# Approximate identities for Ideals in $L(L^{p}))$

W. B. Johnson∗ and G. Schechtman

(Date: January 2024)

###### Abstract.

The main result is that the only non trivial closed ideal in the Banach
algebra $L(L^{p})$ of bounded linear operators on $L^{p}(0,1)$, $1\leq
p<\infty$, that has a left approximate identity is the ideal of compact
operators. The algebra $L(L^{1})$ has at least one non trivial closed ideal
that has a contractive right approximate identity as well as many, including
the unique maximal ideal, that do not have a right approximate identity.

###### 2000 Mathematics Subject Classification:

Primary 47L20; Secondary 46E30.

∗Supported in part by NSF DMS-1900612

##  1\. Introduction

Arguably the most natural examples of $L^{p}$, $1\leq p<\infty$, operator
algebras; i.e., closed subalgebras of the Banach algebra $L(L^{p})$ of (always
bounded, linear) operators on $L^{p}(\mu)$ for some measure $\mu$, are the
closed ideals in $L(L^{p})$. When $p=2$ and the measure $\mu$ is separable, it
is well-known that the only non trivial closed ideal in $L(L^{p})$ is the
ideal ${\mathcal{K}}(L^{p})$ of compact operators. Similarly,
${\mathcal{K}}(\ell^{p})$ is the only non trivial closed ideal in
$L(\ell^{p})$ for other values of $1\leq p<\infty$. Here we focus on separable
$L^{p}(\mu)$, in which case up to Banach space isomorphism the only other
examples are $L^{p}(0,1)$, so in the sequel we let $L^{p}$ denote
$L^{p}(0,1)$.

When $1<p\not=2<\infty$, the structure of the closed ideals in $L(L^{p})$ is
complicated and not well understood. Given any Banach space $X$, every non
trivial ideal in $L(X)$ contains the ideal ${\mathcal{F}}(X)$ of finite rank
operators on $X$, and hence every non trivial closed ideal contains the ideal
${\mathcal{K}}(X)$ of compact operators on $X$ when $X$ has the approximation
property; in particular, when $X$ is any $L^{p}$ space, $1\leq p\leq\infty$.
Sometimes $L(X)$ has a largest non trivial ideal. Let ${\mathcal{M}}(X)$
denote the collection of all operators $T\in L(X)$ such that the identity
operator $I_{X}$ on $X$ does not factor through $T$ (an operator $S:W\to Z$ is
said to factor through an operator $T:X\to Y$ provided there exist operators
$B:W\to X$ and $A:Y\to Z$ such that $S=ATB$). It is easy to see that if
${\mathcal{M}}(X)$ is closed under addition then ${\mathcal{M}}(X)$ is an
ideal, in which case ${\mathcal{M}}(X)$ is obviously the largest ideal in
$L(L^{p})$. It is true but non trivial that ${\mathcal{M}}(L^{p})$ is closed
under addition for $1\leq p<\infty$ (see [ES] for the case $p=1$ and [JMST]
for the cases $1<p<\infty$). Actually more is contained in the cited papers;
namely, that ${\mathcal{M}}(L^{p})$ is the class of $L^{p}$–singular operators
in $L(L^{p})$. (An operator $T:X\to Y$ is said to be $Z$–singular provided
there is no subspace $Z_{1}$ of $X$ such that $Z_{1}$ is isomorphic to $Z$ and
the restriction $T_{|Z_{1}}$ of $T$ to $Z_{1}$ is an isomorphism. So $T$ is
strictly singular if it is $Z$-singular for every infinite dimensional $Z$.)
This last description of ${\mathcal{M}}(L^{p})$ will be used later in this
note.

We take this oportunity to describe some of the ways to build many more
exmaples of closed ideals in $L(L^{p})$. As was pointed out by Pietsch [Pie]
it follows from [Sch] that there are infinitely many different closed ideals
in $L(L^{p})$, $1<p\not=2<\infty$. In a similar way it follows from [BRS] that
there are at least $\aleph_{1}$. These ideals are all large in the sense that
they contain operators that are not strictly singular. Schlumprecht and Zsák
[SZ] proved that there are at least a continuum $c:=2^{\aleph_{0}}$ closed
ideals in $L(L^{p})$, $1<p\not=2<\infty$ . The ideals constructed in [SZ] are
all small; i.e., contained in the strictly singular operators. Recently, the
authors [JS] proved that there are $2^{c}$ large closed ideals and also
$2^{c}$ small closed ideals in $L(L^{p})$ when $1<p\not=2<\infty$. The ideals
constructed in [Sch] and [BRS] look much different from the ideals constructed
in [SZ] and [JS] and illustrate somewhat different approaches to constructing
ideals. In the first approach one constructs a family $\mathcal{S}$ of
complemented subspaces of $L^{p}$ that are mutually non isomorphic (in fact,
for each two members of $\mathcal{S}$ one of them does not embed
complementably into any finite direct sum of the other). Given one of these
spaces $X$, let $\mathcal{I}_{X}$ denote the space of all operators in
$L(L^{p})$ that factor through some finite direct sum of $X$. Then
$\mathcal{I}_{X}$ is obviously a (possibly non closed) ideal in $L(L^{p})$,
and if an idempotent $P$ is in the closure $\overline{\mathcal{I}_{X}}$ of
$\mathcal{I}_{X}$, then it is easy to see that the range of $P$ is isomorphic
to a complemented subspace of some finite direct sum of $X$. Consequently,
$\overline{\mathcal{I}_{X}}\not=\overline{\mathcal{I}_{Y}}$ for distinct $X$,
$Y$ in $\mathcal{S}$. In practice, it is a nuisance to have to consider all
finite direct sums of the spaces $X$ and so one builds examples where it is
easy to check that $X\oplus X$ is isomorphic to $X$. In any case, the ideals
$\overline{\mathcal{I}_{X}}$ are large ideals because, by construction, they
contain projections onto infinite dimensional subspaces, and such operators
are obviously not strictly singular.

In the second approach, one defines a specific operator $T$ from some Banach
space $X_{1}$ into some Banach space $X_{2}$ and considers the collection
$\mathcal{I}_{T}$ of all operators $S$ in $L(L^{p})$ that factor through $T$;
i.e., $S$ can be written as $ASB$ for some bounded linear operators
$B:L^{p}\to X_{1}$ and $A:X_{2}\to L^{p}$. In practice one uses only $T$ for
which the direct sum operator $T\oplus T:X_{1}\oplus X_{1}\to X_{2}\oplus
X_{2}$ factors through $T$ in order to guarantee that $\mathcal{I}_{T}$ is a
(possibly non closed) ideal in $L(L^{p})$ so that the closure
$\overline{\mathcal{I}_{T}}$ is a closed ideal in $L(L^{p})$. Of course, the
first approach is the special case of the second approach where $T$ is an
idempotent in $L(L^{p})$, but in the second approach what have been used are
injections, including strictly singular operators such as the formal
injections $i_{s,r}$ from $\ell_{s}$ to $\ell_{r}$ with $s<r$. When the
injection $T$ is strictly singular, the deal $\overline{\mathcal{I}_{T}}$
consists of strictly singular operators and so is a small ideal. In an earlier
version of [JS] this approach was used to construct $c$ large closed ideals in
$L(L^{p})$, $1<p\not=2<\infty$. The injections used there are not strictly
singular but are very far from being idempotents.

In [JPS], using this approach but with very different operators, it was proved
that $L(L^{1})$ has at least $c$ small closed ideals. It is open whether
$L(L^{1})$ has more than a continuum of closed ideals. There are known to
exist only three large ideals in $L(L^{1})$.

To build $2^{c}$ closed ideals in $L(L^{p})$, $1<p\not=2<\infty$, a yet
somewhat different approach was used in [JS]. Each of these ideals is
generated by a class of operators $\\{T_{\alpha}\\}_{\alpha\in\mathcal{A}}$.
That is the closure of the set of operators of the form
$\sum_{i=1}^{n}A_{i}T_{\alpha_{i}}B_{i}$ where n ranges over the positive
integers and $\\{\alpha_{1},\dots,\alpha_{n}\\}$ over ${\mathcal{A}}^{n}$.
This of course is not very informative as any ideal is of that form. The main
contribution of [JS] is to find a way to produce $2^{c}$ sets ${\mathcal{A}}$
that give rise to different ideals.

Although the classification of the closed ideals in $L(L^{p})$ is far from
being known, we can prove some general facts about them. These facts involve
the notion of approximate identity.

######  Definition 1.1.

Given a Banach space $X$ and an ideal ${\mathcal{I}}$ in $L(X)$, we say that a
net $(T_{\alpha})$ of operators in ${\mathcal{I}}$ is a left; respectively,
right, approximate identity for ${\mathcal{I}}$ provided $\|T_{\alpha}S-S\|\to
0$; respectively, $\|ST_{\alpha}-S\|\to 0$, for every $S$ in ${\mathcal{I}}$.

In particular, we prove in Section 2 that the only non trivial ideal in
$L(L^{p})$, $1\leq p<\infty$, that has a left approximate identity is the
ideal ${\mathcal{K}}(L^{p})$ of compact operators. If follows by duality that
for $1<p<\infty$ the ideal ${\mathcal{K}}(L^{p})$ is the only non trivial
ideal that has a right approximate identity. In contrast to this last fact, in
Section 3 we prove that $L(L^{1})$ has a non trivial ideal other than
${\mathcal{K}}(L^{1})$ that has a right approximate identity consisting of
contractive idempotents. We also point out that no small ideal in $L(L^{1})$
has a right approximate identity and show that at least one of the other two
known large ideals in $L(L^{1})$, the unique maximal ideal
${\mathcal{M}}(L^{1})$, does not have a right approximate identity.

We thank N. C. Phillips for many discussions about the contents herein. In
fact, Phillips raised the question whether there are ideals in $L(L^{p})$
other than ${\mathcal{K}}(L^{p})$ that have a left or right approximate
identity. Part of his motivation was the open problem to determine which
ideals ${\mathcal{I}}$ in $L(L^{p})$ have the property that
$L(L^{p})/{\mathcal{I}}$ are isomorphically homomorphic to a subalgebra of
$L(L^{p}(\mu))$ for some measure $\mu$. The proofs in [BJ] and [BP] that
${\mathcal{K}}(L^{p})$ has this property both use the fact that
${\mathcal{K}}(L^{p})$ has a bounded approximate identity.

##  2\. Left approximate identities for ideals in $L(L^{p})$

Here we prove

######  Theorem 2.1.

Let $\mathcal{I}$ be a non trivial closed ideal in $L(L^{p})$, $1\leq
p<\infty$, and assume that $\mathcal{I}$ has a left approximate identity. Then
$\mathcal{I}$ is ${\mathcal{K}}(L^{p})$, the ideal of compact operators.

Proof: Since the Haar basis is a monotone Schauder basis for $L^{p}$, the
ideal ${\mathcal{K}}(L^{p})$ has a left approximate identity consisting of
finite rank projections of norm one. The converse direction really consists of
three theorems as the proofs for the cases $p=1$, $2<p<\infty$, and $1<p<2$
are very different. To call attention to the fact that we prove something
stronger than an ideal has no left approximate identity, we introduce some new
language. Given a Banach space $X$, an ideal ${\mathcal{I}}$ in $L(X)$, and a
collection ${\mathcal{S}}$ of operators in $L(X)$, we say that a net
$(T_{\alpha})$ of operators in $L(X)$ is an ${\mathcal{S}}$–left;
respectively, ${\mathcal{S}}$–right, approximate identity for ${\mathcal{I}}$
provided each $T_{\alpha}$ is in ${\mathcal{S}}$ and $\|T_{\alpha}T-T\|\to 0$;
respectively, $\|TT_{\alpha}-T\|\to 0$, for every $T$ in ${\mathcal{I}}$.

Case $1$: $p=1$.

Let $\mathcal{I}$ be a non trivial closed ideal in $L(L^{1})$. We treat first
the case that ${\mathcal{I}}$ is contained in the closed ideal
$\mathcal{W}=\mathcal{W}(L^{1})$ of weakly compact operators in $L(L^{1})$. In
this case we show that there is no ${\mathcal{W}}$–left approximate identity
for ${\mathcal{I}}$ (the same argument shows that it also has no
${\mathcal{W}}$–right approximate identity).

We use without further reference the facts that $\mathcal{W}$ is equal to the
ideal $\mathcal{S}=\mathcal{S}(L^{1})$ of strictly singular operators on
$L^{1}$, and that an operator on $L^{1}$ is weakly compact if and only if it
is $\ell^{1}$–singular; i.e., is not an isomorphism when restricted to any
subspace that is isomorphic to $\ell^{1}$. This background is presented in,
for example, [AK, Chapter 5].

Since $L^{1}$ has the Dunford–Pettis property [AK, Theorem 5.4.6], the product
$\mathcal{W}\cdot\mathcal{I}$ is a subset of the compact operators
$\mathcal{K}=\mathcal{K}(L^{1})$. Thus if $(S_{\alpha})$ is a net of weakly
compact operators and $\|S_{\alpha}T-T\|\to 0$, then $T$ must be a compact
operator. This proves, in particular that there is no ${\mathcal{W}}$–left
approximate identity for ${\mathcal{I}}$ unless
${\mathcal{I}}\subset{\mathcal{K}}$. This finishes the proof because
${\mathcal{K}}$ is the smallest non trivial closed ideal in $L(L^{1})$.

