# Schauder-Orlicz decompositions, $\ell_{\Phi}$-decompositions and pseudo-
Daugavet property

Vitalii Marchenko<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The concept of $\ell_{\Phi}$-decomposition, extending the concept of
$\ell_{p}$-decomposition of a Banach space, is presented and basic properties
of Schauder-Orlicz decompositions and $\ell_{\Phi}$-decompositions are
studied. We show that Schauder-Orlicz decompositions are orthogonal in a sense
of Grinblyum-James and Singer. Simple constructions of
$\ell_{p}$-decompositions and Schauder-Orlicz decompositions in $L_{p}$ are
presented. We prove that in the class of spaces possessing pseudo-Daugavet
property, which includes classical $L_{p}$, $1\leq p\neq 2$, and $C$,
Schauder-Orlicz decompositions with at least one finite dimensional subspace
do not exist. It follows that Kato theorem on similarity for sequences of
projections [1] cannot be extended to spaces from this class. Moreover we show
that Banach spaces, possessing Schauder-Orlicz decompositions with at least
one finite dimensional subspace, do not have pseudo-Daugavet property. Thus
for Banach spaces $X$ possessing Schauder-Orlicz decompositions we obtain the
following characterization of pseudo-Daugavet property: $X$ has pseudo-
Daugavet property if and only if there is no Schauder-Orlicz decomposition in
$X$ with at least one finite dimensional subspace if and only if there is no
Schauder-Orlicz decomposition in $X$, which is an FDD.

###### keywords:

Schauder-Orlicz decomposition, Schauder decomposition,
$\ell_{\Phi}$-decomposition, pseudo-Daugavet property

###### MSC:

[2020] 46B15 , 46E30 , 46B20 , 46B50 , 46B04 , 46B45 , 41A65

††journal: ArXiv.org

[l1]organization=Institute of Mathematics, University of Szczecin,
addressline=Wielkopolska 15, city=Szczecin, postcode=70451, state=Poland

[l2]organization=B. Verkin Institute for Low Temperature Physics and
Engineering of the National Academy of Sciences of Ukraine, addressline=Nauky
Avenue 47, city=Kharkiv, postcode=61103, state=Ukraine

##  1 Introduction

Let $X$ be a Banach space with norm $\|\cdot\|.$ The sequence
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ of closed nonzero linear subspaces of
$X$ is called a Schauder decomposition (or basis of subspaces) of $X$ provided
that each $x\in X$ has a unique, norm convergent expansion
$x=\sum\limits_{n=0}^{\infty}x_{n},$ where $x_{n}\in\mathfrak{M}_{n}$ for all
$n,$ i.e.

$X=\bigoplus\limits_{n=0}^{\infty}\mathfrak{M}_{n}.$ (1)

It has long been known that every normed linear space has a decomposition [2]
(the case when all subspaces from (1) need not to be closed), but
$\ell_{\infty}$ has no Schauder decomposition [3], [4]. Nevertheless, if we
denote $E=\ell_{\infty}=c_{0}^{\ast\ast}$, then each of spaces $E$,
$E^{\ast\ast}$, $E^{\ast\ast\ast\ast}$, $\ldots$ has no Schauder decomposition
whereas all of spaces $c_{0}$, $c_{0}^{\ast}=\ell_{1}$, $E^{\ast}$,
$E^{\ast\ast\ast}$, $E^{\ast\ast\ast\ast\ast}$, $\ldots$ have Schauder
decompositions, see Example 15.1 in [4].

If $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a Schauder decomposition of $X$,
the sequence of corresponding bounded linear projections
$\\{P_{n}\\}_{n=0}^{\infty}$ on $X$, defined by $P_{n}x=x_{n},$ where
$x=\sum\limits_{n=0}^{\infty}x_{n},$ $x_{n}\in\mathfrak{M}_{n}$, will be
called, as in [4] and [5], the associated sequence of coordinate projections
(a.s.c.p.). Thus, if $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a Schauder
decomposition of $X$, then each $x\in X$ has a unique expansion

$x=\sum\limits_{n=0}^{\infty}P_{n}x,$

where $\\{P_{n}\\}_{n=0}^{\infty}$ is the a.s.c.p. satisfying
$P_{i}P_{j}=\delta_{i}^{j}P_{i}$ for all indexes $i,j\geq 0.$ It is well known
that only separable Banach space may possess a Schauder basis, whereas there
exist nonseparable Banach spaces with Schauder decompositions, see e.g. [2],
[6]. It is worth noting that every non-separable reflexive Banach space has a
Schauder decomposition [6]. On the other hand, there exists a separable Banach
space without a Schauder decomposition [7].

In case when every $\mathfrak{M}_{n}$ is finite dimensional, the decomposition
is called a finite dimensional decomposition, or simply an FDD. Every FDD is
clearly a Schauder decomposition and it was shown by Johnson that a separable
Banach space $X$ has an FDD if and only if $X$ is a dual $\pi_{\lambda}$ space
[8]. Note that there exist Banach spaces with FDD, which have no Schauder
basis [9].

Schauder decompositions are frequently used in functional analysis and
operator theory. Among the most recent papers we point out one by Hajek and
Medina [10] dealing mainly with separable Banach spaces, where it was proved
that if a Banach space has an FDD then there exists a Lipschitz retraction
onto a small generating convex and compact set. Moreover, it was proved in
[10] that if a Banach space possesses a small generating Lipschitz retract,
then it has the $\pi$-property. It is worth mentioning that a Banach space
with the metric $\pi$-property has an FDD [8], but it is still an open
question whether the $\pi$-property implies the metric $\pi$-property. Thus
these results combining together constitute almost a characterization of the
existence of an FDD for separable Banach spaces in terms of the existence of
Lipschitz retracts. For the case of dual spaces this characterization is
indeed true [10]. Linearization of holomorphic functions and composition
ideals of bounded holomorphic mappings were studied via absolute Schauder
decompositions in [11]. In [12] Schauder decompositions were used to the study
of series representations in spaces of vector-valued functions. In [13] the
authors proved that any separable complex infinite dimensional Banach space
with an unconditional Schauder decomposition has chaotic and frequently
hypercyclic operator. Furthermore, the existence of chaotic and frequently
hypercyclic polynomials of arbitrary positive degree in spaces with mentioned
in previous sentence properties was shown in [14]. Note that there exists a
real Banach space with unconditional basis and without chaotic operators [13].
For more about Schauder decompositions and bases we refer to classical
monographs [15], [16], [17], [18], [4].

In [5] the author generalized a theorem of Kato [1] to the case of isomorphic
Schauder decompositions in Banach spaces possessing certain geometric
structure. To this end $\ell_{\Psi}$-Hilbertian and $\infty$-Hilbertian
Schauder decompositions were used instead of initial orthogonal Schauder
decompositions in a Hilbert space from Kato theorem. Moreover, the concept of
an orthogonal Schauder decomposition was generalized to the case of Banach
spaces (new concept was called in [5] by Schauder-Orlicz decomposition) and a
class of Banach spaces with Schauder-Orlicz decompositions was considered.

######  Definition 1.

[5] A Schauder decomposition $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ of $X$
with the a.s.c.p. $\\{P_{n}\\}_{n=0}^{\infty}$ is called a Schauder-Orlicz
decomposition with an Orlicz function $\Phi$ provided there exists an Orlicz
function $\Phi$ such that for every $x\in X$

$\inf\left\\{\rho>0:\sum\limits_{n=0}^{\infty}\Phi\left(\frac{\|P_{n}x\|}{\rho}\right)\leq
1\right\\}=\|x\|.$ (2)

Note that all separable Orlicz spaces $\ell_{\Phi}$, i.e. such that $\Phi$
satisfies the $\Delta_{2}$-condition at zero [17], have a Schauder-Orlicz
decomposition with an Orlicz function $\Phi$. Indeed, it is sufficient to
consider a Schauder decomposition $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$
corresponding to a canonical basis $\\{e_{n}\\}_{n=0}^{\infty}$, i.e.
$\\{\mathfrak{M}_{n}=Lin\\{e_{n}\\}\\}_{n=0}^{\infty}$. Thus,
$\ell_{p}$-spaces have a Schauder-Orlicz decomposition with an Orlicz function
$t^{p}$ for any $p\geq 1.$ Moreover, by the generalized Parseval identity, we
have that a Schauder decomposition in a Hilbert space $H$ is orthogonal if and
only if it is a Schauder-Orlicz decomposition with an Orlicz function
$\Phi(t)=t^{2}$. Therefore the concept of Schauder-Orlicz decomposition is
natural generalization of the concept of an orthogonal Schauder decomposition
to the case of Banach spaces.

In Section $2$ we present the concept of $\ell_{\Phi}$-decomposition,
extending the concept of $\ell_{p}$-decomposition of a Banach space, see
Definitions 2 and 10. The question on the existence of Schauder-Orlicz
decomposition or $\ell_{\Phi}$-decomposition in a Banach space seems to be new
and perspective, since it is linked with an existence of certain geometric
structure in a Banach space and, as it turns out by Theorems 5 and 8, is
related to the study of infinite $\ell_{\Phi}$-sums of Banach spaces. On the
other hand, an existence of such structure in a Banach space $X$ may be useful
for various applications, similar to the role played by orthogonal and Riesz
bases of subspaces in a Hilbert space, e.g. in operator theory, spectral
theory or in the study of evolution equations on $X$ [19], [20].

The main purpose of this work is the study of concepts of Schauder-Orlicz
decomposition and $\ell_{\Phi}$-decomposition as well as relations between
them. The next section is focused on properties of Schauder-Orlicz
decompositions and $\ell_{\Phi}$-decompositions, see Theorem 3, Corollaries 4,
7, Theorems 5, 6, 8, 11, Propositions 16 and 17. In particular, Theorem 8
yields an answer on our question from [5] about the characterization of Banach
spaces possessing a Schauder-Orlicz decomposition. In Theorem 14 we prove that
each Schauder-Orlicz decomposition is monotone and orthogonal in a sense of
Grinblyum-James and Singer, thereby once more confirm that the concept of
Schauder-Orlicz decomposition is natural extension of orthogonality concept to
the case of Banach spaces. However, it turns out that the concept of
Grinblyum-James orthogonal Schauder decomposition is much more general than
the concept of Schauder-Orlicz decomposition, see Proposition 15.

