|x\|\leq 1$ and
$\|\sum_{s=1}^{M}a_{s}I_{s}\|\leq 40\|\sum_{s=1}^{M}a_{s}I_{s}x\|$. On other
hand

$\displaystyle\|Q\|$
$\displaystyle\geq\|Qx\|=\|\sum_{s=1}^{M}Qb_{s}x_{s}+\sum_{s=N+1}^{N+M}Qb_{s-N}x_{s}\|$
$\displaystyle\geq\|\sum_{s=1}^{M}a_{s}b_{s}x_{s}\|-\|\sum_{s=1}^{M}(Q-a_{s}I_{s})b_{s}x_{s}\|-\|\sum_{s=N+1}^{N+M}a_{s}b_{s-N}x_{s}\|-\|\sum_{s=N+1}^{N+M}(Q-a_{s}I_{s})b_{s-N}x_{s}\|$
$\displaystyle\geq\|\sum_{s=1}^{M}a_{s}I_{s}x\|-1-3M\max_{N+1\leq s\leq
N+M}|a_{s}|-1$ $\displaystyle\geq\|\sum_{s=1}^{M}a_{s}I_{s}x\|-5$

It follows that $\|\sum_{s=1}^{M}a_{s}I_{s}\|\leq 40(\|Q\|+5)$, which ends the
proof. ∎

######  Corollary 5.6.

$\mathscr{L}(X)=\overline{\operatorname{span}}\\{I_{s}:s=0,1,2,\dots\\}\oplus\mathscr{SS}(X)$.

###### Proof.

For arbitrary $T\in\mathscr{L}(X)$ pick suitable $(a_{s})_{s=0}^{\infty}$,
$a_{0}$ by Thm 5.4, $(a_{s})_{s=1}^{\infty}$ by Prop. 5.1 for $T-a_{0}I_{0}$.
Let $R=\sum_{s=0}^{\infty}a_{s}I_{s}\in\mathscr{L}(X)$, well-defined by Prop.
5.5.

By the choice of $(a_{s})_{s=1}^{\infty}$ for any $s\in\mathbb{N}$ the
operator $(T-R)\circ I_{s}=(T-a_{0}I_{0})\circ I_{s}-a_{s}I_{s}$ is strictly
singular. Therefore $T-R\in\mathscr{SS}(X)$ by Prop. 5.3. Therefore Thm 3.3
(1) ends the proof (notice that
$\operatorname{dist}(I_{0},\overline{\operatorname{span}}\\{I_{s}:s=1,2,\dots\\}+\mathscr{SS}(X))=1$).
∎

The main theorem follows from Corollaries 3.4 and 5.6.

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