Next suppose that ${\mathcal{I}}$ is the ideal ${\mathcal{W}}$ of weakly
compact operators on $L^{1}$. We have already observed that ${\mathcal{W}}$
does not have a left approximate identity (since there are weakly compact
operators on $L^{1}$ that are not compact), but we want to show that there is
no ${\mathcal{M}}$–left approximate identity for ${\mathcal{W}}$, where
${\mathcal{M}}={\mathcal{M}}(L^{1})$ is the unique maximal ideal in $L(L^{1})$
described in the introduction. It then follows that there is no
${\mathcal{M}}$–left approximate identity for any closed ideal in $L(L^{1})$
that contains ${\mathcal{W}}$. We then complete the proof of the $p=1$ case of
Theorem 2.1 by recalling that a closed ideal in $L(L^{1})$ that is not
contained in ${\mathcal{W}}$ must contain ${\mathcal{W}}$. This is true
because (1) the identity operator $I_{\ell^{1}}$ on $\ell^{1}$ factors through
every non weakly compact operator in $L(L^{1})$ [KP], and (2) every weakly
compact (even every representable) operator from $L^{1}$ to any Banach space
factors through $\ell^{1}$. Again, this background is in the above mention
chapter in [AK].

To prove that there is no ${\mathcal{M}}$–left approximate identity for
${\mathcal{W}}$, it is enough to find $S\in{\mathcal{W}}$ such that $S$ is not
in the closure of ${\mathcal{M}}\cdot S$. To do this, first observe that
$S:=I_{2,1}Q$ is in ${\mathcal{W}}$, where $I_{2,1}$ is the formal identity
from $L^{2}$ into $L^{1}$ and $Q$ is a quotient mapping from $L^{1}$ onto
$L^{2}$; that is, $Q$ maps the open unit ball $B^{\circ}_{L^{1}}$ of $L^{1}$
onto the open unit ball $B^{\circ}_{L^{2}}$ of $L^{2}$. Thus
$\overline{SB^{\circ}_{L^{1}}}$ contains all measurable functions having
constant modulus $1$. But if $T$ is an operator on $L^{1}$ that maps all
measurable functions having constant modulus $1$ into functions whose norms
are bounded away from zero, then $T$ is not in ${\mathcal{M}}$ by [ES],
[Ros2]. In particular, no operator in ${\mathcal{M}}$ can be close to the
identity on the functions that have constant modulus $1$. This shows that $S$
is not in the closure of ${\mathcal{M}}\cdot S$.

This completes the proof of Theorem 2.1 in the case $p=1$. We just remark that
we do not really need the fact that a closed ideal ${\mathcal{I}}$ in
$L(L^{1})$ that contains a non weakly compact operator must contain
${\mathcal{W}}$, which uses the background facts (1) and (2). Only (1) is
needed. (1) says that ${\mathcal{I}}$ contains a projection whose range is
isomorphic to $\ell^{1}$, and it is easy to construct an operator
$U:\ell^{1}\to L^{1}$ such that $UB_{\ell_{1}}$ contains all measurable
functions that have constant modulus one.

Case 2: $2<p<\infty$.

We assume that ${\mathcal{I}}$ is a non trivial ideal in $L(L^{p})$ different
from ${\mathcal{K}}={\mathcal{K}}(L^{p})$. The main tool needed for the proof
is the Kadec–Pełczyński dichotomy principle [KP], which says that every
normalized weakly null sequence $\\{x_{n}\\}_{n=1}^{\infty}$ in $L^{p}$,
$2<p<\infty$, contains a subsequence $\\{x_{n}\\}_{n=1}^{\infty}$ that is
equivalent either to the unit vector basis of $\ell^{2}$ or to the unit vector
basis of $\ell^{p}$, and moreover that the closed linear space of
$\\{x_{n}\\}_{n=1}^{\infty}$ is complemented in $L^{p}$. (This and other
standard background on $L^{p}$ that we use is contained in the book [AK].)
Both possibilities can occur because $L^{p}$ contains complemented subspaces
isomorphic to $\ell^{2}$ and contractively complemented subspaces
isometrically isomorphic to $\ell_{p}$. By the K–P dichotomy, if $T$ is a non
compact operator on $L^{p}$ then either $i_{p}:=i_{p,p}$, $i_{2}:=i_{2,2}$, or
$i_{2,p}$ factors through $T$, where for $r\leq s$ we denote by $i_{r,s}$ the
formal identity operator from $\ell^{r}$ into $\ell^{s}$. If $T$ lies in the
ideal ${\mathcal{I}}$ and $i_{2,p}$ does not factor through $T$, then the
ideal property implies that $i_{2,p}$ factors through some other operator in
${\mathcal{I}}$. So in all cases we have $T\in{\mathcal{I}}$ such that
$i_{2,p}$ factors through $T$. However, we get more information when either
$i_{2}$ or $i_{p}$ factors through an operator in ${\mathcal{I}}$, so we
discuss all three cases.

The easiest case is that $i_{2}$ factors through some $T\in{\mathcal{I}}$.
Thus in $TB_{L^{p}}$ there is a sequence $\\{x_{n}\\}_{n=1}^{\infty}$ that is
equivalent to the unit vector basis for $\ell^{2}$ and such that the closed
linear span $[\\{x_{n}\\}_{n=1}^{\infty}]$ of $\\{x_{n}\\}_{n=1}^{\infty}$ is
complemented in $L^{p}$. Let $\\{h_{n}\\}_{n=1}^{\infty}$ be the Haar basis
for $L^{p}$, normalized so that each $h_{n}$ has norm one. Since
$\\{h_{n}\\}_{n=1}^{\infty}$ is an unconditional basis for $L^{p}$ and $L^{p}$
has type $2$, the operator $x_{n}\mapsto h_{n}$ extends to a bounded linear
operator from $[\\{x_{n}\\}_{n=1}^{\infty}]$ into $L^{p}$. Since
$[\\{x_{n}\\}_{n=1}^{\infty}]$ is complemented in $L^{p}$, this operator
extends to an operator $S$ on $L^{p}$. Thus the operator $U:=ST$ is an
operator in ${\mathcal{I}}$ such that $UB_{L^{p}}$ contains the Haar basis
$\\{h_{n}\\}_{n=1}^{\infty}$. We claim that $U$ is not in the closure of
${\mathcal{M}}\cdot U$ (where here ${\mathcal{M}}={\mathcal{M}}(L^{p})$). This
claim is an immediate consequence of Andrew’s result [And] that if $S\in
L(L^{p})$ and $\sup_{n}\|Sh_{n}-h_{n}\|<1$, then $S$ is not in
${\mathcal{M}}$. This proves that there is no ${\mathcal{M}}$–left approximate
identity for ${\mathcal{I}}$.

Next assume that $i_{p}$ factors through some $T\in{\mathcal{I}}$. It then
follows from Proposition 2.3 below that there is another operator
$U\in{\mathcal{I}}$ such that $UB_{L^{p}}$ contains the Haar basis
$\\{h_{n}\\}_{n=1}^{\infty}$. The argument in the previous paragraph then
gives that there is no ${\mathcal{M}}$–left approximate identity for
${\mathcal{I}}$.

Finally, assume that only $i_{2,p}$ factors through an operator
$T\in{\mathcal{I}}$. In this case we only prove that if ${\mathcal{J}}$ is a
closed ideal such that there is a ${\mathcal{J}}$–left approximate identity
for ${\mathcal{I}}$, then $i_{p}$ factors through some operator in
${\mathcal{J}}$. We can assume that there is $\\{x_{n}\\}_{n=1}^{\infty}$ in
$TB_{L^{p}}$ so that $\\{x_{n}\\}_{n=1}^{\infty}$ is equivalent to the unit
vector basis of $\ell^{p}$ and there is a projection $P$ from $L^{p}$ onto
$[\\{x_{n}\\}_{n=1}^{\infty}]$. Now suppose that there is a
${\mathcal{J}}$–left approximate identity for ${\mathcal{I}}$. Given
$\varepsilon>0$, there is $S\in{\mathcal{J}}$ such that
$\sup_{n}\|x_{n}-Sx_{n}\|<\varepsilon$. Replacing $S$ with
$PS\in{\mathcal{J}}$, we can (since $\varepsilon>0$ is arbitrary) assume that
$S[\\{x_{n}\\}_{n=1}^{\infty}]\subset[\\{x_{n}\\}_{n=1}^{\infty}]$ and so the
restriction of $S$ to $[\\{x_{n}\\}_{n=1}^{\infty}]$ is equivalent to a
necessarily non compact operator on $\ell_{p}$ and hence $i_{p}$ factors
through $S$.

This completes the proof of the case $2<p<\infty$. Notice that by the K-P
result, an operator on $L^{p}$ is strictly singular if and only if neither
$i_{p}$ nor $i_{2}$ factors through the operator. Thus any closed ideal for
which there is a ${\mathcal{M}}$–approximate identity must be contained in the
ideal of strictly singular operators. However, we do not know whether any of
these $2^{2^{\aleph_{0}}}$ closed ideals other than ${\mathcal{K}}$ has a
${\mathcal{M}}$–left approximate identity.

Case 3: $1<p<2$.

Let ${\mathcal{I}}$ be any non trivial ideal in $L(L^{p})$ other than
${\mathcal{K}}$. We show that ${\mathcal{I}}$ does not have a
${\mathcal{M}}$–left approximate identity. By duality from the $p>2$ case we
know that there is $T\in{\mathcal{I}}$ such that $i_{p,2}$ factors through
$T$. Let $\\{w_{n}\\}_{n=1}^{\infty}$ be the Walsh functions. The mapping
$\delta_{n}\mapsto w_{n}$ extends to a contraction from $\ell^{2}$ into
$L^{p}$ (here $(\\{\delta_{n}\\}_{n=1}^{\infty})$ is the unit vector basis for
$\ell^{2}$). Thus there is another operator $S\in{\mathcal{I}}$ such that
$\\{w_{n}\\}_{n=1}^{\infty}\subset SB_{L^{p}}$. Thus if there is a
${\mathcal{M}}$–left approximate identity for ${\mathcal{I}}$, then for each
$\varepsilon>0$ there is $U\in{\mathcal{M}}$ such that
$\sup_{n}\|w_{n}-Uw_{n}\|<\varepsilon$. Another result from Andrew’s paper
[And] says that such a $U$ cannot be $L^{p}-$-singular if $\varepsilon<1$. But
it was proven in [JMST, Theorem 9.1] that no non $L^{p}$–singular operator on
$L^{p}$ is in ${\mathcal{M}}$ (this just says that an operator on $L^{p}$ that
preserves some isomorph of $L^{p}$ must send some isomorph of $L^{p}$
isomorphically onto a complemented subspace of $L^{p}$). This completes the
proof of the case $1<p<2$, but we remark that one could avoid quoting the
result from [JMST] by understanding what Andrews proved but did not state
explicitly in [And]. ∎

By using duality we can summarize the main things that were proved in Theorem
2.1 as Corollary 2.2. In addition to background that has already been stated,
we need the fact [Weis] that an operator in $L(L^{p})$, $1<p<2$, is
$L^{p}$–singular if it is both $\ell^{p}$–singular and $\ell^{2}$–singular
because this fact does not follow formally by duality from the corresponding
statement for $2<p<\infty$.

######  Corollary 2.2.

Let ${\mathcal{I}}$ be a nontrivial closed ideal in $L(L^{p})$, $1\leq
p<\infty$, other than the ideal of compact operators. Then ${\mathcal{I}}$
does not have a left approximate identity. When $p>1$, ${\mathcal{I}}$ does
not have a right approximate identity. Moreover, if $1<p<2$, then
${\mathcal{I}}$ does not have a ${\mathcal{M}}$–left approximate identity and
when $2<p<\infty$, ${\mathcal{I}}$ does not have a ${\mathcal{M}}$–right
approximate identity. If $1<p<\infty$ and ${\mathcal{I}}$ is not contained in
the ideal of strictly singular operators, then ${\mathcal{I}}$ does not have
either a ${\mathcal{M}}$–left approximate identity or a ${\mathcal{M}}$–right
approximate identity. If $p=1$ and ${\mathcal{I}}$ is not contained in the
ideal of weakly compact operators, then ${\mathcal{I}}$ does not have a either
a ${\mathcal{M}}$–left approximate identity or a ${\mathcal{M}}$–right
approximate identity.

In the proof of Theorem 2.1 we used the fact that there exists $T\in
L(\ell^{p},L^{p})$, $2<p<\infty$, such that $TB_{\ell^{p}}$ contains the
$L^{p}$ normalized Haar basis for $L^{p}$. This fact is immediate from
Proposition 2.3 and the well known fact that for $1<p<\infty$ the space
$\ell^{p}$ is isomorphic to $(\sum_{n=1}^{\infty}\ell^{2}_{n})_{p}$, or to any
other $p$–sum of a sequence of finite dimensional non zero Hilbert spaces
[KP].