In Section $3$ we give a simple construction of Schauder-Orlicz decomposition
with an Orlicz function $t^{p}$ in space $L_{p}$, $1\leq p<\infty$, – a
slicing Schauder decomposition, see Theorem 18. Thus we give a positive answer
to our question from [5] on the existence of Schauder-Orlicz decomposition in
$L_{p}$. However, each subspace of slicing Schauder decomposition of $L_{p}$
is infinite dimensional and isometric to $L_{p}$. Also the construction of
$\ell_{p}$-decomposition in $L_{p}$, based on the construction of Schauder-
Orlicz decomposition from Theorem 18, is given by Corollary 20. Furthermore,
the absence of other types of $\ell_{\Phi}$-decompositions and Schauder-Orlicz
decompositions in $L_{p}$ is proved in Theorem 21. Similar result on the
absence of other types of $\ell_{\Phi}$-decompositions, except
$c_{0}$-decompositions, in space $C$ is given by Theorem 22.

It is clear that spaces $\ell_{p}$, $p\geq 1$, and Hilbert spaces have a
Schauder-Orlicz decomposition, which is at the same time an FDD. This fact was
the cornerstone of extensions of Kato stability theorem [1] to stability
theorems for unconditional Schauder decompositions in $\ell_{p}$ spaces [21]
and to stability theorems for Riesz basis of subspaces in a Hilbert space, at
least one of which is finite dimensional [22]. Note that to obtain stability
theorems for unconditional Schauder decompositions in $\ell_{p}$ an intrinsic
geometric properties of $\ell_{p}$ and the best constants in the Khintchine
inequality were essentially used [21]. Actually to obtain a version of Kato
theorem for a Banach space $X$ one should know that $X$ has a Schauder-Orlicz
decomposition with at least one finite dimensional subspace, see Theorem 7.2
from [5]. However, it is proved in Section $4$ that spaces $L_{p}$, $p\neq 2$,
and $C$, in contrast to spaces $\ell_{p}$, $p\geq 1$, do not have a Schauder-
Orlicz decomposition with at least one finite dimensional subspace, see
Theorems 23 and 28. Thus, it turns out that analogs of Kato theorem cannot be
obtained in these spaces. Theorem 26 shows that normality of a Schauder
decomposition in a Hilbert space means that it is a Schauder-Orlicz
decomposition with an Orlicz function $t^{2}$. However, as a corollary of
Theorem 23 we obtain that there exists a normal Schauder decomposition in
$L_{p}$, $p\neq 2$, which is not a Schauder-Orlicz decomposition (see
Corollary 25). Also in Theorem 27 we deduce from the well-known optimality of
the Haar basis [23], [15] the best estimate for constants of one dimensional
unconditional Schauder decompositions in $L_{p}$, $1<p<\infty.$ The main
result of Section $4$ is Theorem 30, first part of which demonstrates that the
validity of pseudo-Daugavet property in a Banach space $X$ immediately implies
the lack of Schauder-Orlicz decomposition in $X$ with at least one finite
dimensional subspace. It means that Kato theorem on similarity for sequences
of projections [1] cannot be extended to spaces with pseudo-Daugavet property.
Conversely, the second part of Theorem 30 reads that existence in $X$ of the
Schauder-Orlicz decomposition, at least one subspace of which is finite
dimensional, ensures that $X$ does not have pseudo-Daugavet property. For
Banach spaces $X$ possessing Schauder-Orlicz decompositions we obtain the
criterion of $X$ to have pseudo-Daugavet property on the language of
dimensions of elements of Schauder-Orlicz decompositions, which exist in $X$,
see Theorem 33.

Finally, Section $5$ contains concluding remarks and indicates some further
perspectives, including the Conjecture 34 on the uniqueness, up to an
isometry, of Schauder-Orlicz decomposition with preassigned dimensions in
$\ell_{1}$ and $c_{0}$.

##  2 Schauder-Orlicz decompositions and $\ell_{\Phi}$-decompositions

###  2.1 Basic properties of Schauder-Orlicz decompositions and
$\ell_{\Phi}$-decompositions

It is known that every $c_{0}$-decomposition is shrinking, every
$\ell_{1}$-decomposition is boundedly complete and every
$\ell_{p}$-decomposition, where $1<p<\infty$, is both shrinking and boundedly
complete [4]. Also in [24] Sanders proved that a Banach space $X$ with
Schauder decomposition $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is reflexive if
and only if $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is both shrinking and
boundedly complete and each $\mathfrak{M}_{n}$ is reflexive. Finite
dimensional $\ell_{p}$-decompositions, where $1\leq p<\infty$, as well as
$c_{0}$-decompositions were studied by Johnson $\&$ Zippin [25]. Three years
later Chadwick [26] introduced the concept of absolute Schauder decomposition
(coincides with $\ell_{1}$-decomposition), generalizing the concept of
absolute Schauder basis. Also in [26] it was proved that space possessing
$\ell_{1}$-decomposition has a complemented subspace with a Schauder basis.

Throughout the paper we will use the following definition, which substantially
extends the concept of $\ell_{p}$-decomposition and includes the concept of
$c_{0}$-decomposition.

######  Definition 2.

Let $\Phi$ be an Orlicz function. A Schauder decomposition, which is both
$\ell_{\Phi}$-Hilbertian and $\ell_{\Phi}$-Besselian, will be called an
$\ell_{\Phi}$-decomposition.

Obviously every Schauder-Orlicz decomposition is an
$\ell_{\Phi}$-decomposition, but the converse is false. Moreover, every Riesz
basis of subspaces in a Hilbert space is an $\ell_{2}$-decomposition and vice
versa [5].

For $\ell_{\Phi}$-decompositions we can prove the following result.

######  Theorem 3.

Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be a Schauder decomposition of
$X$.

(i) If $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is $\ell_{\Phi}$-decomposition
with non-degenerate Orlicz function $\Phi$, then
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is boundedly complete.

(ii) If $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is $\ell_{\Phi}$-decomposition
with degenerate Orlicz function $\Phi$, then
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is shrinking.

###### Proof.

Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be $\ell_{\Phi}$-decomposition
with non-degenerate Orlicz function $\Phi$. Consider the a.s.c.p.
$\\{P_{n}\\}_{n=0}^{\infty}$ of $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$. Then
[5], [4] there exist constants $c_{2}\geq c_{1}>0$ such that for each
$x=\sum\limits_{n=0}^{\infty}P_{n}x\in X$ we have

$c_{1}\|\\{\|P_{n}x\|\\}_{n=0}^{\infty}\|_{\ell_{\Phi}}\leq\|x\|\leq
c_{2}\|\\{\|P_{n}x\|\\}_{n=0}^{\infty}\|_{\ell_{\Phi}}.$ (3)

Now take any sequence $\\{x_{i}\\}_{i=0}^{\infty}\in X$,
$x_{i}\in\mathfrak{M}_{i}$ for each $i$, such that

$\left\\{\left\|\sum\limits_{i=0}^{n}x_{i}\right\|\right\\}_{n=0}^{\infty}$

is bounded, i.e. there exists a constant $c>0$ such that for all $n$ we have

$\left\|\sum\limits_{i=0}^{n}x_{i}\right\|\leq c.$

Applying of left-hand the side of (3) yields

$c_{1}\|\>\\{\|x_{i}\|\\}_{i=0}^{n}\>\|_{\ell_{\Phi}}=c_{1}\|\>\\{\|P_{i}x\|\\}_{i=0}^{n}\>\|_{\ell_{\Phi}}\leq\left\|\sum\limits_{i=0}^{n}x_{i}\right\|\leq
c,$

hence the sequence of norms
$\\{\|\>\\{\|x_{i}\|\\}_{i=0}^{n}\>\|_{\ell_{\Phi}}\\}_{n=0}^{\infty}$ is
bounded and increasing. Consequently the sequence
$\\{\|\>\\{\|x_{i}\|\\}_{i=0}^{n}\>\|_{\ell_{\Phi}}\\}_{n=0}^{\infty}$ is
convergent. Combining this with the right-hand the side of (3) yields that for
any $m>n$

$\|x_{m+n}-x_{n}\|=\left\|\sum\limits_{i=n+1}^{m+n}x_{i}\right\|\leq
c_{2}\|\>\\{\|P_{i}x\|\\}_{i=n+1}^{m+n}\>\|_{\ell_{\Phi}}=c_{2}\|\>\\{\|x_{i}\|\\}_{i=n+1}^{m+n}\>\|_{\ell_{\Phi}}\to
0$

as $n,m\to\infty.$ Thus $\\{x_{i}\\}_{i=0}^{\infty}$ is a Cauchy sequence in
$X$, hence $\\{x_{i}\\}_{i=0}^{\infty}$ is convergent and (i) is proved.

To prove (ii) we note that if the Orlicz function $\Phi$ is degenerate, then
$\ell_{\Phi}$ is isomorphic to $\ell_{\infty}$ [17]. It follows that if
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is an $\ell_{\Phi}$-decomposition with
degenerate Orlicz function $\Phi$, then
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is $c_{0}$-decomposition. Hence
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is shrinking [4]. ∎

For Schauder-Orlicz decompositions we have the following corollary of Theorem
2.

######  Corollary 4.

Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be a Schauder-Orlicz decomposition
with non-degenerate Orlicz function $\Phi$. Then
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is boundedly complete.

By result of Sanders [24] we have that every Schauder-Orlicz decomposition in
reflexive space $X$ is shrinking. The construction of non-shrinking Schauder-
Orlicz decomposition of $L_{1}(0,1)$ is given by Theorem 18.

Further we have the following natural generalization of Corollary 15.6 of [4].

######  Theorem 5.

A Banach space $X$ has an $\ell_{\Phi}$-decomposition if and only if there
exists a sequence of Banach spaces $\\{X_{n}\\}_{n=0}^{\infty}$ such that

$X\cong\left(X_{0},X_{1},X_{2},\ldots\right)_{\ell_{\Phi}},$

where $\left(X_{0},X_{1},X_{2},\ldots\right)_{\ell_{\Phi}}$ stands for a
Banach space of all sequences $\mathcal{X}=\\{x_{n}\\}_{n=0}^{\infty}$, such
that $x_{n}\in X_{n}$ and

$|||\mathcal{X}|||=\inf\left\\{\rho>0:\sum\limits_{n=0}^{\infty}\Phi\left(\frac{\|x_{n}\|}{\rho}\right)\leq
1\right\\}<\infty.$

###### Proof.

To prove this result one can use, e.g., arguments from the proof of Corollary
15.6 of [4] together with two-sided estimate (3). ∎

Note that for the case of $\ell_{1}$-decompositions, i.e. when $\Phi(t)=t$,
Theorem 5 was first proved by Chadwick [26].