Recall the definition of the Haar functions: For $n=0,1,\dots$ and
$i=1,\dots,2^{n}$, denote

$h_{n,i}(t)=\begin{cases}1&\frac{i-1}{2^{n}}\leq t<\frac{2i-1}{2^{n+1}}\\\
-1&\frac{2i-1}{2^{n+1}}\leq t<\frac{i}{2^{n}}.\end{cases}$

As is well known, the Haar system, ${\bf
1}\cup\\{h_{n,i}\\}_{n=0,i=1}^{\infty\ \ \ 2^{n}}$, in its lexicographic
order, forms a Schauder basis for $L^{p}$, $1\leq p<\infty$ and this basis is
unconditional iff $p>1$.

We also introduce the space $(\sum_{n=0}^{\infty}\oplus\ell_{2^{n}}^{2})_{p}$;
i.e., the $\ell_{p}$ sum of Euclidean spaces of dimensions $2^{n}$,
$n=0,1,\dots$. Its natural basis will be denoted
$\\{e_{n,i}\\}_{n=0,i=1}^{\infty\ \ \ 2^{n}}$. In the next proposition we show
that, for $p>2$, the basis to basis map $e_{n,i}\to h_{n,i}/\|h_{n,i}\|_{p}$
extends to a bounded operator.

######  Proposition 2.3.

Let $2<p<\infty$ and define $T$ from the finitely supported sequences in
$(\sum\oplus\ell_{2^{n}}^{2})_{p}$ to $L^{p}$ as the linear extension of
$Te_{n,i}=h_{n,i}/\|h_{n,i}\|_{p}$. Then $T$ is bounded and thus naturally
extends to an operator from $(\sum\oplus\ell_{2^{n}}^{2})_{p}$ to $L^{p}$.

Proof: Recall ([Mul], 1.2.6, page 37) that dyadic BMO is the space of all
formal expansions $\sum_{n,i}b_{n,i}h_{n,i}$ whose norm

$\|\sum_{n,i}b_{n,i}h_{n,i}\|_{BMO}=\sup_{0\leq n<\infty,1\leq i\leq
2^{n}}(2^{n}\sum_{m=n}^{\infty}\sum_{\\{j;|h_{m,j}|\leq|h_{n,i}|\\}}b_{m,j}^{2}2^{-m})^{1/2}$

Is finite. So,

$\|\sum_{n,i}b_{n,i}h_{n,i}\|_{BMO}\leq\sup_{0\leq
n<\infty}(2^{n}\sum_{m=n}^{\infty}2^{-m}\sum_{j=1}^{2^{m}}b_{m,j}^{2})^{1/2}\\\
\leq\sup_{0\leq n<\infty}(2\sup_{m\geq
n}\sum_{j=1}^{2^{m}}b_{m,j}^{2})^{1/2}=2^{1/2}\|\\{b_{m,j}\\}\|_{(\sum\oplus\ell_{2^{n}}^{2})_{\infty}}.$

Recall that, for $1\leq p<\infty$, dyadic $H^{p}$ is the space of all
functions $\sum_{n,i}a_{n,i}h_{n,i}$ for which

$\|\sum_{n,i}a_{n,i}h_{n,i}\|_{H^{p}}=\|(\sum_{n,i}a_{n,i}^{2}h_{n,i}^{2})^{1/2}\|_{H^{p}}$

is finite.

By the duality between dyadic $H^{1}$ and dyadic BMO (see section 1.2 in
[Mul]) we get that for some universal constant $C$ and all coefficients
$\\{a_{n,i}\\}$,

(2.1) $\sum_{n=0}^{\infty}(\sum_{i=1}^{2^{n}}a_{n,i}^{2})^{1/2}\\\ \leq
C\|\sum_{n,i}a_{n,i}h_{n,i}/\|h_{n,i}\|_{L^{1}}\|_{H^{1}}=C\|(\sum_{n,i}a_{n,i}^{2}h_{n,i}^{2}/\|h_{n,i}\|_{L^{1}}^{2})^{1/2}\|_{L^{1}}.$

We now interpolate between $H^{1}$ and $L^{2}$ to get that for some absolute
constant $C$, for all $1<p<2$ and all coefficients $\\{a_{n,i}\\}$,

(2.2)
$\left(\sum_{n=0}^{\infty}(\sum_{i=1}^{2^{n}}a_{n,i}^{2})^{p/2}\right)^{1/p}\\\
\leq
C\left\|\sum_{n,i}a_{n,i}h_{n,i}/\|h_{n,i}\|_{L^{p}}\right\|_{H^{p}}=C\left\|\big{(}\sum_{n,i}a_{n,i}^{2}h_{n,i}^{2}/\|h_{n,i}\|_{L^{p}}^{2}\big{)}^{1/2}\right\|_{L^{p}}.$

More precisely, we use

######  Theorem (2.3.4 in [Mul]).

For all $v=\sum_{n,i}v_{n,i}h_{n,i}\in H^{p},1<p<2$, there exists
$x=\sum_{n,i}x_{n,i}h_{n,i}\in H^{1}$ and $y=\sum_{n,i}y_{n,i}h_{n,i}\in
L^{2}$ satisfying $x_{n,i},y_{n,i}\geq 0$ for all $n,i$,

$|v_{n,i}|=x_{n,i}^{1-\theta}y_{n,i}^{\theta}\ \ \mbox{for all}\ \
n=0,1,\dots,\ \mbox{and}\ i=1,\dots,2^{n}$

and

$\|x\|_{H^{1}}^{1-\theta}\|y\|_{L^{2}}^{\theta}\leq 32\|v\|_{H^{p}},$

where $\theta$ is defined by
$\frac{1}{p}=\frac{1-\theta}{1}+\frac{\theta}{2}$; i.e.,
$\theta=2-\frac{2}{p}$.

To prove (2.2) we use this theorem with
$v_{n,i}=a_{n,i}/\|h_{n,i}\|_{L^{p}}=2^{n/p}a_{n,i}$. So there are non
negative $x_{n,i},y_{n,i}$ such that

$2^{n/p}|a_{n,i}|=x_{n,i}^{1-\theta}y_{n,i}^{\theta}\ \ \mbox{for all}\ \
n=0,1,\dots,\ \mbox{and}\ i=1,\dots,2^{n}$

and

(2.3)
$\|(\sum_{n,i}x_{n,i}^{2}h_{n,i}^{2})^{1/2}\|_{L^{1}}^{1-\theta}\|(\sum_{n,i}y_{n,i}^{2}h_{n,i}^{2})^{1/2}\|_{L^{2}}^{\theta}\leq
32\|\sum_{n,i}a_{n,i}h_{n,i}/\|h_{n,i}\|_{L^{p}}\|_{H^{p}}.$

Now,

$\displaystyle(\sum_{n=1}^{\infty}(\sum_{i=1}^{2^{n}}a_{n,i}^{2})^{p/2})^{1/p}=(\sum_{n}2^{-n}(\sum_{i}x_{n,i}^{2(1-\theta)}y_{n,i}^{2\theta})^{p/2})^{1/p}$
$\displaystyle\phantom{aaaaa}\leq(\sum_{n}2^{-n}(\sum_{i}x_{n,i}^{2})^{\frac{(1-\theta)p}{2}}(\sum_{i}y_{n,i}^{2})^{\frac{\theta
p}{2}})^{1/p}\ \ \mbox{by H\"{o}lder with}\ \
\frac{1}{1-\theta},\frac{1}{\theta}$
$\displaystyle\phantom{aaaaa}=(\sum_{n}(2^{-2n}\sum_{i}x_{n,i}^{2})^{\frac{(1-\theta)p}{2}}(2^{-n}\sum_{i}y_{n,i}^{2})^{\frac{\theta
p}{2}})^{1/p}$
$\displaystyle\phantom{aaaaa}\leq(\sum_{n}(2^{-2n}\sum_{i}x_{n,i}^{2})^{1/2})^{1-\theta}(\sum_{n}2^{-n}\sum_{i}y_{n,i}^{2})^{\frac{\theta}{2}}$
$\displaystyle\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}\ \ \mbox{by
H\"{o}lder with}\ \ \frac{1}{(1-\theta)p},\frac{2}{\theta p}$
$\displaystyle\leq
C^{1-\theta}\|\sum_{n,i}2^{-n}x_{n,i}h_{n,i}/\|h_{n,i}\|_{L^{1}}\|_{H^{1}}^{1-\theta}\|\sum_{n,i}2^{-n/2}y_{n,i}h_{n,i}/\|h_{n,i}\|_{L^{2}}\|_{L^{2}}^{\theta}\
\ \mbox{by}\ (\ref{eq:h1})$
$\displaystyle=C^{1-\theta}\|\sum_{n,i}x_{n,i}h_{n,i}\|_{H^{1}}^{1-\theta}\|\sum_{n,i}y_{n,i}h_{n,i}\|_{L^{2}}^{\theta}$
$\displaystyle<32C\|\sum_{n,i}a_{n,i}h_{n,i}/\|h_{n,i}\|_{L^{p}}\|_{H^{p}}\ \
\mbox{by}\ (\ref{eq:int}).$

The unconditionality of the Haar system in $L^{p}$ implies that the last
quantity is equivalent, with constant depending on $p$ only, to

$\|\sum_{n,i}a_{n,i}h_{n,i}/\|h_{n,i}\|_{L^{p}}\|_{L^{p}}.$

Dualizing again we get the required inequality showing that $T$ is bounded. ∎

Although we focus here on separable $L^{p}(\mu)$ spaces, it is worth remarking
that if $L^{p}(\mu)$ is non separable, then there is a non trivial ideal
${\mathcal{I}}$ in $L(L^{p}(\mu))$ with
${\mathcal{I}}\not={\mathcal{K}}(L^{p}(\mu))$ such that ${\mathcal{M}}$ has a
left approximate identity consisting of contractive projections; namely, let
${\mathcal{I}}$ be the ideal of operators that have separable ranges. Since
every closed sublattice of $L^{p}$ is contractively complemented [LT2, Lemma
1.b.9], the collection of contractive projections onto separable sublattices
forms a left approximate identity for ${\mathcal{I}}$.

##  3\. Right approximate identities for ideals in $L(L^{1})$

It was stated in Corollary 2.2 that no small ideal (that is, contained in the
ideal of strictly singular operators) in $L(L^{1})$) other than the compact
operators has a ${\mathcal{W}}(L^{1})$-right approximate identity. In this
section we consider two of the three known large closed ideals in $L(L^{1})$.
We do not know whether the remaining large ideal, the ideal of Dunford-Pettis
operators, has a right approximate identity. Our guess is that it does not.

Background we use is contained in [DU, section III]. The main facts needed are
that RNP (representable) operators with domain an $L^{1}$ space are the same
as the operators that factor through $\ell^{1}$, that every operator $B$ from
$L^{1}$ to $\ell^{1}$ is an RNP operator, and if $B$ is an RNP operator with
domain $L^{1}$, then there is a partition $(E_{n})_{n=1}^{\infty}$ of $(0,1)$
such that for each $n$ the operator $B_{n}:=B_{|L^{1}(E_{n})}$ is compact
(this is how Lewis and Stegall [LS] showed that RNP operators from $L^{1}$
factor through $\ell^{1}$–that operators from $L^{1}$ into $\ell^{1}$ are RNP
operators was known before their work).

The two large ideals in $L(L^{1})$ that we consider are
$\Gamma_{\ell^{1}}(L^{1})$, the ideal of operators in $L(L^{1})$ that factor
through $\ell^{1}$; and ${\mathcal{M}}(L^{1})$, the unique maximal ideal in
$L(L^{1})$. The ideal ${\mathcal{M}}(L^{1})$ is the class of the operators $T$
in $L(L^{1})$ such that the identity $I_{L^{1}}$ does not factor through $T$.
It was already mentioned in the introduction that ${\mathcal{M}}(L^{1})$ is
indeed an ideal (so that it obviously is maximal and the only maximal ideal)
and that ${\mathcal{M}}(L^{1})$ can also be described as the $L^{1}$-singular
operators on $L^{1}$.

######  Theorem 3.1.

The ideal ${\mathcal{I}}:=\Gamma_{\ell^{1}}(L^{1})$ in $L(L^{1})$ has a right
contractive approximate identity consisting of contractive idempotents.

Proof: Notice that ${\mathcal{I}}$ is a large ideal and is closed because the
RNP operators are closed. To prove that $\mathcal{I}$ has a right approximate
identity consisting of contractive idempotents it is enough to show that there
is a net $\mathcal{N}$ of contractive idempotents (i.e., conditional
expectations) in $\mathcal{I}$ such that for all $B:L^{1}\to\ell^{1}$ and each
$\epsilon>0$ there is a contractive idempotent $P$ in $\mathcal{N}$ such that
$\|BP-B\|\leq\epsilon$.

Let $\mathcal{N}$ be the net of all $P\in L(L^{1})$ such that there exist a
partition $\\{E_{n}\\}_{n=1}^{\infty}$ of $(0,1)$ and, for each $n$, a finite
rank idempotent $P_{n}$ on $L^{1}(E_{n})$ such that, for each $x\in L^{1}$ and
each $n$, $(Px)_{|E_{n}}=P_{n}(1_{E_{n}}x)$.

Since each $P_{n}$, being of finite rank, is in $\mathcal{I}$ (formally,
$P_{n}$ preceded by the restriction to $E_{n}$ and followed by the natural
embedding of $L^{1}(E_{n})$ into $L^{1}$) it is easy to see that $P$ is in
$\mathcal{I}$ (for if $A_{1},A_{2},\dots$ all factor through $\ell^{1}$, then
so does $A:=(A_{1},A_{2},\dots,):L^{1}\to(\sum_{k=1}^{\infty}L^{1})_{1}$,
where $Ax:=(A_{1}x,A_{2}x,\dots)$). Also, $\mathcal{I}$ is clearly a net under
the natural order on idempotents.

Finally, given a $B:L^{1}\to\ell^{1}$ and $\epsilon>0$ take a partition
$(E_{n})$ as in the result of Lewis and Stegall mentioned above and define
$B_{n}:=B_{|L^{1}(E_{n})}$. Since $B_{n}$ is compact there is a contractive
finite rank idempotent $P_{n}$ on ${L^{1}(E_{n})}$ such that
$\|B_{n}-B_{n}P_{n}\|\leq\epsilon$. Define $P$ on $L^{1}$ by
$(Px)_{|E_{n}}:=P_{n}(1_{E_{n}}x)$. ∎

Y. Choi [Choi] has used Theorem 3.1 to give a simple proof that $L(L^{1})$ is
not amenable as a Banach algebra.

In order to see that ${\mathcal{M}}(L^{1})$ does not have a right approximate
identity, we need some further background and notation. Let
$r_{1},r_{2},\dots$, denote the Rademacher functions, considered either as
functions on $[0,1]$ or the coordinate projections on the Cantor group
$\\{-1,1\\}^{N}$ endowed with Haar measure. The ($L^{1}$-normalized) Haar
system (less the constant function), used already in the previous section, can
also be defined by:

$h_{\phi}=r_{1},\quad\quad h_{\delta}=r_{2}(1+\delta r_{1}),\quad\delta=\pm
1,$

and for $n=2,3,\dots$, by induction,

$h_{\delta}=r_{n+1}\prod_{i=1}^{n}(1+\delta_{i}r_{i}),\quad\delta=(\delta_{1},\dots,\delta_{n}),\quad\delta_{i}=\pm
1.$

This system union a non zero constant function spans $L^{1}$ and is a
martingale difference sequence in any order such that
$(\delta_{1},\dots,\delta_{n})$ comes after $(\gamma_{1},\dots,\gamma_{n-1})$
for every $n>1$ and every $\delta_{1},\dots,\delta_{n}$ and
$\gamma_{1},\dots,\gamma_{n-1}$ in $\\{-1,1\\}$ (here and elsewhere when
$n=1$, $(\gamma_{1},\dots,\gamma_{n-1})$ is interpreted as $\phi$). Hence the
Haar system forms a monotone basis of $L^{1}$ in any such order.

######  Claim 3.2.

Let $\varphi:\\{\delta_{1},\dots,\delta_{n}\\}_{n=1,\delta_{i}=\pm
1}^{\infty}\to\\{2,3,\dots\\}$ be a function such that

$\varphi(\delta_{1},\dots,\delta_{n})>\varphi(\gamma_{1},\dots,\gamma_{n-1})$

for every $n>1$ and every $\delta_{1},\dots,\delta_{n}$ and
$\gamma_{1},\dots,\gamma_{n-1}$ in $\\{-1,1\\}$. Define

$k_{\phi}=r_{1},\quad\quad k_{\delta}=r_{\varphi(\delta)}(1+\delta
r_{1}),\quad\delta=\pm 1,$

and for $n=2,3,\dots$, by recursion,

$k_{\delta}=r_{\varphi(\delta_{1},\dots,\delta_{n})}\prod_{i=1}^{n}(1+\delta_{i}r_{\varphi(\delta_{1},\dots,\delta_{i-1})}),\quad\delta=(\delta_{1},\dots,\delta_{n}),\quad\delta_{i}=\pm
1.$

Then $k_{\phi},\\{k_{(\delta_{1},\dots,\delta_{n})}\\}$ has the same joint
distribution as the Haar system
$h_{\phi},\\{h_{(\delta_{1},\dots,\delta_{n})}\\}$.

Proof: Define a map from the $\sigma$-field generated by $r_{1},\dots,r_{n+1}$
into (but not necessarily onto) the $\sigma$-field generated by
$\\{r_{1},r_{\varphi(\eta_{1})},\dots,r_{\varphi(\eta_{1},\dots,\eta_{n})}\\}_{\eta_{1},\dots,\eta_{n}\in\\{-1,1\\}}$
by

$\psi(\\{t;\
r_{1}(t)=\delta_{1},\dots,r_{n}(t)=\delta_{n},r_{n+1}(t)=\delta_{n+1}\\})\\\
=\\{t;\
r_{1}(t)=\delta_{1},r_{\varphi(\delta_{1})}=\delta_{2},\dots,r_{\varphi(\delta_{1},\dots,\delta_{n})}(t)=\delta_{n+1}\\}.$

This is a measure preserving map and
$k_{(\delta_{1},\dots,\delta_{m})}(\psi(t))=h_{(\delta_{1},\dots,\delta_{m})}(t)$
for all $t$, $m=1,\dots,n$ and $\delta_{1},\dots,\delta_{n}\in\\{-1,1\\}$. ∎

######  Lemma 3.3.

Let $\\{x_{k}\\}_{k=1}^{\infty}$ be a sequence in $L^{1}$ such that there is a
sequence $\\{x^{*}_{k}\\}_{k=1}^{\infty}$ of norm one functionals with
$x^{*}_{k}(x_{k})\geq a>0$ for all $k$ and $x^{*}_{k}(x_{l})=0$ for $k\not=l$.
Let $\\{h_{k}\\}_{k=1}^{\infty}$ be an enumeration of the Haar system in
$L^{1}$ and assume that $S(h_{k})=x_{k}$ extends to an operator in $L(L^{1})$.
Assume also that $T$ in $L(L^{1})$ satisfies

$\|ST-S\|=b<a.$

Then $T$ is an Enflo operator; that is, there is a subspace $Y$ of $L^{1}$ so
that the restriction $T_{|Y}$ of $T$ to $Y$ is an isomorphism.

Proof: By the assumptions,

$x^{*}_{k}(STh_{k})\geq x^{*}_{k}(x_{k})-b\geq a-b.$

Write

$Th_{k}=\alpha h_{k}+\sum_{l\not=k}\alpha_{l}h_{l}$

so that

$x^{*}_{k}(STh_{k})=\alpha x^{*}_{k}(Sh_{k})\leq|\alpha|\|S\|$

and hence

$|\alpha|\geq(a-b)/\|S\|.$

It now follows from [LMMS] that $T$ is an Enflo operator. ∎

For $0<\varepsilon<1$, the biased coin operator $T_{\varepsilon}$ is
convolution with the measure $\prod_{i=1}^{\infty}(1+\varepsilon r_{i})$. The
basic source for information about biased coin operators is Bonami’s thesis
[Bon]. Biased coin operators were introduced into Banach space theory by
Rosenthal [Ros1].

######  Theorem 3.4.

Suppose that ${\mathcal{I}}$ is a closed ideal in $L(L^{1})$ and there is
$0<\varepsilon<1$ such that the biased coin operator $T_{\varepsilon}$ is in
${\mathcal{I}}$. Then ${\mathcal{I}}$ does not have a
${\mathcal{M}}(L^{1})$–right approximate identity. In particular, the largest
ideal ${\mathcal{M}}(L^{1})$ does not have a right approximate identity.

Proof: Recall that ${\mathcal{M}}(L^{1})$ is the collection of non Enflo
operators. $T_{\varepsilon}$ is not an Enflo operator [Ros1], so
$T_{\varepsilon}$ is contained in ${\mathcal{M}}(L^{1})$; so the “In
particular” statement is clear. To prove the Theorem itself, by Lemma 3.3 it
is enough to find an operator $S$ in ${\mathcal{I}}$ that sends the Haar
system to a biorthogonal sequence whose biorthogonal functionals are bounded.
Note that $T_{\varepsilon}h_{(\delta_{1},\dots,\delta_{n})}=\varepsilon
r_{n+1}\prod_{n=1}^{n}(1+\varepsilon\delta_{i}r_{i})$, and
$\|T_{\varepsilon}h_{(\delta_{1},\dots,\delta_{n})}\|=\varepsilon$ is indeed
bounded away from zero. However, this sequence does not have a bounded
sequence of biorthogonal functionals. We overcome this difficulty by
considering the action of $T_{\varepsilon}$ on an equivalent system of the
kind discussed in Claim 3.2.

Let the function $\varphi:\\{\delta_{1}\dots,\delta_{n}\\}_{n=1,\delta_{i}=\pm
1}^{\infty}\to\\{2,3,\dots\\}$ in the statement of Claim 3.2 also be one to
one. Note that

$T_{\varepsilon}k_{\delta}=\varepsilon
r_{\varphi(\delta_{1},\dots,\delta_{n})}\prod_{i=1}^{n}(1+\varepsilon\delta_{i}r_{\varphi(\delta_{1},\dots,\delta_{i-1})}).$

It is still the case that $\|T_{\varepsilon}k_{\delta}\|=\varepsilon$, but
now, due to the fact that $\varphi(\delta_{1},\dots,\delta_{n})$ are all
different, the sequence $\\{T_{\varepsilon}k_{\delta}\\}$ is a martingale
difference sequence and thus a montone basic sequence. Consequently, the
biorthogonal functionals to $\\{T_{\varepsilon}k_{\delta}\\}$ all have norm at
most $2/\varepsilon$. Thus we can apply Lemma 3.3, using for the operator $S$
the basis to basis map from the Haar system to the $k$-system followed by
$T_{\varepsilon}$. ∎

## References

  * [AK] F. Albiac and N. J. Kalton, _Topics in Banach space theory_ , Graduate Texts in Mathematics, 233 (2nd edition). Springer, New York, 2016.
  * [And] A. D. Andrew, _Perturbations of Schauder bases in the spaces $C(K)$ and $L^{p}$, $p>1$_, Studia Mathematica 65 (1979), 287–298.
  * [BJ] M. T. Boedihardjo and W. B. Johnson, _On mean ergodic convergence in the Calkin algebras_ , Proc. Amer. Math. Soc. 143 (2015), no. 6, 2451–2457.
  * [Bon] A. Bonami, _Etude des coefficients de Fourier des fonctions de $L^{p}(G)$_, (French) Ann. Inst. Fourier (Grenoble) 20 1970 fasc. 2 335–402 (1971).
  * [BP] D. Blecher and N. C. Phillips, _$L^{p}$ operator algebras with approximate identities I_, Pacific J. Math. 303 (2019) 401–457.
  * [BRS] J. Bourgain, H. P. Rosenthal, and G. Schechtman, _An ordinal $L_{p}$-index for Banach spaces, with application to complemented subspaces of $L_{p}$_, Ann. of Math. (2) 114 (1981), no. 2, 193–228.
  * [Choi] Y. Choi, _A short proof that $B(L_{1})$ is not amenable_, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 6, 1758–1767.
  * [DU] J. Diestel and J. Uhl, _Vector Measures_ , Math. Surveys 15, American Mathematical Society, Providence, RI, 1977, xiii+322 pp.
  * [ES] P. Enflo and T. W. Starbird, _Subspaces of $L^{1}$ containing $L^{1}$_, Studia Math. 65 (1979), no. 2, 203–225.
  * [Joh] W. B. Johnson, _Operators into $L_{p}$ which factor through $\ell_{p}$_, J. London Math. Soc. (2) 14 (1976), no. 2, 333–339.
  * [JMST] W. B. Johnson, B Maurey, G. Schechtman, and L. Tzafriri, _Symmetric structures in Banach spaces_ ’ Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298 pp.
  * [JO] W. B. Johnson and E. W. Odell, _Subspaces of $L_{p}$ which embed into $\ell_{p}$_, Compositio Math. 28 (1974), 37–49.
  * [JPS] William B. Johnson, Gilles Pisier, and Gideon Schechtman, _Ideals in $L(L_{1})$_, Math. Ann. 376 (2020), no. 1–2, 693–-705.
  * [JS] William B. Johnson and Gideon Schechtman, _The number of closed ideals in $L(L_{p})$_, Acta Math. 227 (2021), no. 1, 103–113.
  * [KP] M. I Kadec and A. Pełczyński, _Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$_, Studia Math. 21 1961/1962 161–176.
  * [LMMS] R. Lechner, P. Müller, P. Motakis, and Th. Schlumprecht, _Strategically reproducible bases and the factorization property_ , Israel J. Math. 238 (2020), no. 1, 13–60.
  * [LS] D. .R. Lewis and C. Stegall, _Banach spaces whose duals are isomorphic to $\ell_{1}(\Gamma)$_, J. Functional Analysis 12 (1973), 177–187.
  * [LT2] Joram Lindenstrauss and Lior Tzafriri, _Classical Banach spaces. II. Function spaces_ , Ergebnisse der Mathematik und ihrer Grenzgebiete 97, Springer-Verlag, Berlin-New York, 1979.
  * [Mul] Paul F. X. Müller, _Hardy martingales—stochastic holomorphy, $L^{1}$1 -embeddings, and isomorphic invariants_, New Math. Monogr., 43 Cambridge University Press, Cambridge, 2022, xv+500 pp.
  * [Pie] Pietsch, A. Operator Ideals. Volume 16 of Mathematische Monographien. Deutscher Verlag der Wissenschaften (1978).
  * [Ros1] H. P. Rosenthal, _C_ onvolution by a biased coin, The Altgeld Book 1975/76.
  * [Ros2] H. P. Rosenthal, _Embeddings of $L_{1}$ in $L_{1}$_, Contemp. Math. 26 (1984), 335–349.
  * [Sch] Gideon Schechtman, _Examples of $\mathcal{L}_{p}$-spaces ($1<p\neq 2<\infty$)_, Israel J. Math. 22 (1975), 138–147.
  * [SZ] Thomas Schlumprecht and András Zsák, _The algebra of bounded linear operators on $\ell_{p}\oplus\ell_{q}$ has infinitely many closed ideals_, J. Reine Angew. Math. 735 (2018), 225–247.
  * [Weis] L. Weis, _Approximation by weakly compact operators in $L^{1}$_, Math. Nachr. 119 (1984), 321–326.