It turns out that $\ell_{\Phi}$-decomposition is invariant under isomorphism
and Schauder-Orlicz decomposition is invariant under isometry.

######  Theorem 6.

Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be a Schauder decomposition in a
Banach space $X$.

  1. 1.

Suppose that $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a Schauder-Orlicz
decomposition with an Orlicz function $\Phi$ and the a.s.c.p.
$\\{P_{n}\\}_{n=0}^{\infty}$ in $X$ and $X$ is isometric to a Banach space $Y$
with the operator of isometry $T:X\mapsto Y$. Then
$\\{T\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is also a Schauder-Orlicz
decomposition with an Orlicz function $\Phi$ in $Y$.

  2. 2.

Suppose that $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is an
$\ell_{\Phi}$-decomposition with an Orlicz function $\Phi$ and the a.s.c.p.
$\\{P_{n}\\}_{n=0}^{\infty}$ in $X$ and $X$ is isomorphic to a Banach space
$Y$ with the isomorphism $S:X\mapsto Y$. Then
$\\{S\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is also an
$\ell_{\Phi}$-decomposition with an Orlicz function $\Phi$ in $Y$.

###### Proof.

1\. For all $x\in X$ we have that $Tx=y\in Y$, $x=T^{-1}y$ and, moreover,

$\|Tx\|_{Y}=\|x\|_{X},\qquad\|P_{n}x\|_{X}=\|TP_{n}x\|_{Y},\>n\geq 0.$

Since $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a Schauder decomposition of
$X$ with the a.s.c.p. $\\{P_{n}\\}_{n=0}^{\infty}$ we have that every $x\in X$
has a unique expansion

$\sum\limits_{n=0}^{\infty}P_{n}x=x.$

Hence

$\sum\limits_{n=0}^{\infty}TP_{n}x=Tx.$

Thus every $y=Tx\in Y$ has a unique expansion

$\sum\limits_{n=0}^{\infty}TP_{n}T^{-1}y=TT^{-1}y=y,$

which means that $\\{TP_{n}T^{-1}Y\\}_{n=0}^{\infty}$ is a Schauder
decomposition of $Y$.

To show that $\\{TP_{n}T^{-1}Y\\}_{n=0}^{\infty}$ is a Schauder-Orlicz
decomposition with an Orlicz function $\Phi$ in a Banach space $Y$ we note
that for any $y\in Y$ there exists $x=T^{-1}y\in X$ such that

$\displaystyle\|y\|_{Y}=\|Tx\|_{Y}=\|x\|_{X}=$
$\displaystyle\inf\left\\{\rho>0:\sum\limits_{n=0}^{\infty}\Phi\left(\frac{\|P_{n}x\|_{X}}{\rho}\right)\leq
1\right\\}$ $\displaystyle=$
$\displaystyle\inf\left\\{\rho>0:\sum\limits_{n=0}^{\infty}\Phi\left(\frac{\|TP_{n}x\|_{Y}}{\rho}\right)\leq
1\right\\}$ $\displaystyle=$
$\displaystyle\inf\left\\{\rho>0:\sum\limits_{n=0}^{\infty}\Phi\left(\frac{\|TP_{n}T^{-1}y\|_{Y}}{\rho}\right)\leq
1\right\\}.$

Hence [5] $\\{T\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is also a Schauder-Orlicz
decomposition with an Orlicz function $\Phi$ and the a.s.c.p.
$\\{TP_{n}T^{-1}\\}_{n=0}^{\infty}$ in $Y$.

The proof of the second statement goes similar to the proof of the first. ∎

######  Corollary 7.

Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be a Schauder-Orlicz decomposition
with an Orlicz function $\Phi$ in a Banach space $X$ and $X$ is isomorphic to
a Banach space $Y$ with the isomorphism $S:X\mapsto Y$. Then
$\\{S\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is an $\ell_{\Phi}$-decomposition in
$Y$.

###### Proof.

The arguments of the proof are the same as in the proof of Theorem 6. ∎

One can prove a partial case of Theorem 5.

######  Theorem 8.

A Banach space $X$ has a Schauder-Orlicz decomposition with an Orlicz function
$\Phi$ if and only if there exists a sequence of Banach spaces
$\\{X_{n}\\}_{n=0}^{\infty}$ such that space $X$ is isometric to

$\left(X_{0},X_{1},X_{2},\ldots\right)_{\ell_{\Phi}}.$

In what follows we are using the following definitions.

######  Definition 9.

[5] A Schauder decomposition $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ in a
Banach space $X$ is called unconditional with constant $M$ provided that
$\exists M\geq 1$ such that

$\left\|\sum\limits_{i=0}^{n}\delta_{i}y_{i}\right\|\leq
M\left\|\sum\limits_{i=0}^{n}y_{i}\right\|$

for any $n\geq 0$, $y_{i}\in\mathfrak{M}_{i}$ and selectors
$\delta_{i}\in\\{0,1\\}.$

######  Definition 10.

We will call an $\ell_{\Phi}$-decomposition
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ of a Banach space $X$ with the
a.s.c.p. $\\{P_{n}\\}_{n=0}^{\infty}$ by $\ell_{\Phi}$-decomposition with
constants $c_{1},$ $c_{2}$ provided there exist constants $c_{2}\geq c_{1}>0$
such that for each $x=\sum\limits_{n=0}^{\infty}P_{n}x\in X$ we have (3), i.e.

$c_{1}\|\\{\|P_{n}x\|\\}_{n=0}^{\infty}\|_{\ell_{\Phi}}\leq\|x\|\leq
c_{2}\|\\{\|P_{n}x\|\\}_{n=0}^{\infty}\|_{\ell_{\Phi}}.$

Clearly any Schauder-Orlicz decomposition with an Orlicz function $\Phi$ is an
$\ell_{\Phi}$-decomposition with constants $1$, $1$.

######  Theorem 11.

Every $\ell_{\Phi}$-decomposition $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ with
constants $c_{1},$ $c_{2}$ in $X$ is unconditional with constant
$\frac{c_{2}}{c_{1}}$. Every Schauder-Orlicz decomposition is unconditional
with constant $1.$

###### Proof.

Fix any set of selectors $\delta_{i}\in\\{0,1\\}$. Since
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ with the a.s.c.p.
$\\{P_{n}\\}_{n=0}^{\infty}$ is an $\ell_{\Phi}$-decomposition in $X$, for
each $x=\sum\limits_{n=0}^{\infty}x_{n}=\sum\limits_{n=0}^{\infty}P_{n}x\in X$
we have applying (3) that

$\displaystyle\left\|\sum\limits_{i=0}^{\infty}\delta_{i}x_{i}\right\|$
$\displaystyle=\left\|\sum\limits_{i=0}^{\infty}\delta_{i}P_{i}x\right\|\leq
c_{2}\left\|\left\\{\left\|P_{j}\sum\limits_{i=0}^{\infty}\delta_{i}P_{i}x\right\|\right\\}_{j=0}^{\infty}\right\|_{\ell_{\Phi}}=c_{2}\left\|\left\\{\left\|\delta_{j}P_{j}x\right\|\right\\}_{j=0}^{\infty}\right\|_{\ell_{\Phi}}$
$\displaystyle=c_{2}\left\|\left\\{\delta_{j}\left\|P_{j}x\right\|\right\\}_{j=0}^{\infty}\right\|_{\ell_{\Phi}}\leq
c_{2}\left\|\left\\{\left\|P_{j}x\right\|\right\\}_{j=0}^{\infty}\right\|_{\ell_{\Phi}}\leq\frac{c_{2}}{c_{1}}\|x\|.$

It follows that $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is unconditional with
constant $\frac{c_{2}}{c_{1}}$.

To prove that every Schauder-Orlicz decomposition in a Banach space is
unconditional with constant $1$ it is enough to note that Schauder-Orlicz
decomposition is an $\ell_{\Phi}$-decomposition such that $c_{1}=c_{2}=1$ in
(3). ∎

Note that spaces $C[0,1]$ and $L_{1}(0,1)$ have no unconditional basis, but
they still have unconditional Schauder decompositions [4]. An example of
unconditional Schauder decomposition (even Schauder-Orlicz decomposition) in
space $L_{1}(0,1)$ may be found in Theorem 18. Moreover, $L_{1}(0,1)$ has no
absolutely convergent basis [18] although it has an absolute Schauder
decomposition (coincides with notion of $\ell_{1}$-decomposition), see Theorem
18. The reader can find more about various properties of unconditional
Schauder decompositions in Banach spaces in, e.g., [4], [17], [5], [22], [21].

###  2.2 Inclinations of subspaces, Grinblyum-James orthogonality and
Schauder-Orlicz decompositions

Let $\mathfrak{M}$ and $\mathfrak{N}$ be two subspaces of a Banach space $X$.
Recall that an inclination of $\mathfrak{M}$ to $\mathfrak{N}$ is the number
[27]

$\left(\widehat{\mathfrak{M},\mathfrak{N}}\right)=\inf\limits_{x\in\mathfrak{M},y\in\mathfrak{N},\|x\|=1}\|x-y\|.$

We mention here some facts about inclinations of subspaces and their
importance in Schauder decompositions theory. It is easy to see that

$\left(\widehat{\mathfrak{M},\mathfrak{N}}\right)=\frac{1}{\|P\|},$

where $P$ is a projection from $\mathfrak{M}\oplus\mathfrak{N}$ onto
$\mathfrak{M}$, parallel to $\mathfrak{N}$ [28]. Since $\|P\|\geq 1$ it
follows that for any $\mathfrak{M}$, $\mathfrak{N}$ we have

$0\leq\left(\widehat{\mathfrak{M},\mathfrak{N}}\right)\leq 1.$

Note that the notion of inclination is not symmetric in general [29]: there
exists $\mathfrak{M}$, $\mathfrak{N}$ such that
$\left(\widehat{\mathfrak{M},\mathfrak{N}}\right)\neq\left(\widehat{\mathfrak{N},\mathfrak{M}}\right)$.
Moreover, the following is true [29]:

  1. 1.

If $\left(\widehat{\mathfrak{M},\mathfrak{N}}\right)=\delta$, then
$\left(\widehat{\mathfrak{N},\mathfrak{M}}\right)\geq\frac{\delta}{1+\delta}$.
The latter inequality is exact for any $\delta\in[0,1].$

  2. 2.