William B. Johnson
Department Mathematics
Texas A&M University
College Station, TX, USA
E-mail<EMAIL_ADDRESS>

Gideon Schechtman
Department of Mathematics
Weizmann Institute of Science
Rehovot, Israel
E-mail<EMAIL_ADDRESS>


A flat FLRW dark energy model in $f(Q,C)$-gravity theory with observational
constraints
Anirudh Pradhan1, Archana Dixit2, M. Zeyauddin3, S. Krishnannair4

1Centre for Cosmology, Astrophysics and Space Science (CCASS), GLA University,
Mathura-281 406, Uttar Pradesh, India

2Department of Mathematics, Institute of Applied Sciences and Humanities, GLA
University, Mathura-281 406, Uttar Pradesh, India

3Department of General Studies (Mathematics) Jubail Industrial College Jubail
31961, Saudi Arabia

4Department of Mathematical Sciences, University of Zululand Private Bag X1001
Kwa-Dlangezwa 3886 South Africa

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
3E-mail:uddin<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

In the recently suggested modified non-metricity gravity theory with boundary
term in a flat FLRW spacetime universe, dark energy scenarios of cosmological
models are examined in this study. An arbitrary function, $f(Q,C)=Q+\alpha
C^{2}$, has been taken into consideration, where $Q$ is the non-metricity
scalar, $C$ is the boundary term denoted by $C=\mathring{R}-Q$, and $\alpha$
is the model parameter, for the action that is quadratic in $C$. The Hubble
function $H(z)=H_{0}[c_{1}(1+z)^{n}+c_{2}]^{\frac{1}{2}}$, where $H_{0}$ is
the current value of the Hubble constant and $n,c_{1}$ and $c_{2}$ are
arbitrary parameters with $c_{1}+c_{2}=1$, has been used to examine the dark
energy characteristics of the model. We discovered a transit phase expanding
universe model that is both decelerated in the past and accelerated in the
present, and we discovered that the dark energy equation of state (EoS)
$\omega^{(de)}$ behaves as $(-1\leq\omega^{(de)}<2)$. The Om diagnostic
analysis reveals the quintessence behavior in the present and the cosmological
constant scenario in the late-time universe. Finally, we calculated the
universe’s current age, which was found to be quite similar to recent data.

Keywords: Flat FLRW universe; Dark Energy; $f(Q,C)$-gravity; Transit phase
universe; Observational constraints.

##  1 Introduction

Numerous observational observations [1]-[4] support the so-called $\Lambda$CDM
model, which is the established theory of cosmology. They depict a universe
that is only now beginning to expand at an accelerated rate. The easiest way
to understand dark energy within the context of general relativity (GR), which
is responsible for the late-time acceleration, is through the cosmological
constant $\Lambda$, with a negative-constant equation of state [5]. Despite
Lambda’s remarkable effectiveness in describing current data, its fundamental
interpretation is still far from being understood, largely because of the
fine-tuning problem inherent in the energy of the vacuum [6]. Modified gravity
theories, such as $f(R)$, $f(T)$, and $f(Q)$ theories [7]-[11], have drawn a
lot of attention recently among all potential explanations for the effects of
dark energy (see, for example, [12, 13]).

A fascinating possibility that has just lately been researched is the
gravitational interaction mediated by non-metricity, when curvature and
torsion are disappearing [14, 15, 16]. Since gravity can be regarded as a
gauge theory without explicitly assuming the validity of the Equivalence
Principle, this approach can be vital for understanding gravity at its most
fundamental level. The cosmic speed increase that results from the intrinsic
implications of a different geometry than the Riemannian one can be viewed
from new angles by looking into the $f(Q)$ theories, where $Q$ is the non-
metricity scalar. A power-law function is assumed in the analysis of the
connecting matter in $f(Q)$ gravity by [17] and a model-independent
reconstruction approach is recently employed to investigate cosmological
characteristics in [18]. Nonmetricity formulation of general relativity and
its scalar-tensor extension is presented in [19] and General relativity with
spin and torsion is investigated in [20]. A Covariant formulation of $f(Q)$
theory is given in [21] and an extension of $f(Q)$ gravity in the form of
$f(Q,T)$ theory is proposed in [22].

The above-mentioned novel classes of modified gravity exist in the curvature,
torsional, and non-metricity situations, despite similarities between the non-
modified theories at the level of equations. The reason for this is that when
the torsion scalar $T$ and the non-metricity scalar $Q$ are used in place of
the standard Levi-Civita Ricci scalar $\mathring{R}$ of general relativity,
arbitrary functions $f(\mathring{R})$, $f(T)$, and $f(Q)$ no longer differ by
a complete derivative. Recently, a new extension theory of $f(Q)$ gravity is
proposed in [23] by adding the boundary terms $C=\mathring{R}-Q$ and is called
as $f(Q,C)$ gravity theory. Recently, the role of boundary terms in $f(Q)$
theory is investigated by [24] in details. These initial studies suggest that
the boundary term may behaves role of dark energy factor. Therefore, motivated
by above newly proposed theory, in this paper, we will investigate behaviour
of dark energy cosmological models in $f(Q,C)$ gravity with perfect-fluid.

Here, we analyze the dynamical dark energy behavior of dark energy models in
the current research by investigating the EoS parameters $\omega$ in various
contexts by employing a particular Hubble function, which was inspired by the
aforementioned studies. For this, a Lagrangian action with the formula
$L=\frac{1}{16\pi}[f(Q,C)+16\pi L_{m}]\sqrt{-g}$, and any arbitrary function
$f(Q,C)=Q+\alpha C^{2}$ have been taken into consideration. The $f(Q,C)$
gravity formulation was briefly described in section 2, and the field
equations were derived in section 3. Section 4 displays cosmological solutions
using a specific Hubble function, while Section 5 examines observational
constraints using three datasets, $H(z)$, Union 2.1 and Bined datasets of SNe
Ia. The result analysis and comparison with various cosmological parameters
are covered in section 6, and finally conclusions are mentioned in section 7.

##  2 $f(Q,C)$ Gravity

To examine the cosmological properties of nonmetric gravity, let’s start with
the most general form of the affine connections [19]

${\Gamma^{\lambda}}_{\mu\nu}={\mathring{\Gamma}^{\lambda}}_{\mu\nu}+{K^{\lambda}}_{\mu\nu}+{L^{\lambda}}_{\mu\nu}$
(1)

The metric tensor $g_{\mu\nu}$’s Levi-Civita connection is provided here by

${\mathring{\Gamma}^{\lambda}}_{\mu\nu}\equiv\frac{1}{2}g^{\lambda\beta}(\partial_{\mu}g_{\beta\nu}+\partial_{\nu}g_{\beta\mu}-\partial_{\beta}g_{\mu\nu}),$
(2)

whereas

${K^{\lambda}}_{\mu\nu}\equiv\frac{1}{2}g^{\lambda\beta}(\mathcal{T}_{\mu\beta\nu}+\mathcal{T}_{\nu\beta\mu}+\mathcal{T}\beta\mu\nu)$
(3)
${L^{\lambda}}_{\mu\nu}\equiv\frac{1}{2}g^{\lambda\beta}(-Q_{\mu\beta\nu}-Q_{\nu\beta\mu}+Q_{\beta\mu\nu})$
(4)

are, respectively, the disformation and contortion tensors. The torsion
tensor,
${\mathcal{T}^{\lambda}}_{\mu\nu}\equiv{\Gamma^{\lambda}}_{\mu\nu}-{\Gamma^{\lambda}}_{\nu\mu}$,
and the non-metricity tensor, given by

$Q_{\rho\mu\nu}\equiv\nabla_{\rho}g_{\mu\nu}=\partial_{\rho}g_{\mu\nu}-{\Gamma^{\beta}}_{\rho\mu}g_{\beta\nu}-{\Gamma^{\beta}}_{\rho\nu}g_{\mu\beta}.$
(5)

Thus, some choices made on the linkages will determine the properties of the
metric-affine spacetime. Torsion and curvature are assumed to be disappearing
in our analysis, leading to non-metricity as the source of geometry. The non-
metricity tensor is represented by two different traces,
${Q_{\mu}={Q_{\mu}}^{\alpha}}_{\alpha}$ and
$\tilde{Q}^{\mu}={Q_{\alpha}}^{\mu\alpha}$, depending on the contraction
order. The non-metricity scalar can therefore be defined as [11].

$Q=-\frac{1}{4}Q_{\alpha\beta\mu}Q^{\alpha\beta\mu}+\frac{1}{2}Q_{\alpha\beta\mu}Q^{\beta\mu\alpha}+\frac{1}{4}Q_{\alpha}Q^{\alpha}-\frac{1}{2}Q_{\alpha}\tilde{Q}^{\alpha}.$
(6)

which, for general diffeomorphisms, is an invariant quadratic combination.
Last but not least, the relations can be further obtained by applying the
torsion-free and curvature-free constraints (all values with a $\mathring{()}$
are calculated with respect to the Levi-Civita connection
$\mathring{\Gamma}$):

$\mathring{R}_{\mu\nu}+\mathring{\nabla}_{\alpha}{L^{\alpha}}_{\mu\nu}-\mathring{\nabla}_{\nu}\tilde{L}_{\mu}+\tilde{L}_{\alpha}{L^{\alpha}}_{\mu\nu}-L_{\alpha\beta\nu}{L^{\alpha\beta}}_{\mu}=0,$
(7)
$\mathring{R}+\mathring{\nabla}_{\alpha}(L^{\alpha}-\tilde{L}^{\alpha})-Q=0.$
(8)

Thus, it is evident that since
$Q^{\alpha}-\tilde{Q}^{\alpha}=L^{\alpha}-\tilde{L}^{\alpha}$, we consider the
boundary term from the previous relation [17, 18].

$C=\mathring{R}-Q=-\mathring{\nabla}_{\alpha}(Q^{\alpha}-\tilde{Q}^{\alpha})=-\frac{1}{\sqrt{-g}}\partial_{\alpha}[\sqrt{-g}(Q^{\alpha}-\tilde{Q}^{\alpha})].$
(9)

Now, we consider the following action for $f(Q,C)$ gravity theory [23, 24]

$S=\int{\frac{1}{16\pi}[f(Q,C)+16\pi\mathcal{L}_{m}]\sqrt{-g}d^{4}x}$ (10)

where $f(Q,C)$ is an arbitrary function of non-metricity scalar $Q$ and $C$ is
the boundary term, and $L_{m}$ denotes the matter Lagrangian and $g$ is the
determinant of the metric tensor $g_{\mu\nu}$.

Field equations emerge from variations in the action with respect to the
metric as follows:

$\frac{2}{\sqrt{-g}}\partial_{\lambda}(\sqrt{-g}f_{Q}{P^{\lambda}}_{\mu\nu})+(P_{\mu\alpha\beta}{Q_{\nu}}^{\alpha\beta}-2P_{\alpha\beta\nu}{Q^{\alpha\beta}}_{\mu})f_{Q}-\frac{f}{2}g_{\mu\nu}+\left(\frac{C}{2}g_{\mu\nu}-\mathring{\nabla}_{\mu}\mathring{\nabla}_{\nu}+g_{\mu\nu}\mathring{\nabla}^{\alpha}\mathring{\nabla}_{\alpha}-2{P^{\lambda}}_{\mu\nu}\partial_{\lambda}\right)f_{C}=8\pi
T_{\mu\nu},$ (11)

where $f_{Q}=\frac{\partial{f}}{\partial{Q}}$ and
$f_{C}=\frac{\partial{f}}{\partial{C}}$, and where $T_{\mu\nu}$ is the matter
energy momentum tensor. We observe here that by altering the action with
respect to the affine connection, we may obtain the connection field equation
because the affine connection is independent of the metric tensor [23].

$(\nabla_{\mu}-\tilde{L}_{\mu})(\nabla_{\nu}-\tilde{L}_{\nu})[4(f_{Q}-f_{C}){P^{\mu\nu}}_{\lambda}+{\Delta_{\lambda}}^{\mu\nu}]=0$
(12)

where
${\Delta_{\lambda}}^{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta{\Gamma^{\lambda}}_{\mu\nu}}$
is the hyper momentum tensor [22]. As

$\partial_{\nu}\sqrt{-g}=-\sqrt{-g}\tilde{L}_{\nu}.$ (13)

The previous connection field equation can be re-expressed as while taking the
coincident gauge as

$\partial_{\mu}\partial_{\nu}(\sqrt{-g}[4(f_{Q}-f_{C}){P^{\mu\nu}}_{\lambda}+{\Delta_{\lambda}}^{\mu\nu}])=0,$
(14)

if this is the case, the field equation that is equivalent to the $f(Q)$
gravity equation will be found [17].
The second and third terms on the right-hand side make up the $f(Q)$ theory,
as we can see from the field equation (11). Following the conventional
computation (for an example, see [21, 22]), we arrive at

$\frac{2}{\sqrt{-g}}\partial_{\lambda}(\sqrt{-g}f_{Q}{P^{\lambda}}_{\mu\nu})+(P_{\mu\alpha\beta}{Q_{\nu}}^{\alpha\beta}-2P_{\alpha\beta\nu}{Q^{\alpha\beta}}_{\mu})f_{Q}=\left(\frac{Q}{2}g_{\mu\nu}+\mathring{G}_{\mu\nu}+2{P^{\lambda}}_{\mu\nu}\partial_{\lambda}\right)f_{Q},$
(15)

where the Einstein tensor associated with the Levi-Civita link is
$\mathring{G}_{\mu\nu}$. Consequently, we may covariantly rewrite the metric
field equation as

$-\frac{f}{2}g_{\mu\nu}+2{P^{\lambda}}_{\mu\nu}\nabla_{\lambda}(f_{Q}-f_{C})+\left(\frac{Q}{2}g_{\mu\nu}+\mathring{G}_{\mu\nu}\right)f_{Q}+\left(\frac{C}{2}g_{\mu\nu}-\mathring{\nabla}_{\mu}\mathring{\nabla}_{\nu}+g_{\mu\nu}\mathring{\nabla}^{\alpha}\mathring{\nabla}_{\alpha}\right)f_{C}=8\pi
T_{\mu\nu}.$ (16)

The effective stress energy tensor is defined as follows:

$T^{eff}_{\mu\nu}=T_{\mu\nu}+\frac{1}{8\pi}\left[\frac{f}{2}g_{\mu\nu}-2{P^{\lambda}}_{\mu\nu}\nabla_{\lambda}(f_{Q}-f_{C})-\frac{Qf_{Q}}{2}g_{\mu\nu}-\left(\frac{C}{2}g_{\mu\nu}-\mathring{\nabla}_{\mu}\mathring{\nabla}_{\nu}+g_{\mu\nu}\mathring{\nabla}^{\alpha}\mathring{\nabla}_{\alpha}\right)f_{C}\right]$
(17)

and as a result, we get

$\mathring{G}_{\mu\nu}=\frac{8\pi}{f_{Q}}T^{eff}_{\mu\nu}$ (18)

As a result, we acquire an additional, effective sector of geometrical origin
within the context of $f(Q,C)$ gravity.

If the function $f$ is linear in $C$, then (16) reduces to the standard field
equation for $f(Q)$ gravity: $f(Q)=f(Q)+\beta C$.

$-\frac{f}{2}g_{\mu\nu}+2{P^{\lambda}}_{\mu\nu}\nabla_{\lambda}f_{Q}+\left(\frac{Q}{2}g_{\mu\nu}+\mathring{G}_{\mu\nu}\right)f_{Q}=8\pi
T_{\mu\nu}.$ (19)

##  3 Field Equations

In this part, $f(Q,C)$ cosmology is introduced, and $f(Q,C)$ gravity is
applied to a cosmological framework. We study the homogeneous and isotropic
flat Friedmann-Robertson-Walker (FRW) spacetime given by the line element in
Cartesian coordinates.

$ds^{2}=-dt^{2}+a^{2}(t)[dx^{2}+dy^{2}+dz^{2}],$ (20)

where the scale factor $a(t)$ is given by its first time derivative, which
yields the Hubble parameter $H(t)=\frac{\dot{a}}{a}$. The corresponding
stress-energy momentum tensor is given by

$T_{\mu\nu}=(\rho^{(m)}+p^{(m)})u_{\mu}u_{\nu}+p^{(m)}g_{\mu\nu}.$ (21)

where $\rho^{(m)},p^{(m)}$ are matter energy density and matter fluid
pressure, respectively, and $u_{\mu}u^{\mu}=-1$ and $u^{\mu}=(0,0,0,-1)$ is
the four-velocity vectors, $g_{\mu\nu}$ is the metric tensor.
As we said in the previous section, the framework of $f(Q,C)$ gravity enables
us to obtain an additional, useful sector of geometrical origin, as shown in
(17). As a result, in a cosmic context, this phrase will be comparable to an
effective dark-energy sector with an energy-momentum tensor.

$T^{(de)}_{\mu\nu}=\frac{1}{f_{Q}}\left[\frac{f}{2}g_{\mu\nu}-2{P^{\lambda}}_{\mu\nu}\nabla_{\lambda}(f_{Q}-f_{C})-\frac{Qf_{Q}}{2}g_{\mu\nu}-\left(\frac{C}{2}g_{\mu\nu}-\mathring{\nabla}_{\mu}\mathring{\nabla}_{\nu}+g_{\mu\nu}\mathring{\nabla}^{\alpha}\mathring{\nabla}_{\alpha}\right)f_{C}\right]$
(22)

In this paper, we consider the case with vanishing affine connection
${\Gamma^{\alpha}}_{\mu\nu}=0$. Hence, one can obtain

$\mathring{G}_{\mu\nu}=-(3H^{2}+2\dot{H})h_{\mu\nu}+3H^{2}u_{\mu}u_{\nu},$
(23) $\mathring{R}=6(2H^{2}+\dot{H}),$ (24) $Q=-6H^{2},$ (25)
$C=\mathring{R}-Q=6(3H^{2}+\dot{H}),$ (26)

where $h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$, and $u_{\nu}=(dt)_{\nu}$. We get
the Friedmann-like equations by incorporating these into the general field
equations (11) as

$3H^{2}=8\pi(\rho^{(m)}+\rho^{(de)})$ (27)
$-(2\dot{H}+3H^{2})=8\pi(p^{(m)}+p^{(de)})$ (28)

where the energy density and pressure of the area of matter considered to be a
perfect fluid are $\rho^{(m)}$ and $p^{(m)}$, respectively, and where the
effective dark-energy density and pressure are given as

$\rho^{(de)}=\frac{1}{8\pi}\left[3H^{2}(1-2f_{Q})-\frac{f}{2}+(9H^{2}+3\dot{H})f_{C}-3H\dot{f}_{C}\right],$
(29)

and

$p(de)=\frac{1}{8\pi}\left[-2\dot{H}(1-f_{Q})-3H^{2}(1-2f_{Q})+\frac{f}{2}+2H\dot{f}_{Q}-(9H^{2}+3\dot{H})f_{C}+\ddot{f}_{C}\right],$
(30)

respectively.
Now, we define the energy conservation equation as

$\dot{\rho}^{(m)}+3H[\rho^{(m)}+p^{(m)}]=0,$ (31)

and

$\dot{\rho}^{(de)}+3H[\rho^{(de)}+p^{(de)}]=0.$ (32)

##  4 Cosmological Solutions

We have two linearly independent field equations (27) and (28) in four
unknowns $H,\rho^{(m)},p^{(m)},f$, hence we need at least two constraints on
these unknowns in order to discover accurate solutions of the field equations.
First, we take the arbitrary quadratic in $C$ gravity function, $f(Q,C)$, as

$f(Q,C)=Q+\alpha C^{2}$ (33)

where $\alpha$ is an arbitrary model free parameter.
Due to the non-linear complexity of its field equations and the requirement to
examine the dynamical behavior of dark energy models under $f(Q,C)$ gravity,
we must adopt a parametrization of either the scale factor $a(t)$ or the
Hubble function $H(t)$. Model independent strategy to examine dark energy
features of expanding universe models is the name of this technique. In this
instance, we apply the parametrization of $H(t)$ as shown in [25], $H\propto
H(a)$, and as a result, we take it to be:

$H(a)=H_{0}\left(\frac{c_{1}}{a^{n}}+c_{2}\right)^{\frac{1}{2}}$ (34)

where $H_{0}$ is the current value of the Hubble parameter $H$ and
$c_{1},c_{2},n$ with $c_{1}+c_{2}=1$ are three positive arbitrary constants.
The recent observational datasets [1]-[4] support this choice of Hubble
function $H$ parametrization since it produces a scale factor that is
consistent with an expanding universe in the transit phase (moving from a
decelerating to an accelerating state). Now that we have the Hubble function
defined as $H=\frac{\dot{a}}{a}$, we can rewrite equation (34) as

$\frac{\dot{a}}{a}=H_{0}\left(\frac{c_{1}}{a^{n}}+c_{2}\right)^{\frac{1}{2}}$
(35)

Integrating Eq. (35), we obtain

$a(t)=\left[\frac{c_{3}e^{H_{0}c_{2}nt}-c_{1}}{c_{2}}\right]^{\frac{1}{n}}$
(36)

where $c_{3}$ is an integrating constant and taking $t=0$, we obtain

$a(0)=\left[\frac{c_{3}-c_{1}}{c_{2}}\right]^{\frac{1}{n}}$ (37)

In standard convention, we take $a(0)=1$ and hence, we find
$c_{3}=c_{1}+c_{2}=1$, that implies $c_{1}<1$.
Now, we express the Hubble function $H$ in terms of redshift $z$ using the
relation [26]

$\frac{a_{0}}{a}=1+z,$ (38)

with standard convention $a_{0}=1$, as given below

$H(z)=H_{0}[c_{1}(1+z)^{n}+c_{2}]^{\frac{1}{2}}$ (39)

Also, from Eq. (38), we can obtain the differential coefficients of Hubble
function $\dot{H}$, $\ddot{H}$ and $\dddot{H}$ in terms of $z$ and diffential
coefficients of $H(z)$ with respect to $z$, as

$\dot{H}=-(1+z)HH^{\prime},$ (40)
$\ddot{H}=(1+z)H^{2}H^{\prime}+(1+z)^{2}HH^{\prime
2}+(1+z)^{2}H^{2}H^{\prime\prime},$ (41)
$\dddot{H}=-(1+z)H^{3}H^{\prime}-4(1+z)^{2}H^{2}H^{\prime
2}-3(1+z)^{2}H^{3}H^{\prime\prime}-(1+z)^{3}HH^{\prime
3}-4(1+z)^{3}H^{2}H^{\prime}H^{\prime\prime}-(1+z)^{3}H^{3}H^{\prime\prime\prime}.$
(42)

Now, from Eq. (33), we have $f_{Q}=1$, $f_{C}=2\alpha C$ and using these
expressions in Eq. (29) & (30), we obtain the following expressions dark
energy density $\rho^{(de)}$ and dark fluid pressure $p^{(de)}$ as:

$\rho^{(de)}=\frac{9\alpha
H_{0}^{4}}{16\pi}\left[(36-8n-3n^{2})c_{1}^{2}(1+z)^{2n}+(72-8n-4n^{2})c_{1}c_{2}(1+z)^{n}+36c_{2}^{2}\right]$
(43)

and

$p^{(de)}=-\frac{3\alpha
H_{0}^{4}}{16\pi}\left[(108-36n-33n^{2}+6n^{3})c_{1}^{2}(1+z)^{2n}+(216-36n-24n^{2}+4n^{3})c_{1}c_{2}(1+z)^{n}+108c_{2}^{2}\right]$
(44)

respectively. Hence, the EoS parameter for dark energy is obtained as

$\omega^{(de)}=-1-\frac{6n(n^{2}-4n-2)c_{1}^{2}(1+z)^{2n}+4n(n^{2}-3n-3)c_{1}c_{2}(1+z)^{n}}{3[(36-8n-3n^{2})c_{1}^{2}(1+z)^{2n}+(72-8n-4n^{2})c_{1}c_{2}(1+z)^{n}+36c_{2}^{2}]}$
(45)

Now, we define the equation of state (EoS) parameter $\omega$ for perfect
fluid and dark energy source as $p^{(m)}=\omega^{(m)}\rho^{(m)}$ and
$p^{(de)}=\omega^{(de)}\rho^{(de)}$, respectively, and solving Eq. (31) with
$\omega^{(m)}=constant$, we obtain the matter energy density as

$\rho^{(m)}=\rho_{0}^{(m)}(1+z)^{3(1+\omega^{(m)})}$ (46)

Now, we define the effective EoS parameter $\omega^{eff}$ as

$\omega^{eff}=\frac{p^{(m)}+p^{(de)}}{\rho^{(m)}+\rho^{(de)}}$ (47)

The deceleration parameter $q$ is derived as

$q=-1+\frac{nc_{1}(1+z)^{n}}{2[c_{1}(1+z)^{n}+c_{2}]},$ (48)

##  5 Observational Constraints

###  5.1 Hubble Function

For the best fit values of model parameters in $H(z)$, we considered $46$
Hubble constant datasets of $H(z)$ across redshift $z$ with errors in $H(z)$
that are observed in [27]-[42] using the method of differential age (DA) time
to time (see Table 1). For this investigation, we employed the $\chi^{2}$-test
formula shown below:

$\chi^{2}(n,c_{1},H_{0})=\sum_{i=1}^{i=N}\frac{[(H_{ob})_{i}-(H_{th})_{i}]^{2}}{\sigma_{i}^{2}}$

Where $N$ denotes the total amount of data, $H_{ob},~{}H_{th}$, respectively,
the observed and hypothesized datasets of $H(z)$ and standard deviations are
displayed by $\sigma_{i}$.