$\left(\widehat{\mathfrak{M},\mathfrak{N}}\right)=\left(\widehat{\mathfrak{N},\mathfrak{M}}\right)$
for any two subspaces $\mathfrak{M}$, $\mathfrak{N}$ in a Banach space $X$
with $\dim X>2$ if and only if $X$ is a space with a scalar product.

The direct sum $\mathfrak{M}\oplus\mathfrak{N}$ of two subspaces in a Banach
space is closed if and only if [30], [29]

$\left(\widehat{\mathfrak{M},\mathfrak{N}}\right)>0.$

The following geometric criterion of Schauder decomposition was proved by
Grinblyum in [31], [30].

######  Theorem 12.

[31, 30, 27] Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be a complete
sequence of subspaces in a Banach space $X$. Then
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a Schauder decomposition of $X$ if
and only if there exists a constant $\delta>0$ such that for all $n\geq 0$ and
$m\geq 1$ we have

$\left(\widehat{\mathfrak{G}_{n},\mathfrak{L}_{n,m}}\right)\geq\delta.$ (4)

Here

$\mathfrak{G}_{n}=\overline{Lin}\left\\{\mathfrak{M}_{0},\mathfrak{M}_{1},\ldots,\mathfrak{M}_{n}\right\\},$
$\mathfrak{L}_{n,m}=\overline{Lin}\left\\{\mathfrak{M}_{n+1},\mathfrak{M}_{n+2},\ldots,\mathfrak{M}_{n+m}\right\\}.$

The supremum of all $\delta$, satisfying (4), was called in [31] the index of
the sequence $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$. The index
$\gamma\left(\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}\right)$ is uniquely defined
the sequence $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ and satisfy [31]

$0\leq\gamma\left(\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}\right)\leq 1.$

Moreover, if

$\gamma\left(\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}\right)=1,$ (5)

then the sequence $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is called orthogonal
[27].

If $\left(\widehat{\mathfrak{M},\mathfrak{N}}\right)=1$ then it is said that
$\mathfrak{M}$ is orthogonal to $\mathfrak{N}$ [27]. In [27] Gurarii proved
the following interesting geometric result: There exists Banach spaces of
arbitrary finite dimension $n>2$, for each two subspaces $\mathfrak{M}$,
$\mathfrak{N}$ of which, where $\dim\mathfrak{M}>1$, we have

$\left(\widehat{\mathfrak{M},\mathfrak{N}}\right)<1.$

It follows that there exists Banach spaces of arbitrary finite dimension $n>2$
without an orthogonal Schauder basis [27], i.e. a basis, linear spans of
elements of which satisfy (5), see also [18], p. 233. This notion of
orthogonality of bases in Banach spaces is due to Grinblyum [31] and James
[32].

######  Remark 1.

Note that for any Schauder basis in Banach space $X$ one can define a new
norm, which depends on this basis and equivalent to the original norm of $X$,
with respect to which this basis become orthogonal [33].

######  Proposition 13.

There exists an orthogonal Schauder decomposition in $C[a,b]$ which is not
Schauder-Orlicz decomposition.

###### Proof.

There exists an orthogonal basis in space $C[a,b]$ [34], e.g. a classical
basis of Schauder [35], see also [18]. Since unconditional bases do not exist
in $C[a,b]$, this basis is conditional. Denote by
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ its one dimensional linear spans. Then
obviously $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is, on the one hand,
orthogonal. On the other hand, by virtue of Theorem 11 we have that
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is not a Schauder-Orlicz
decomposition. ∎

For any Schauder-Orlicz decomposition in a Banach space $X$ we have the
following result.

######  Theorem 14.

Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be a Schauder-Orlicz decomposition
with an Orlicz function $\Phi$ in $X$. Then the following is true.

  1. 1.

$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is monotone.

  2. 2.

For any $n,m$ and all $x_{i_{j}}\in\mathfrak{M}_{i_{j}}$ $(j=0,\ldots,n)$,
$x_{l_{j}}\in\mathfrak{M}_{l_{j}}$ $(j=0,\ldots,m)$, satisfying
$\\{i_{1},\ldots,i_{n}\\}\cap\\{l_{1},\ldots,l_{m}\\}=\emptyset$, we have

$\left\|\sum\limits_{j=0}^{n}x_{i_{j}}\right\|\leq\left\|\sum\limits_{j=0}^{n}x_{i_{j}}+\sum\limits_{j=0}^{m}x_{l_{j}}\right\|$

(i.e. $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is Singer-orthogonal, see remark
3 of the last section).

  3. 3.

For any $n,m$ such that $n\neq m$ we have
$(\widehat{\mathfrak{M}_{n},\mathfrak{M}_{m}})=1.$

  4. 4.

For any $n\geq 0$ and $m\geq 1$ we have

$\left(\widehat{\mathfrak{G}_{n},\mathfrak{L}_{n,m}}\right)=1,$

where subspaces $\mathfrak{G}_{n}$ and $\mathfrak{L}_{n,m}$ are defined in
Theorem 12.

Thus $\gamma\left(\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}\right)=1$, i.e.
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is orthogonal.

###### Proof.

Consider any Schauder-Orlicz decomposition
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ with an Orlicz function $\Phi$ and the
a.s.c.p. $\\{P_{n}\\}_{n=0}^{\infty}$ in a Banach space $X$. Then, by
definition of Schauder-Orlicz decomposition for all $n,m$ we have

$\displaystyle\left\|\sum\limits_{j=0}^{n}y_{j}\right\|$
$\displaystyle=\inf\left\\{\rho>0:\sum\limits_{j=0}^{n}\Phi\left(\frac{\|y_{j}\|}{\rho}\right)\leq
1\right\\}$
$\displaystyle\leq\inf\left\\{\rho>0:\sum\limits_{j=0}^{n+m}\Phi\left(\frac{\|y_{j}\|}{\rho}\right)\leq
1\right\\}=\left\|\sum\limits_{j=0}^{n+m}y_{j}\right\|$

for any $y_{j}\in\mathfrak{M}_{j},$ $j=1,\ldots,n+m$, which means that
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is monotone.

The proof of the second statement is straightforward.

Denote by $\\{P_{n}\\}_{n=0}^{\infty}$ the a.s.c.p. of
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$. To prove the third statement we note
that for any $j,m$ such that $j\neq m$

$\displaystyle(\widehat{\mathfrak{M}_{j},\mathfrak{M}_{m}})$
$\displaystyle=\inf\limits_{x\in\mathfrak{M}_{j},y\in\mathfrak{M}_{m},\|x\|=1}\|x-y\|$
$\displaystyle=\inf\limits_{x\in\mathfrak{M}_{j},y\in\mathfrak{M}_{m},\|x\|=1}\|\\{\|P_{n}(x-y)\|\\}_{n=0}^{\infty}\|_{\ell_{\Phi}}$
$\displaystyle=\inf\limits_{x\in\mathfrak{M}_{j},y\in\mathfrak{M}_{m},\|x\|=1}\|\\{\|P_{n}(P_{j}x-P_{m}y)\|\\}_{n=0}^{\infty}\|_{\ell_{\Phi}}$
$\displaystyle=\inf\limits_{x\in\mathfrak{M}_{j},y\in\mathfrak{M}_{m},\|x\|=1}\|\left(0,\ldots,0,\smash[b]{\\!\underbrace{\|x\|\,}_{\text{$j^{th}$
place}}},0,\ldots,0,\smash[b]{\\!\underbrace{\|y\|\,}_{\text{$m^{th}$
place}}},0,\ldots\right)\|_{\ell_{\Phi}}$ $\displaystyle=1.$

Furthermore, since $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a Schauder-
Orlicz decomposition in $X$ with the a.s.c.p. $\\{P_{n}\\}_{n=0}^{\infty}$ we
have for any $j\geq 0$ and $m\geq 1$ that

$\displaystyle\left(\widehat{\mathfrak{G}_{j},\mathfrak{L}_{j,m}}\right)=\inf\limits_{x\in\mathfrak{G}_{j},y\in\mathfrak{L}_{j,m},\|x\|=1}\|x-y\|$
$\displaystyle\qquad\>=\inf\limits_{x\in\mathfrak{G}_{j},y\in\mathfrak{L}_{j,m},\|x\|=1}\|\\{\|P_{n}(x-y)\|\\}_{n=0}^{\infty}\|_{\ell_{\Phi}}$
$\displaystyle=\inf\limits_{x\in\mathfrak{M}_{j},y\in\mathfrak{M}_{m},\|x\|=1}\|\left(\|P_{0}x\|,\ldots,\|P_{j}x\|,\|P_{j+1}y\|,\ldots,\|P_{j+m}y\|,0,0,\ldots\right)\|_{\ell_{\Phi}}$
$\displaystyle\qquad\>=1,$

where subspaces $\mathfrak{G}_{n}$ and $\mathfrak{L}_{n,m}$ are defined in
Theorem 12. Hence $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is orthogonal. ∎

Combining Theorem 14 with Proposition 13 we arrive at the following
observation.

######  Proposition 15.

The set of all Schauder-Orlicz decompositions is strictly contained in the set
of all Grinblyum-James orthogonal Shauder decompositions.

###  2.3 To the question of uniqueness of $\ell_{\Phi}$-decompositions and
Schauder-Orlicz decompositions

For any $\ell_{\Phi}$-decomposition in a Banach space $X$ we have, using ideas
of the proof of Theorem 14, the following result.

######  Proposition 16.

Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be an $\ell_{\Phi}$-decomposition
with constants $c_{1},$ $c_{2}$ in $X$. Then the following is true.

  1. 1.

For any $n,m$ and all $x_{i_{j}}\in\mathfrak{M}_{i_{j}}$ $(j=0,\ldots,n)$,
$x_{l_{j}}\in\mathfrak{M}_{l_{j}}$ $(j=0,\ldots,m)$, satisfying
$\\{i_{1},\ldots,i_{n}\\}\cap\\{l_{1},\ldots,l_{m}\\}=\emptyset$, we have

$\left\|\sum\limits_{j=0}^{n}x_{i_{j}}\right\|\leq\frac{c_{2}}{c_{1}}\left\|\sum\limits_{j=0}^{n}x_{i_{j}}+\sum\limits_{j=0}^{m}x_{l_{j}}\right\|.$

  2. 2.