Figure 1: The best fit curve of Hubble function $H(z)$. “S.No. | $z$ | $H(z)$ | $\sigma_{H}$ | Reference | S.No. | $z$ | $H(z)$ | $\sigma_{H}$ | Reference
---|---|---|---|---|---|---|---|---|---
1 | $0$ | $67.77$ | $1.30$ | [27] | 24 | $0.4783$ | $80.9$ | $9$ | [41]
2 | $0.07$ | $69$ | $19.6$ | [28] | 25 | $0.48$ | $97$ | $60$ | [29]
3 | $0.09$ | $69$ | $12$ | [40] | 26 | $0.51$ | $90.4$ | $1.9$ | [31]
4 | $0.10$ | $69$ | $12$ | [29] | 27 | $0.57$ | $96.8$ | $3.4$ | [42]
5 | $0.12$ | $68.6$ | $26.2$ | [28] | 28 | $0.593$ | $104$ | $13$ | [39]
6 | $0.17$ | $83$ | $8$ | [29] | 29 | $0.60$ | $87.9$ | $6.1$ | [33]
7 | $0.179$ | $75$ | $4$ | [39] | 30 | $0.61$ | $97.3$ | $2.1$ | [31]
8 | $0.1993$ | $75$ | $5$ | [39] | 31 | $0.68$ | $92$ | $8$ | [39]
9 | $0.2$ | $72.9$ | $29.6$ | [28] | 32 | $0.73$ | $97.3$ | $7$ | [33]
10 | $0.24$ | $79.7$ | $2.7$ | [30] | 33 | $0.781$ | $105$ | $12$ | [39]
11 | $0.27$ | $77$ | $14$ | [29] | 34 | $0.875$ | $125$ | $17$ | [39]
12 | $0.28$ | $88.8$ | $36.6$ | [28] | 35 | $0.88$ | $90$ | $40$ | [29]
13 | $0.35$ | $82.7$ | $8.4$ | [32] | 36 | $0.9$ | $117$ | $23$ | [29]
14 | $0.352$ | $83$ | $14$ | [39] | 37 | $1.037$ | $154$ | $20$ | [30]
15 | $0.38$ | $81.5$ | $1.9$ | [31] | 38 | $1.3$ | $168$ | $17$ | [29]
16 | $0.3802$ | $83$ | $13.5$ | [32] | 39 | $1.363$ | $160$ | $33.6$ | [35]
17 | $0.4$ | $95$ | $17$ | [40] | 40 | $1.43$ | $177$ | $18$ | [29]
18 | $0.004$ | $77$ | $10.2$ | [41] | 41 | $1.53$ | $140$ | $14$ | [29]
19 | $0.4247$ | $87.1$ | $11.2$ | [41] | 42 | $1.75$ | $202$ | $40$ | [35]
20 | $0.43$ | $86.5$ | $3.7$ | [30] | 43 | $1.965$ | $186.5$ | $50.4$ | [30]
21 | $0.44$ | $82.6$ | $7.8$ | [33] | 44 | $2.3$ | $224$ | $8$ | [38]
22 | $0.44497$ | $92.8$ | $12.9$ | [41] | 45 | $2.34$ | $222$ | $7$ | [36]
23 | $0.47$ | $89$ | $49.6$ | [34] | 46 | $2.36$ | $226$ | $8$ |  [37]”
Table 1: Observed values of $H(z)$. Parameter | Value
---|---
$H_{0}$ | $68\pm 0.84653$
$c_{1}$ | $0.36934\pm 0.04197$
$n$ | $2.76902\pm 0.08594$
Reduced $\chi^{2}$ | $0.50647$
Table 2: The best fit values of cosmological parameters in the $\chi^{2}$-test
of $H(z)$ with observed $H(z)$ data with a $95\%$ confidence level of bounds.

Table 2 lists the approximations for the cosmological parameters used in the
$\chi^{2}$-test of $H(z)$, and Figure 1 depicts the best fit shape. In Eq.
(39), the mathematical expression for $H(z)$ is shown. We calculated the
Hubble constant ($H_{0}$) for this analysis as $H_{0}=68\pm
0.84653~{}Km/s/Mpc$, which is in good agreement with recently measured values.
Recently in 2018, the Plank Collaboration estimated the present value of
Hubble constant $H_{0}=67.4\pm 0.5$ km/s/Mpc [43] whereas in 2021, Riess et
al.[44] have obtained as $H_{0}=73.2\pm 1.3$ km/s/Mpc. For a lower $\chi^{2}$
value of $0.50647$, we calculated the best fit values of the model parameters
$c_{1}=0.36934\pm 0.04197$ and $n=2.76902\pm 0.08594$, which are listed in
Table 2.

###  5.2 Apparent Magnitude $m(z)$

One of the main observational tools for researching the evolution of the
cosmos is the relationship between redshift and luminosity distance. Since the
universe is expanding and distant luminous objects’ light is experiencing a
redshift, the luminosity distance ($D_{L}$) is calculated in terms of the
cosmic redshift ($z$). It is provided as

$D_{L}=a_{0}r(1+z).$ (49)

where the radial coordinate of the source $r$, is established by

$r=\int^{r}_{0}dr=\int^{t}_{0}\frac{cdt}{a(t)}=\frac{1}{a_{0}}\int^{z}_{0}\frac{cdz}{H(z)}$
(50)

where we have used $dt=dz/\dot{z},\dot{z}=-H(1+z).$

As a result, the luminosity distance is calculated as follows:

$D_{L}=c(1+z)\int^{z}_{0}\frac{dz}{H(z)}$ (51)

Hence, the apparent magnitude $m(z)$ of a supernova [26, 45, 46] is defined
as:

$m(z)=16.08+5log_{10}\left[\frac{(1+z)H_{0}}{.026}\int^{z}_{0}\frac{dz}{H(z)}\right].$
(52)

We use the most recent collection of supernovae pantheon samples $40$ Bined
data in the ($0.01\leq z\leq 1.7$) range [47, 48] plus union 2.1 compilation
of supernovae data in the range ($0.1\leq z\leq 1.414$) [49]. We have used the
following $\chi^{2}$ formula to constrain different model parameters:

$\chi^{2}(n,c_{1})=\sum_{i=1}^{i=N}\frac{[(m_{ob})_{i}-(m_{th})_{i}]^{2}}{\sigma_{i}^{2}}$

The entire amount of data is denoted by $N$, the observed and theoretical
datasets of $m(z)$ are represented by $m_{ob}$ and $m_{th}$, respectively, and
standard deviations are denoted by $\sigma_{i}$.

a. b.

Figure 2: Best fit curve of apparent magnitude $m(z)$ over redshift $z$ with observational SNe Ia data sets, Union 2.1 and Bined, respectively. Parameter | Value for Union 2.1 data | Value for Bined data
---|---|---
$c_{1}$ | $0.3670\pm 0.03877$ | $0.40\pm 0.032996$
$n$ | $2.83802\pm 0.65173$ | $2.72294\pm 1.34523$
Reduced $\chi^{2}$ | $0.03497$ | $0.05498$
Table 3: The best fit values of cosmological parameters in the $\chi^{2}$-test
of $m(z)$ with observational data from Union 2.1 and Bined data at $95\%$
confidence level of bounds.

The best fit shapes of apparent magnitude $m(z)$ over $z$ are shown in Figure
2a, 2b and its expression is given in Eq. (52). The best fit estimated values
of various cosmological parameters in $\chi^{2}$-test of $m(z)$ with SNe Ia
data are given in Table 3.

##  6 Result Analysis and Discussions

### Deceleration parameter

Figure 3 depicts the geometric evolution of the deceleration parameter $q(z)$,
and its mathematical expression is represented in equation (48). Figure 3
illustrates how $q(z)$ evolves across $(-1\leq z\leq 3)$ and exhibits a
signature-flipping point (transition point). $q(z)$ is an increasing function
of $z$. Our universe model is now in an accelerating phase, according to the
estimated values of the deceleration parameter along the three observational
datasets $H(z)$, Union $2.1$, and Bined, respectively, which are
$q_{0}=-0.4886,-0.4792,-0.4534$. We can observe that as $z\to-1$, $q\to-1$
that reveals the accelerating scenarios of late-time universe and as $z\to 3$,
the $q$ tends to a positive value that reveals the early decelerating
scenarios of the expanding universe. Also, from Eq. (48), we can find as
$z\to\infty$, $q\to-1+\frac{n}{2}$, which reveals that early universe is
decelerating for $n>2$, and accelerating for $n<2$. And also, one can see that
as $z\to 0$, $q\to-1+\frac{nc_{1}}{2}$, that shows the accelerating current
universe for $n<\frac{2}{c_{1}}$ and decelerating current universe for
$n>\frac{2}{c_{1}}$. But the recent observations [1]-[4] suggests the
existence of an universe which is decelerating in past and accelerating in
current and future, and therefore, we can say that $2<n<\frac{2}{c_{1}}$ and
this implies that $c_{1}$ should be $c_{1}<1$ which supports to our values
obtained in observational constraints in Table 2, 3. The transition point is
generally denoted by $z_{t}$ and expressed as

$z_{t}=\left[\frac{c_{2}}{(n-2)c_{1}}\right]^{\frac{1}{n}}-1$ (53)

Figure 3: Variation of deceleration parameter $q(z)$ versus $z$.

According to three observational datasets, the estimated transition values are
$z_{t}=0.7132,0.6463,0.6864$, respectively, which are quite close to the most
recent observed values. The expansion phase transition point is reached at
$z=z_{t}$, $q=0$, when the universe is expanding in an accelerating phase for
$-1\leq z<z_{t}$ and a decelerating phase for $z=z_{t}$, $q=0$. The “High-z
Supernova Search (HZSNS) team” recently discovered $z_{t}=0.46\pm 0.13$ [2, 3,
50] in 2004 and $z_{t}=0.43\pm 0.07$ [50] in 2007. The transition redshift
determined by [26, 51] from the “Supernova Legacy Survey (SNLS)” is $z_{t}\sim
0.6$, which is more consistent with the flat $\Lambda$CDM model
($z_{t}=(2\Omega_{\Lambda}/\Omega_{m})^{\frac{1}{3}}-1\sim 0.66$). $0.60\leq
z_{t}\leq 1.18$, in [52] is an extra limitation.

### Dark energy density and pressure

Dark energy $\rho^{(de)}$ and dark energy pressure $p^{(de)}$ are
mathematically expressed as Eqs. (43) and (44), respectively, and their
geometrical behavior is depicted in Figures 4a and 4b, respectively. Figure 4a
shows that $\rho^{(de)}$ is an increasing function of $z$ over $(-1,0.4)$ and
that its value drops as redshift increases from $z\geq 0.4$. According to
Figure 4b, $p^{(de)}$ is an increasing function of both $z$ and $p^{(de)}<0$
at the present and late stages of its evolution. This negative pressure in the
dark energy component of the cosmos is what drives the acceleration of the
expanding universe. From Eqs. (43) & (44), we may deduce that the density and
pressure of dark energy are given by

$\rho^{(de)}\to\frac{81\alpha
c_{2}^{2}H_{0}^{4}}{4\pi},~{}~{}~{}~{}p^{(de)}\to-\frac{81\alpha
c_{2}^{2}H_{0}^{4}}{4\pi}.$

Here, we have chosen the value of model parameter $\alpha$ for which
$\rho^{(de)}=\Lambda$ and this this gives
$\alpha=\frac{4\pi\Lambda}{81c_{2}^{2}H_{0}^{4}}$, where we have used the
value of $\Lambda=2.036\times 10^{-35}s^{-2}$ which is supported by [5]-[7],
the characteristics of dark energy and pressure, cause of acceleration in
expanding universe. Thus, at late-time universe $\rho^{(de)}\to\Lambda$ and
$p^{(de)}\to-\Lambda$ which reveals the consistency of our universe model with
$\Lambda$CDM model.

a. b.

Figure 4: Behaviour of Dark energy $\rho^{(de)}$ and pressure $p^{(de)}$ over
$z$, respectively.