For all $n,m$ and for any $y_{j}\in\mathfrak{M}_{j},$ $j=1,\ldots,n+m$, we
have that

$\left\|\sum\limits_{j=0}^{n}y_{j}\right\|\leq\frac{c_{2}}{c_{1}}\left\|\sum\limits_{j=0}^{n+m}y_{j}\right\|.$

Next we discuss the uniqueness of Schauder-Orlicz decompositions and
$\ell_{\Phi}$-decompositions in Hilbert spaces.

######  Proposition 17.

(i) Every Schauder-Orlicz decomposition with preassigned dimensions in a
Hilbert space is unique, up to an isometry.

(ii) Every $\ell_{2}$-decomposition with preassigned dimensions in a Hilbert
space is unique, up to an isomorphism.

###### Proof.

Every Schauder decomposition in a Hilbert space is unconditional if and only
if it is a Riesz basis of subspaces if and only if it is an
$\ell_{2}$-decomposition [5]. Since unconditional Schauder decomposition with
preassigned dimensions in a Hilbert space is unique, up to an isomorphism [5],
[22], we have (ii).

To prove (i) it remains to note that Schauder decomposition in a Hilbert space
is orthogonal if and only if it is a Schauder-Orlicz decomposition with Orlicz
function $t^{2}$ [5]. ∎

######  Remark 2.

Note that by virtue of Theorem 11 in a Hilbert space from the set of all
$\ell_{\Phi}$-decompositions only $\ell_{2}$-decompositions exist.
Furthermore, by virtue of Theorem 11 in a Hilbert space only Schauder-Orlicz
decompositions with an Orlicz function $t^{2}$ exist.

######  Remark 3.

In [36] Lindenstrauss and Pelczynski found that all unconditional Schauder
bases in $\ell_{1}$ (resp. $c_{0}$) are equivalent to the canonical basis of
$\ell_{1}$ (resp. $c_{0}$). Moreover, one year later Lindenstrauss and Zippin
established that the only Banach spaces, which have a unique, up to
equivalence, normalized unconditional basis are $\ell_{2}$, $\ell_{1}$ and
$c_{0}$ [37].

Also in [36] the following remarkable result was proved (see Corollary $8$ in
[36]): If $X$ is an $\mathcal{L}_{1}$-space (respectively
$\mathcal{L}_{\infty}$-space) and if $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$
is an unconditional Schauder decomposition of $X$, then

$X\cong\left(\mathfrak{M}_{0},\mathfrak{M}_{1},\mathfrak{M}_{2},\ldots\right)_{\ell_{1}},$

or

$X\cong\left(\mathfrak{M}_{0},\mathfrak{M}_{1},\mathfrak{M}_{2},\ldots\right)_{c_{0}},$

respectively. Particularly it means that all unconditional Schauder
decompositions in $\ell_{1}$ or $L_{1}(0,1)$ are $\ell_{1}$-decompositions and
all unconditional Schauder decompositions in $c_{0}$ or $C[0,1]$ are
$c_{0}$-decompositions.

##  3 Existence of Schauder-Orlicz decompositions and
$\ell_{p}$-decompositions in spaces $L_{p}$, $1\leq p<\infty$

Since all separable $L_{p}$, $1\leq p<\infty,$ with purely non-atomic measures
are isometric to $L_{p}(0,1)$ with standard Lebesgue measure [38] and since by
Proposition 6 Schauder-Orlicz decompositions are invariant under isometry, we
focus ourselves on spaces $L_{p}(0,1)$, $1\leq p<\infty,$ with standard
Lebesgue measure $\mu$. Construction of Schauder-Orlicz decompositions in
other types of spaces $L_{p}$ may be done similarly to the following.

######  Theorem 18.

Consider the family of sets $\left\\{A_{n}\right\\}_{n=0}^{\infty}$ with the
following properties:

  1. 1.

$A_{i}\cap A_{j}=\emptyset$ if $i\neq j$;

  2. 2.

$\bigcup\limits_{n=0}^{\infty}A_{n}=(0,1)$;

  3. 3.

$\mu(A_{n})>0$ for all $n$ and $\mu(A_{n})\to 0$ as $n\to\infty$.

Further fix any $p\geq 1$ and consider the corresponding projections $P_{n}$
in $L_{p}(0,1)$. For every $f\in L_{p}(0,1)$ we define

$P_{n}f(t)=\left\\{\begin{array}[]{l}f(t),\quad t\in A_{n},\\\
0,\quad\text{otherwise}.\\\ \end{array}\right.$

Then $\left\\{P_{n}L_{p}(0,1)\right\\}_{n=0}^{\infty}$ is a Schauder-Orlicz
decomposition with an Orlicz function $t^{p}$ in $L_{p}(0,1)$.

###### Proof.

Indeed, it is easy to check that each linear subspace $P_{n}L_{p}(0,1)$,
$n\geq 0,$ is closed. Further, the fact that for every $f\in L_{p}(0,1)$

$\sum\limits_{n=0}^{\infty}P_{n}f=f$

is obvious by construction. Finally, by the additivity of Lebesgue integral we
have that for all $f\in L_{p}(0,1)$

$\sum\limits_{n=0}^{\infty}\left\|P_{n}f\right\|^{p}=\sum\limits_{n=0}^{\infty}\int\limits_{A_{n}}|f(t)|^{p}dt=\int\limits_{0}^{1}|f(t)|^{p}dt=\|f\|^{p}.$

This means that $\left\\{P_{n}L_{p}(0,1)\right\\}_{n=0}^{\infty}$ is a
Schauder-Orlicz decomposition with an Orlicz function $t^{p}$ in $L_{p}(0,1)$.
∎

Taking into account their nature it is convenient to call Schauder
decompositions from Theorem 18 slicing Schauder decompositions.

######  Remark 4.

Each subspace $P_{n}L_{p}(0,1)$, $n\geq 0,$ is isometric to $L_{p}(0,1)$ and,
hence, infinite dimensional. So we give a simple construction of Schauder-
Orlicz decomposition in $L_{p}$, each subspace of which is infinite
dimensional.

Applying Theorem 8 we have the following simple corollary of Theorem 18.

######  Corollary 19.

The space $L_{p}$, $1\leq p<\infty,$ is isometric to

$\left(\mathfrak{M}_{0},\mathfrak{M}_{1},\mathfrak{M}_{2},\ldots\right)_{\ell_{p}},$

where $\left\\{\mathfrak{M}_{n}\right\\}_{n=0}^{\infty}$ is slicing Schauder-
Orlicz decomposition with an Orlicz function $t^{p}$ in $L_{p}$.

######  Remark 5.

While writing this paper the author found out that the simplest slicing
Schauder decomposition appeared first in a paper [6] (Example 3.7) but in a
different context. It was constructed as a boundedly complete Schauder
decomposition in $L_{1}(0,1)$.

Theorem 18 yields also the constructions of $\ell_{p}$-decompositions in
$L_{p}(0,1)$, $1\leq p<\infty$.

######  Corollary 20.

Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be a Schauder-Orlicz decomposition
from Theorem 18 in $L_{p}(0,1)$. Assume that $S$ is an isomorphism in
$L_{p}(0,1)$. Then $\\{S\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is an
$\ell_{p}$-decomposition in $L_{p}(0,1)$.

###### Proof.

The proof follows from Corollary 7. ∎

We can obtain the following result on the existence of
$\ell_{\Phi}$-decompositions and Schauder-Orlicz decompositions in spaces
$L_{p}$, $1\leq p<\infty$, similar to the situation in Hilbert spaces, see
Remark 2.

######  Theorem 21.

From the set of all $\ell_{\Phi}$-decompositions in $L_{p}$ only
$\ell_{p}$-decompositions exist. Also in $L_{p}$ only Schauder-Orlicz
decompositions with an Orlicz function $t^{p}$ exist.

###### Proof.

The proof of this theorem is based on the combination of Theorem 11 with
Proposition 4.8 from [5] on two-sided estimates for unconditional Schauder
decompositions in $L_{p}$ spaces, see also Corollary 2.4 in [21]. ∎

Combining Theorem 11 with Corollary 8 of [36] we arrive at the following
conclusion.

######  Theorem 22.

From the set of all $\ell_{\Phi}$-decompositions in $C$ only
$c_{0}$-decompositions exist.

However, the construction of Schauder-Orlicz decomposition in the space $C$ of
continuous functions looks as a non-trivial problem, since the slicing
decomposition in $C[0,1]$ is obviously not a Schauder decomposition in
$C[0,1]$. Thus, even the existence of a Schauder-Orlicz decomposition in
$C[0,1]$ is unclear, although $C[0,1]$ has a $c_{0}$-decomposition [4].

##  4 Nonexistence of Schauder-Orlicz decompositions with at least one finite
dimensional subspace

###  4.1 Nonexistence of Schauder-Orlicz decompositions with at least one
finite dimensional subspace in spaces $L_{p}$, $1\leq p\neq 2$ and $C$

Since $L_{p}\ncong\ell_{p}$ for $p\neq 2$, it is clear that there is no basis
in $L_{p}$, linear spans of elements of which form a Schauder-Orlicz
decomposition. Further we prove much more strong statement that there is no
Schauder-Orlicz decomposition in $L_{p}$, $p\neq 2$, with at least one finite
dimensional subspace.

######  Theorem 23.

There is no Schauder-Orlicz decomposition in $L_{p}$, $1\leq p\neq 2$, with at
least one finite dimensional subspace.

###### Proof.

By Theorem 21 only Schauder-Orlicz decomposition with an Orlicz function
$t^{p}$ can exist in $L_{p}$.

First we consider the case when $p\in(1,2)\cup(2,\infty)$ and assume that
there exists the Schauder-Orlicz decomposition
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ with an Orlicz function $t^{p}$ in
$L_{p}$, such that

$\dim\mathfrak{M}_{0}<\infty.$ (6)

Denote by $\\{P_{n}\\}_{n=0}^{\infty}$ the a.s.c.p. of
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$. Then we note that (6) means that
$P_{0}$ is compact. Clearly $\|P_{0}\|\geq 1.$ Let

$\varepsilon=\|P_{0}\|\geq 1.$

Then, on the one hand, by Theorem 2 from [39] for chosen above $\varepsilon>0$
there exists $\delta(\varepsilon)>0$ such that

$\left\|I-P_{0}\right\|=\left\|\sum\limits_{n=0}^{\infty}P_{n}-P_{0}\right\|=\left\|\sum\limits_{n=1}^{\infty}P_{n}\right\|\geq
1+\delta(\varepsilon)>1.$ (7)

On the other hand, since $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a
Schauder-Orlicz decomposition with an Orlicz function $t^{p}$, for any $x\in
L_{p}$ we have

$\sum\limits_{n=0}^{\infty}\left\|P_{n}x\right\|^{p}=\|x\|^{p}.$

Hence for any $x\in L_{p}$ we have

$\left\|\sum\limits_{n=1}^{\infty}P_{n}x\right\|^{p}=\sum\limits_{j=0}^{\infty}\left\|P_{j}\left(\sum\limits_{n=1}^{\infty}P_{n}x\right)\right\|^{p}=\sum\limits_{n=1}^{\infty}\left\|P_{n}x\right\|^{p}\leq\sum\limits_{n=0}^{\infty}\left\|P_{n}x\right\|^{p}=\|x\|^{p}.$

Thus we arrive at

$\left\|\sum\limits_{n=1}^{\infty}P_{n}\right\|\leq 1.$ (8)

Combining (7) with (8) we arrive at a contradiction.

For the case $p=1$ the proof goes the same as above. It is based on the
Daugavet equation in $L_{1}$, proved in [40]. It reads that for any compact
operator $K$ in $L_{1}$ one has

$\|I+K\|=1+\|K\|.$

∎

######  Remark 6.

Theorem 23 has an essential geometric nature, since for the case $1<p\neq 2$
it is based on Theorem 2 from [39], which in turn is essentially based on
Clarkson’s inequalities for norms in $L_{p}$ (see Lemma 3 in [39]).

Also we have the following obvious corollary of Theorem 23 concerning FDDs in
$L_{p}$.

######  Corollary 24.

There is no Schauder-Orlicz decomposition in $L_{p}$, $1\leq p\neq 2$, which
is an FDD.

Clearly each Schauder-Orlicz decomposition with the a.s.c.p.
$\\{P_{n}\\}_{n=0}^{\infty}$ is normal, i.e. for all $n$ we have

$\|P_{n}\|=1,$

but the converse if false, in general. We can prove this fact as a corollary
of Theorem 23.

######  Corollary 25.

There exists a normal Schauder decomposition in $L_{p}(0,1)$, $1\leq p\neq 2$
which is not a Schauder-Orlicz decomposition.

###### Proof.

To show this it enough to consider the normalized Haar basis
$\\{h_{n}\\}_{n=0}^{\infty}$ in $L_{p}(0,1)$, $1\leq p\neq 2$, and
corresponding to it one dimensional projections $\\{J_{n}\\}_{n=0}^{\infty}$.
It is known that $\\{h_{n}\\}_{n=0}^{\infty}$ is a normal basis of
$L_{p}(0,1)$ for any $p\geq 1$ [18], so

$\|J_{n}\|=1$

for each $n$. Let $\mathfrak{M}_{n}=Lin\\{h_{n}\\}$ for any $n.$ Then clearly
$\dim\mathfrak{M}_{n}=1$ for any $n.$ Suppose that
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a Schauder-Orlicz decomposition in
$L_{p}(0,1)$, $1\leq p\neq 2$. Then we arrive at a contradiction with Theorem
23. ∎

However, in a Hilbert space class of normal Schauder decompositions coincides
with the class of Schauder-Orlicz decompositions with an Orlicz function
$t^{2}$.

######  Theorem 26.

A Schauder decomposition in a Hilbert space is normal if and only if it is a
Schauder-Orlicz decomposition with an Orlicz function $t^{2}$.

###### Proof.

Clearly, a Schauder decomposition $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ of a
Hilbert space is a Schauder-Orlicz decomposition with an Orlicz function
$t^{2}$ if and only if $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is orthogonal.
Further, $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is orthogonal if and only if
the a.s.c.p. $\\{P_{n}\\}_{n=0}^{\infty}$ of
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ consists only of orthogonal
projections. Since the criterion for a projection $P$ in a Hilbert space to be
an orthogonal projection is

$\|P\|=1,$

$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is orthogonal if and only if it is
normal. ∎

Note that if $C_{p}(\\{h_{n}\\}_{n=0}^{\infty})$ is an unconditional constant
of the Haar basis in $L_{p}(0,1)$, $1<p<\infty$, and
$C_{p}(\\{u_{n}\\}_{n=0}^{\infty})$ is an unconditional constant of any other
unconditional basis $\\{u_{n}\\}_{n=0}^{\infty}$ of $L_{p}(0,1)$,
$1<p<\infty$, then

$C_{p}(\\{h_{n}\\}_{n=0}^{\infty})\leq C_{p}(\\{u_{n}\\}_{n=0}^{\infty}).$

This result was first obtained by Olevskii [23] and reads that the Haar basis
in $L_{p}(0,1)$ has minimal unconditional constant among the others
unconditional bases. Almost two decades later the precise value of
unconditional constant of the Haar basis was determined by Burkholder [41]:

$C_{p}(\\{h_{n}\\}_{n=0}^{\infty})=\max\left\\{p,\frac{p}{p-1}\right\\}-1,\>\>1<p<\infty,$

see also [15]. Next we deduce on the basis of these remarkable results the
following best estimate for constants for one dimensional unconditional
Schauder decompositions in $L_{p}(0,1)$, $1<p<\infty$.

######  Theorem 27.

Let $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ be an unconditional Schauder
decomposition of $L_{p}(0,1)$, $1<p<\infty$, with constant $M$ (see Definition
9), such that for any $n$ we have

$\dim\mathfrak{M}_{n}=1.$

Then

$M\geq M_{p}=\frac{1}{2}\max\left\\{p,\frac{p}{p-1}\right\\},$

where $M_{p}$ is the constant of unconditional Schauder decomposition
corresponding to the Haar basis.

###### Proof.

Since the Haar basis is the ”best” in $L_{p}(0,1)$, $1<p<\infty$, in the
mentioned above sense, we have for the a.s.c.p. $\\{P_{n}\\}_{n=0}^{\infty}$
of $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$, any choice of signs
$\\{\epsilon_{i}=\pm 1\\}_{i=0}^{\infty}$ and any $x\in L_{p}$ that

$\left\|\sum\limits_{i=0}^{\infty}\epsilon_{i}P_{i}x\right\|\leq C\|x\|,$ (9)

where $C\geq C_{p}(\\{h_{n}\\}_{n=0}^{\infty})$. Thus for any choice of
selectors $\\{\delta_{i}^{\prime}\in\\{0,1\\}\\}_{i=0}^{\infty}$ there exist
the set of signs $\\{\epsilon_{i}^{\prime}=\pm 1\\}_{i=0}^{\infty}$ such that

$\delta_{i}^{\prime}=\frac{1+\epsilon_{i}^{\prime}}{2}$

for all $i$. Hence, taking into account (9), for any $x\in L_{p}$ we have that

$\displaystyle\left\|\sum\limits_{i=0}^{\infty}\delta_{i}^{\prime}P_{i}x\right\|$
$\displaystyle=\left\|\frac{1}{2}\sum\limits_{i=0}^{\infty}\epsilon_{i}^{\prime}P_{i}x+\frac{1}{2}\sum\limits_{i=0}^{\infty}P_{i}x\right\|$
$\displaystyle\leq\frac{1}{2}\left\|\sum\limits_{i=0}^{\infty}\epsilon_{i}^{\prime}P_{i}x\right\|+\frac{1}{2}\|x\|\leq\frac{C+1}{2}\|x\|.$

Since $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is unconditional Schauder
decomposition with constant $M$, we have on the other hand for any $x\in
L_{p}$ that

$\left\|\sum\limits_{i=0}^{\infty}\delta_{i}^{\prime}P_{i}x\right\|\leq
M\|x\|.$

Subtracting from the latter inequality the previous inequality we arrive at

$M-\frac{C+1}{2}\geq 0.$

Thus

$M\geq\frac{C+1}{2}\geq\frac{C_{p}(\\{h_{n}\\}_{n=0}^{\infty})+1}{2}=\frac{1}{2}\max\left\\{p,\frac{p}{p-1}\right\\}=M_{p}.$

∎

Theorem 27 immediately yields that for any one dimensional unconditional
Schauder decomposition of $L_{p}$ with constant $M$ we have

$M\geq M_{p}=\frac{1}{2}\max\left\\{p,\frac{p}{p-1}\right\\}>1$

for any $1<p\neq 2$. Furthermore, it follows that the space $L_{p}$ has an
unconditional basis with constant $1$ if and only if $p=2,$ despite the fact
that the ”best” Haar basis in $L_{p}(0,1)$, $1<p<\infty$, is monotone and
unconditional, hence Grinblyum-James orthogonal. On the other hand, any
slicing Schauder decomposition in $L_{p}$, $p\geq 1,$ each element of which is
infinite dimensional, is by Theorem 11 unconditional with constant $1.$

For Schauder-Orlicz decompositions in the space $C$ we have the following
result, analogous to the Theorem 23.

######  Theorem 28.

There is no Schauder-Orlicz decomposition in space $C$ with at least one
finite dimensional subspace.

###### Proof.

Since any Schauder-Orlicz decomposition is unconditional by Theorem 11 and any
unconditional Schauder decomposition in $C$ is $c_{0}$-decomposition by [36],
we have for Schauder-Orlicz decomposition
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ with the a.s.c.p.
$\\{P_{n}\\}_{n=0}^{\infty}$ and any $x\in C$ that

$\max_{n}\|P_{n}x\|=\|x\|.$

The rest of the proof goes by lines of the proof of Theorem 23 and is based on
the validity of the Daugavet equation in the space $C$, see [42]. ∎

Finally we have the following corollary of Theorem 28 concerning FDDs in space
$C$.

######  Corollary 29.

There is no Schauder-Orlicz decomposition in space $C$, which is an FDD.

###  4.2 Nonexistence of Schauder-Orlicz decompositions with at least one
finite dimensional subspace in Banach spaces with pseudo-Daugavet property

A Banach space $X$ has a pseudo-Daugavet property [43], [44] if there exists a
strictly increasing function $\varphi_{X}:$ $(0,\infty)\mapsto(0,\infty)$ such
that for any nonzero compact operator $K$ on $X$ we have

$\|I+K\|\geq 1+\varphi_{X}\left(\|K\|\right).$

In particular, if $X$ has a pseudo-Daugavet property with $\varphi_{X}(t)=t$,
then it has the Daugavet property. Historically first appeared examples of
spaces with Daugavet property are classical spaces $C$ and $L_{1}$, see [42]
and [40]. It is worth mentioning that Kadets first proved that spaces with
Daugavet property have no unconditional basis [45]. For more about the
Daugavet property, its relation to the geometry of a Banach space and its
variations we refer, e.g. to monographs [46] and [44]. Note that a pseudo-
Daugavet property is related to the problem of best compact approximation in
$X$ [47], [39]. In [39] Benyamini and Lin proved that spaces $L_{p}$, $1<p\neq
2$ enjoy a pseudo-Daugavet property. The validity of pseudo-Daugavet property
was also proved for Lorentz function spaces $L_{w,p}\left([0,1],\mu\right)$,
$p>2$, without any restrictions on function $w$ [43], [44], for non-atomic
non-commutative $L_{p}$-spaces, where $1<p\neq 2$ [48], and for certain spaces
of operators on $\ell_{p}$ [49].

We generalize our Theorems 23 and 28 to the case of Banach spaces with pseudo-
Daugavet property as follows.

######  Theorem 30.

The following statements are true.

(i) Let $X$ be a Banach space satisfying pseudo-Daugavet property. Then there
is no Schauder-Orlicz decomposition in space $X$ with at least one finite
dimensional subspace.

(ii) Conversely, let $X$ be a Banach space with Schauder-Orlicz decomposition,
at least one subspace of which is finite dimensional. Then $X$ does not have
pseudo-Daugavet property.

###### Proof.

To prove (i) suppose that there exists the Schauder-Orlicz decomposition
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ with an Orlicz function $\Phi$ in $X$,
such that

$\dim\mathfrak{M}_{0}<\infty.$ (10)

Denote by $\\{P_{n}\\}_{n=0}^{\infty}$ the a.s.c.p. of
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$. Note fisrt that (6) means that
$P_{0}$ is compact. Then, on the one hand, since $X$ has pseudo-Daugavet
property we have that there exists a strictly increasing function
$\varphi_{X}:$ $(0,\infty)\mapsto(0,\infty)$ such that

$\left\|I-P_{0}\right\|=\left\|\sum\limits_{n=0}^{\infty}P_{n}-P_{0}\right\|=\left\|\sum\limits_{n=1}^{\infty}P_{n}\right\|\geq
1+\varphi_{X}\left(\|P_{0}\|\right)>1.$ (11)

On the other hand, since $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a
Schauder-Orlicz decomposition with an Orlicz function $\Phi$ and the a.s.c.p.
$\\{P_{n}\\}_{n=0}^{\infty}$, for any $x\in X$ we have

$\inf\left\\{\rho>0:\sum\limits_{n=0}^{\infty}\Phi\left(\frac{\|P_{n}x\|}{\rho}\right)\leq
1\right\\}=\|x\|.$

Hence for any $x\in X$ we have

$\displaystyle\left\|\sum\limits_{n=1}^{\infty}P_{n}x\right\|$
$\displaystyle=\inf\left\\{\rho>0:\sum\limits_{j=0}^{\infty}\Phi\left(\frac{\left\|P_{j}\left(\sum\limits_{n=1}^{\infty}P_{n}x\right)\right\|}{\rho}\right)\leq
1\right\\}$
$\displaystyle=\inf\left\\{\rho>0:\sum\limits_{n=1}^{\infty}\Phi\left(\frac{\|P_{n}x\|}{\rho}\right)\leq
1\right\\}$
$\displaystyle\leq\inf\left\\{\rho>0:\sum\limits_{n=0}^{\infty}\Phi\left(\frac{\|P_{n}x\|}{\rho}\right)\leq
1\right\\}=\|x\|.$

It means that

$\left\|\sum\limits_{n=1}^{\infty}P_{n}\right\|\leq 1.$ (12)

Combining (11) with (12) we observe a contradiction.

To prove (ii) we consider Schauder-Orlicz decomposition
$\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ with an Orlicz function $\Phi$ in $X$,
such that condition (10) holds. By $\\{P_{n}\\}_{n=0}^{\infty}$ we denote the
a.s.c.p. of $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$. Assume that $X$ has
pseudo-Daugavet property. Hence there exists a strictly increasing function
$\varphi_{X}:$ $(0,\infty)\mapsto(0,\infty)$ such that

$\left\|I-P_{0}\right\|\geq 1+\varphi_{X}\left(\|P_{0}\|\right)>1.$ (13)

On the other hand, since $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a
Schauder-Orlicz decomposition with an Orlicz function $\Phi$ and the a.s.c.p.
$\\{P_{n}\\}_{n=0}^{\infty}$, we have, as in the proof of (i), that condition
(12) holds. Thus we arrive at a contradiction between (13) and (12). ∎

Combining Theorem 30 with [43], [48] and [49] we have the following immediate
corollary.

######  Corollary 31.

Lorentz function spaces $L_{w,p}\left([0,1],\mu\right)$, $p>2$, non-atomic
non-commutative $L_{p}$-spaces, where $1<p\neq 2$ and certain spaces of
operators on $\ell_{p}$ from [49] have no Schauder-Orlicz decomposition, at
least one subspace of which is finite dimensional.

We have the following corollary, extending Corollaries 24 and 29.

######  Corollary 32.

Let $X$ be a Banach space with pseudo-Daugavet property. Then there is no
Schauder-Orlicz decomposition in $X$, which is an FDD. Conversely, if there
exists a Schauder-Orlicz decomposition in $X$, which is an FDD, then $X$ has
no pseudo-Daugavet property.

Finally, in the class of spaces with Schauder-Orlicz decompositions
combination of Theorem 30 with Corollary 32 turns into the following
criterion.

######  Theorem 33.

Let $X$ be a Banach space, which has a Schauder-Orlicz decomposition. Then the
following assertions are equivalent.

  1. 1.

A Banach space $X$ has pseudo-Daugavet property.

  2. 2.

There is no Schauder-Orlicz decomposition in $X$ with at least one finite
dimensional subspace.

  3. 3.

There is no Schauder-Orlicz decomposition in $X$, which is an FDD.

##  5 Concluding remarks and perspectives

Since the concept of Schauder-Orlicz decomposition in Banach spaces seems to
be new there are many interesting questions for further study. Among them is
also a question what the existence of Schauder-Orlicz decomposition in a
Banach space $X$ mean to geometric properties of $X$? Theorem 33 gives only
partial answer and characterizes pseudo-Daugavet property, which has of course
geometric meaning, for Banach spaces possessing Schauder-Orlicz decompositions
on a language of sizes of dimensions of these decompositions.

We note some of questions for further study, which lie on a surface.

1) Does non-separable Orlicz spaces $\ell_{\Phi}$, i.e. such that $\Phi$ does
not satisfy the $\Delta_{2}$-condition at zero [17], have Schauder
decomposition? By [6] we have that all reflexive non-separable $\ell_{\Phi}$
have a Schauder decomposition, but what with non-reflexive non-separable
$\ell_{\Phi}$? Moreover, does non-separable Orlicz spaces $\ell_{\Phi}$ have
$\ell_{\Phi}$-decomposition and Schauder-Orlicz decomposition?

2) It is proved in [36] that all unconditional Schauder decompositions in
$\ell_{1}$ are $\ell_{1}$-decompositions and all unconditional Schauder
decompositions in $c_{0}$ are $c_{0}$-decompositions. Thus the class of
unconditional Schauder decompositions in $\ell_{1}$ (resp. $c_{0}$) coincides
with the class of all $\ell_{1}$-decompositions (resp.
$c_{0}$-decompositions). However, an interesting question on the uniqueness,
up to an isomorphism, of $\ell_{1}$-decomposition with preassigned dimensions
in $\ell_{1}$ and of $c_{0}$-decomposition with preassigned dimensions in
$c_{0}$ is still open. On the way of finding the answer we propose the
following.

######  Conjecture 34.

Every Schauder-Orlicz decomposition with preassigned dimensions in $\ell_{1}$
is unique, up to an isometry. Every Schauder-Orlicz decomposition with
preassigned dimensions in $c_{0}$ is unique, up to an isometry.

3) Note that in [4] Schauder decomposition satisfying the condition $2$ of
Theorem 14 also is called orthogonal. This notion of orthogonality is due to
Singer, dates back to 1962 [34] and has a geometric meaning in view of the
notion of Birkhoff-orthogonality [50], [34], see also [18]. However, in view
of the proof of Proposition 13, notions of Grinblyum-James orthogonality and
Singer orthogonality are distinct, since every Singer-orthogonal basis is
unconditional [34] and unconditional bases do not exist in $C[a,b]$.
Nevertheless a basis is Singer-orthogonal if and only if every its permutation
is Grinblyum-James orthogonal [18]. So Singer-orthogonality is much stronger,
then Grinblyum-James orthogonality. On the other hand by Theorem 14 each
Schauder-Orlicz decomposition is Singer-orthogonal. However, the whole picture
of mutual relations of this two concepts of orthogonality and the concept of
Schauder-Orlicz decomposition looks not complete.

4) Clearly if $\\{\mathfrak{M}_{n}\\}_{n=0}^{\infty}$ is a Schauder-Orlicz
decomposition in $X$ with an Orlicz function $\Phi$, then
$\\{S\mathfrak{M}_{n}\\}_{n=0}^{\infty}$, where $S:X\mapsto X$ is an
isomorphism, is an $\ell_{\Phi}$-decomposition in $X$. However, does every
space with $\ell_{\Phi}$-decomposition has a Schauder-Orlicz decomposition?
The space $C[0,1]$ looks as the key pointer here, since it has a
$c_{0}$-decomposition [4], but the construction of Schauder-Orlicz
decomposition in $C[0,1]$ does not look easy. Taking into account Theorems 11
and 28, the first step may be the construction of unconditional Schauder
decomposition in $C[0,1]$ with constant $1,$ each subspace of which is
infinite dimensional.

5) One can prove, applying Theorem 15.3 from [41] (see also [15]) and
arguments from the proof of Theorem 27, that every monotone Schauder
decomposition in $L_{p}$, $1<p<\infty$, is unconditional with constant

$M_{p}=\frac{1}{2}\max\left\\{p,\frac{p}{p-1}\right\\}.$

Combining this fact with Theorem 27 we have that any Grinblyum-James
orthogonal basis in $L_{p}$, $1<p<\infty$, is unconditional with constant
$M_{p}$ and this constant is the best possible. Therefore it is natural to ask
about the best possible unconditional constant of any unconditional FDD in
$L_{p}$, $1<p\neq 2$, in an intermediate situation between Grinblyum-James
orthogonal unconditional bases in $L_{p}$ with best possible constant $M_{p}$
and slicing Schauder-Orlicz decompositions with unconditional constant $1$.

6) Does the Lorentz space $L_{w,p}\left([0,1],\mu\right)$, where $1\leq p\neq
2$ and $w(t)\neq 1$, have a Schauder-Orlicz decomposition? By Theorem 30 it
may have only Schauder-Orlicz decompositions, each subspace of which is
infinite dimensional. The same question concerns non-atomic non-commutative
$L_{p}$-spaces, where $1\leq p\neq 2$, and spaces of operators on $\ell_{p}$,
considered in [49].

## Acknowledgments

The author is grateful to Prof. Vladimir Kadets for the discussion during the
International Conference in Functional Analysis dedicated to $125^{th}$
anniversary of S. Banach in Lviv and pointing out paper [39].

## Data availability

This manuscript has no associated data.

## Conflict of interest

The author state that there is no conflict of interest.

## References

  * [1] T. Kato, Similarity for sequences of projections, Bull. Amer. Math. Soc. 73(6) (1967) 904–905.
  * [2] B. L. Sanders, On the existence of [Schauder] decompositions in Banach spaces, Proc. Amer. Math. Soc. 16(5) (1965) 987–990.  doi:10.1090/S0002-9939-1965-0180835-2.
  * [3] D. W. Dean, Schauder decompositions in $(m)$, Proc. Amer. Math. Soc. 18 (1967) 619–623.  doi:10.1090/S0002-9939-1967-0217575-9.
  * [4] I. Singer, Bases in Banach Spaces II, Springer-Verlag, Berlin, 1981.
  * [5] V. Marchenko, Isomorphic Schauder decompositions in certain Banach spaces, Open Mathematics 12(11) (2014) 1714–1732.  doi:10.2478/s11533-014-0441-y.
  * [6] J. J. M. Chadwick, Schauder decompositions in non-separable Banach spaces, Bull. Aust. Math. Soc. 6(1) (1972) 133–144.  doi:10.1017/S0004972700044336.
  * [7] G. Allexandrov, D. Kutzarova, A. Plichko, A separable space with no Schauder decomposition, Proc. Amer. Math. Soc. 127(9) (1999) 2805–2806.  doi:10.1090/S0002-9939-99-05370-8.
  * [8] W. B. Johnson, Finite-dimensional Schauder decompositions in $\pi_{\lambda}$ and dual $\pi_{\lambda}$ spaces, Illinois J. Math. 14(4) (1970) 642–647.  doi:10.1215/ijm/1256052956.
  * [9] S. J. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1987) 81–98.  doi:10.1007/BF02392555.
  * [10] P. Hajek, R. Medina, Compact retractions and Schauder decompositions in Banach spaces, Trans. Amer. Math. Soc. 376(2) (2023) 1343–1372.  doi:10.1090/tran/8807.
  * [11] G. Botelho, V. Favaro, J. Mujica, Absolute Schauder decompositions and linearization of holomorphic mappings of bounded type, Publ. Res. Inst. Math. Sci. 55(3) (2019) 651–664.  doi:10.4171/PRIMS/55-3-6.
  * [12] K. Kruse, Series representations in spaces of vector-valued functions via Schauder decompositions, Math. Nachr. 294(2) (2021) 354–376.
  * [13] M. De la Rosa, L. Frerick, S. Grivaux, A. Peris, Frequent hypercyclicity, chaos, and unconditional Schauder decompositions, Israel J. Math. 190 (2012) 389–399.  doi:10.1007/s11856-011-0210-6.
  * [14] N. Bernandes, A. Peris, On the existence of polynomials with chaotic behaviour, Journal of Function Spaces and Applications Article ID 320961 (2013).  doi:10.1155/2013/320961.
  * [15] W. B. Johnson, J. Lindenstrauss, Handbook of the Geometry of Banach Spaces, Volume 1, Elsevier, 2001.
  * [16] W. B. Johnson, J. Lindenstrauss, Handbook of the Geometry of Banach Spaces, Volume 2, Elsevier, 2003.
  * [17] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I and II, Reprint of the 1977, 1979 ed., Springer-Verlag, Berlin, 1996.
  * [18] I. Singer, Bases in Banach Spaces I, Grundlehren der mathematischen Wissenschaften, vol. 154, Springer-Verlag, Berlin, 1970.
  * [19] H. Zwart, Riesz basis for strongly continuous groups, J. Differential Equations 249 (2010) 2397–2408.  doi:10.1016/j.jde.2010.07.020.
  * [20] G. Sklyar, V. Marchenko, Resolvent of the generator of the $C_{0}$-group with non-basis family of eigenvectors and sharpness of the XYZ theorem, J. Spectr. Theory 11 (2021) 369–386.  doi:10.4171/JST/344.
  * [21] V. Marchenko, Stability of unconditional Schauder decompositions in $\ell_{p}$ spaces, Bull. Aust. Math. Soc. 92 (2015) 444–456.  doi:10.1017/S0004972715000775.
  * [22] V. Marchenko, Stability of Riesz bases, Proc. Amer. Math. Soc. 146(8) (2018) 3345–3351.  doi:10.1090/proc/14056.
  * [23] A. M. Olevskii, Fourier series and Lebesgue functions, Uspehi Mat. Nauk 22(3) (1967) 237–239.
  * [24] B. L. Sanders, Decompositions and reflexivity in Banach spaces, Proc. Amer. Math. Soc. 16(2) (1965) 204–208.  doi:10.2307/2033846.
  * [25] W. B. Johnson, M. Zippin, On subspaces of quotients of $\left(\sum G_{n}\right)_{\ell_{p}}$ and $\left(\sum G_{n}\right)_{c_{0}}$, Israel J. Math. 13 (1972) 311–316.  doi:10.1007/BF02762805.
  * [26] J. J. M. Chadwick, Cofinal biorthogonal systems and absolute Schauder decompositions, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences 75 (1975) 33–36.
  * [27] V. I. Gurarii, On inclinations of subspaces and the existence of orthogonal basis in a Banach space, Comm. Inst. Sci. Math. Mec. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] 30(4) (1964) 34–37.
  * [28] V. I. Gurarii, On some geometric characteristics of subspaces and bases in Banach spaces, Colloquium Math. 13(1) (1964) 59–63.  doi:10.4064/cm-13-1-59-63.
  * [29] V. I. Gurarii, On openings and inclinations of subspaces of a Banach space, Teoria Funk., Funkt. Anal. Prilozh. (Kharkov University press) 1 (1965) 194–204.
  * [30] M. M. Grinblyum, On the representation of a space of type B in the form of a direct sum of subspaces, Dokl. Akad. Nauk (N. S.) 70 (1950) 749–752.
  * [31] M. M. Grinblyum, Certain theoremes sur la base dans un espace du type (B), Dokl. Akad. Nauk (N. S.) 31(5) (1941) 428–432.
  * [32] R. C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Natl. Acad. Sci. U.S.A. 37(3) (1951) 174–177.  doi:10.1073/pnas.37.3.174.
  * [33] S. Banach, Théorie des Opérations Linéaires, Warsaw, 1932.
  * [34] I. Singer, On a theorem of I. M. Gelfand, Uspehi Mat. Nauk 17(1) (1962) 169–176.
  * [35] J. Schauder, Zur Theorie stetiger Abbildungen in Funktionalräumen, Mathematische Zeitschrift 26 (1927) 47–65.
  * [36] J. Lindenstrauss, A. Pelczynski, Absolutely summing operators in $\mathcal{L}_{p}$-spaces and their applications, Studia Math. 29(3) (1968) 275–326.
  * [37] J. Lindenstrauss, M. Zippin, Banach spaces with a unique unconditional basis, J. Funct. Anal. 3 (1969) 115–125.  doi:10.1016/0022-1236(69)90054-8.
  * [38] B. Beauzamy, Introduction to Banach Spaces and their Geometry, North-Holland Mathematics Studies, vol. 68, North-Holland Publishing Company, Amsterdam, 1982\.
  * [39] Y. Benyamini, P. K. Lin, An operator on $L^{p}$ without best compact approximation, Israel J. Math. 51(4) (1985) 298–304.  doi:10.1007/BF02764722.
  * [40] V. F. Babenko, S. A. Pichugov, A property of compact operators in the space of integrable functions, Ukrainian Math. J. 33(4) (1981) 374–376.  doi:10.1007/BF01085739.
  * [41] D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12(3) (1984) 647–702.  doi:10.1214/aop/1176993220.
  * [42] I. K. Daugavet, On a property of compact operators in the space $C$, Uspehi Mat. Nauk 18(5) (1963) 157–158.  doi:10.1090/S0002-9939-1967-0217575-9.
  * [43] M. Popov, B. Randrianantoanina, A pseudo-Daugavet property for narrow projections in Lorentz spaces, Illinois J. Math. 46(4) (2002) 1313–1338.  doi:10.1215/ijm/1258138482.
  * [44] M. Popov, B. Randrianantoanina, Narrow Operators on Function Spaces and Vector Lattices, De Gruyter Studies in Mathematics, vol. 45, De Gruyter, Berlin, 2012\.
  * [45] V. Kadets, Some remarks concerning the Daugavet equation, Quaestiones Math. 19 (1996) 222–235.  doi:10.1080/16073606.1996.9631836.
  * [46] V. Kadets, M. Martin, J. Meri, A. Perez, Spear Operators Between Banach Spaces, Lecture Notes in Mathematics, vol. 2205, Springer, Cham, 2018.
  * [47] S. Axler, I. Berg, N. Jewell, A. Shields, Approximation by compact operators and the space $H^{\infty}+C$, Ann. of Math. 109 (1979) 601–612.
  * [48] T. Oikberg, The Daugavet property of $C^{\ast}$-algebras and non-commutative $L_{p}$-spaces, Positivity 6 (2002) 59–73.  doi:10.1023/A:1012023526639.
  * [49] T. Oikberg, Spaces of operators, the $\psi$-Daugavet property, and numerical indices, Positivity 9 (2005) 607–7623.  doi:10.1007/s11117-005-2779-7.
  * [50] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. Journ. 1(2) (1935) 169–172.  doi:10.1215/S0012-7094-35-00115-6.