### Dark energy EoS parameter

The EoS parameter $\omega^{(de)}$ for dark energy factor is defined as

$\omega^{(de)}=\frac{p^{(de)}}{\rho^{(de)}}$ (54)

where $\rho^{(de)}$ and $p^{(de)}$ are given by Eqs. (43) and (44),
respectively. The geometrical evolution of dark energy EoS $\omega^{(de)}$
over redshift $z$ is shown in Figure 5a. From Figure 5a, one can observe that
the values of $\omega^{(de)}$ varies as $(-1<\omega^{(de)}<2)$ over the
redshift $(-1\leq z\leq 0.6)$, and it tends to cosmological constant value
($\omega=-1$) as $z\to-1$ during its evolution. We can easily found that the
present model shows various stages of dark energy models like matter dominated
($\omega=0$), quintessence $\omega>-1$, cosmological constant $\omega=-1$ dark
energy models. We can see that as $z\to-1$, the EoS $\omega^{(de)}\to-1$ along
all three datasets, that shows the tendency of the model to $\Lambda$CDM model
at late-time universe.

a. b.

Figure 5: Evolution of dark energy EoS parameter $\omega^{(de)}$ versus $z$
and $\omega^{(de)}$ versus deceleration parameter $q(z)$, respectively.

Figure 5b represents the evolution of EoS parameter $\omega^{(de)}$ over
deceleration parameter $q(z)$ and we observe that as $q\to-1$, the EoS
$\omega^{(de)}\to-1$. The Figure 5b shows the evolution of $\omega^{(de)}$ in
various phases of the expanding universe. The estimated present values of dark
energy EoS parameter $\omega^{(de)}=-0.5885,-0.5811,-0.5157$, respectively
along three datasets $H(z)$, Union $2.1$, and Bined data. Thus, the current
values of $\omega^{(de)}$ lies in the range of $(-\frac{1}{3},-1)$ that
suggests our universe model is currently in quintessence phase of evolution.
From equation (47), we can obtain the effective EoS parameter $\omega^{eff}$
as

$\omega^{eff}=-1+\frac{2}{3}\frac{nc_{1}(1+z)^{n}}{2[c_{1}(1+z)^{n}+c_{2}]},$
(55)

Figure 6: Evolution of effective EoS parameter $\omega^{eff}$ over $z$.

The geometrical evolution of the effective EoS parameter $\omega^{eff}$ is
shown in figure 6 and we see that as $z\to-1$, then $\omega^{eff}\to-1$ that
depicted the tendency of the model towards $\Lambda$CDM model. Also, the
estimated present values of effective EoS parameter is found as
$\omega^{eff}=-0.6591,-0.6528,-0.6369$ along three observational datasets
respectively, that represents the consistency of the model with observational
datasets. Also, we can find the value of EoS parameter $\omega^{(m)}$ for
matter source as $\omega^{eff}=\omega^{(m)}+\omega^{(de)}$ and this implies
$\omega^{(m)}=-0.0706,-0.0717,-0.1212$, respectively for three observational
datasets.

### Om diagnostic analysis

The way the Om diagnostic function behaves [53] makes it easier to categorize
ideas about cosmic dark energy. The Om diagnostic function for a spatially
homogeneous world is given as

$Om(z)=\frac{\left(\frac{H(z)}{H_{0}}\right)^{2}-1}{(1+z)^{3}-1},$ (56)

where $H_{0}$ represents the current Hubble parameter $H(z)$ value as given in
Eq. (39). Quintessence motion is indicated by a negative slope of $Om(z)$,
whereas phantom motion is indicated by a positive slope. The constant $Om(z)$
stands in for the $\Lambda$CDM model.

Using Eq. (39) in (56), we obtain

$Om(z)=\frac{[c_{1}(1+z)^{n}+c_{2}]-1}{(1+z)^{3}-1}$ (57)

Figure 7 depicts the geometric development of the $Om(z)$ function, while Eq.
(57) provides its mathematical expression. According to Figure 7, the
evolution of the Om diagnostic function has negative slopes for all three
datasets, indicating that the model’s current state is essential. Its trend to
a constant value also indicates the $\Lambda$CDM dark energy model. This
demonstrates the accuracy and usefulness of our model.

Figure 7: Evolution of Om diagnostic function $Om(z)$ over $z$.

### Age of the universe

The age of the cosmic universe is obtained by

$t_{0}-t=\int_{0}^{z}\frac{dz}{(1+z)H(z)}$ (58)

where $H(z)$ is given by Eq. (39). Using this in (58), we have

$H_{0}(t_{0}-t)=\lim_{z\to\infty}\int_{0}^{z}\frac{dz}{(1+z)[\lambda(1+z)^{n}+k]^{\frac{1}{2}}}$
(59)

We can find that as $z\to\infty$, $H_{0}(t_{0}-t)$ tends to a constant value
that represents the cosmic age of the universe, $H_{0}(t_{0}-t)\to
H_{0}t_{0}=0.98458,0.96240,0.97831$, respectively, along three datasets. The
present cosmic age of the universe is estimated as $t_{0}=14.16,13.84,14.07$
Gyrs, respectively along three observational datasets, which are very closed
to observational estimated values.

##  7 Conclusions

In the recently suggested modified non-metricity gravity theory with boundary
term in a flat FLRW spacetime universe, dark energy scenarios of cosmological
models are examined in this study. An arbitrary function, $f(Q,C)=Q+\alpha
C^{2}$, has been taken into consideration, where $Q$ is the non-metricity
scalar, $C$ is the boundary term denoted by $C=\mathring{R}-Q$, and $\alpha$
is the model parameter, for the action that is quadratic in $C$. The Hubble
function $H(z)=H_{0}[c_{1}(1+z)^{n}+c_{2}]^{\frac{1}{2}}$, where $H_{0}$ is
the current value of the Hubble constant and $n,c_{1}$ and $c_{2}$ are
arbitrary parameters with $c_{1}+c_{2}=1$, has been used to examine the dark
energy characteristics of the model. We discovered a transit phase expanding
universe model that is both decelerated in the past and accelerated in the
present, and we discovered that the dark energy equation of state (EoS)
$\omega^{(de)}$ behaves as $(-1\leq\omega^{(de)}<2)$. The Om diagnostic
analysis reveals the quintessence behavior in the present and the cosmological
constant scenario in the late-time universe. Finally, we calculated the
universe’s current age, which was found to be quite similar to recent data.
The estimated values of various parameters are shown in below Table 4.

Parameter |  $H(z)$ data | Union 2.1 data | Bined data
---|---|---|---
$c_{1}$ | $0.36934\pm 0.04197$ | $0.3670\pm 0.03877$ | $0.40\pm 0.032996$
$n$ | $2.76902\pm 0.08594$ | $2.83802\pm 0.65173$ | $2.72294\pm 1.34523$
$c_{2}$ | $0.63066$ | $0.6330$ | $0.60$
$H_{0}$ | $68\pm 0.84653$ | $--$ | $--$
$q_{0}$ | $-0.4886$ | $-0.4792$ | $-0.4534$
$t_{0}$ |  $14.16$ Gyrs |  $13.84$ Gyrs |  $14.07$ Gyrs
$z_{t}$ | $0.7132$ | $0.6463$ | $0.6864$
$\omega^{(de)}$ | $-0.5885$ | $-0.5811$ | $-0.5157$
$\omega^{eff}$ | $-0.6591$ | $-0.6528$ | $-0.6369$
$\alpha$ | $2.7291\times 10^{35}$ | $2.8272\times 10^{35}$ | $2.8814\times 10^{35}$
Table 4: Estimated values of various parameters.

The main features of the model are as follows:

  * •

We have obtained a transit phase accelerating universe model with present
value of deceleration parameter $q_{0}=-0.4886,-0.4792,-0.4534$ along three
observational datasets, respectively and found that $q\to-1$ at late-time
universe.

  * •

We have estimated the transition redshift $z_{t}=0.7132,0.6463,0.6864$ along
three datasets, respectively which very closed to recent observed values
[50]-[52].

  * •

We have found that the boundary term behaves just like dark energy candidate,
cause of acceleration in expanding universe.

  * •

We have investigated the dynamical behaviour of dark energy parameter
$\omega^{(de)}$ and found as quintessence region at present
$(\omega^{(de)}>-1)$ and tends to cosmological constant value
$(\omega^{(de)}=-1)$ at late-time universe.

  * •

The Om diagnostic analysis shows the behaviour of quintessence dark energy at
present and tends to $\Lambda$CDM dark energy model at late-time universe.

  * •

The estimated present age of the universe $t_{0}=14.16,13.84,14.07$ Gyrs along
three datasets, respectively and the present value of Hubble constant
$H_{0}=68\pm 0.84653~{}Km/s/Mpc$ are in good agreement with recent observed
values.

Thus, our derived model is interesting to readers and researchers for more
investigations.

## Acknowledgments

The author (A. Pradhan ) is grateful for the assistance and facilities
provided by the University of Zululand, South Africa during a visit where a
part of this article was completed.

##  8 Declarations

### Funding and/or Conflicts of interests/Competing interests

The authors of this article have no conflict of interests. The authors have no
competing interests to declare that are relevant to the content of this
article. The author did not receive support from any organization for the
submitted work.

## References

  * [1] S. Perlmutter et al., Discovery of a supernova explosion at half the age of the Universe,  Nature 391 51 (1998); A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 1009 (1998).
  * [2] D. J. Eisenstein, W. Hu, and M. Tegmark, Cosmic Complementarity: $H_{0}$ and $\Omega_{m}$ from Combining Cosmic Microwave Background Experiments and Redshift Surveys, Astrophys. J. 504 L57 (1998).
  * [3] N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 A6 (2020).
  * [4] B. S. Haridasu, et al., Strong evidence for an accelerating universe, Astron. Astrophys. 600 L1 (2017).
  * [5] V. Sahni and A. Starobinsky, The case for a positive cosmological $\Lambda$-term, Int. J. Mod. Phys. D 9 373 (2000); S. M. Carroll, The cosmological constant, Living Rev. Rel. 4 1 (2001); P. J. E. Peebles and B. Ratra, The cosmological constant and dark energy, Rev. Mod. Phys. 75 559 (2003); E. J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15 1753 (2006).
  * [6] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61 1 (1989); T. Padmanabhan, Cosmological constant-the weight of the vacuum, Phys. Rept. 380 235 (2003).
  * [7] S. Capozziello, R. D’Agostino and O. Luongo, Extended gravity cosmography, Int. J. Mod. Phys. D 28 1930016 (2019); S. Nojiri and S. D. Odintsov, Unified cosmic history in modified gravity: from $F(R)$ theory to Lorentz non-invariant models, Phys. Rept. 505 59 (2011); S. Capozziello, M. De Laurentis, Extended theories of gravity, Phys. Rept. 509 167 (2011); K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov, Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests, Astrophys. Space Sci. 342 155 (2012).
  * [8] A. De Felice, S. Tsujikawa, $f(R)$ Theories, Living Rev. Rel. 13 3 (2010); S. Nojiri, S. D. Odintsov, V. K. Oikonomou, Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution, Phys. Rept. 692 1 (2017).
  * [9] G. R. Bengochea and R. Ferraro, Dark torsion as the cosmic speed-up, Phys. Rev. D 79 124019 (2009); E. Linder, Einstein’s other gravity and the acceleration of the universe, Phys. Rev. D 82 109902 (2010); Y. F. Cai, S. Capozziello, M. De Laurentis, E. N. Saridakis, $f(T)$ teleparallel gravity and cosmology, Rept. Prog. Phys. 79 106901 (2016); R. D’Agostino, O. Luongo, Growth of matter perturbations in nonminimal teleparallel dark energy, Phys. Rev. D 98 124013 (2018).
  * [10] R. D’Agostino, R. C. Nunes, Measurements of $H_{0}$ in modified gravity theories: The role of lensed quasars in the late-time Universe, Phys. Rev. D 101 103505 (2020); A. Bonilla, R. D’Agostino, R. C. Nunes, J. C. N. de Araujo, Forecasts on the speed of gravitational waves at high z, J. Cosm. Astrop. Phys. 03 015 (2020); R. D’Agostino, R. C. Nunes, Probing observational bounds on scalar-tensor theories from standard sirens, Phys. Rev. D 100 044041 (2019).
  * [11] J. Beltrán Jiménez, L. Heisenberg, T. Koivisto, Coincident general relativity, Phys. Rev. D 98, 044048 (2018); J. Beltran Jimenez, L. Heisenberg, T. S. Koivisto and S. Pekar, Cosmology in $f(Q)$ geometry, Phys. Rev. D 101 (2020) 103507.; E. Saridakis et al. Modified Gravity and Cosmology, Springer, Cham, Switzerland (2021).
  * [12] P. J. Peebles and B. Ratra, Cosmology with a Time-Variable “Cosmological Constant”, Astrophys. J. 325 1220 (1988); R. R. Caldwell, R. Dave and P. J. Steinhardt, Cosmological imprint of an energy component with general equation of state, Phys. Rev. Lett. 80 1582 (1998); R. D’Agostino, Holographic dark energy from nonadditive entropy: cosmological perturbations and observational constraints, Phys. Rev. D 99 103524 (2019); R. D’Agostino and O. Luongo, Cosmological viability of a double field unified model from warm inflation, (2021) e-Print:2112.12816 [astro-ph.CO].
  * [13] S. Capozziello, R. D’Agostino and O. Luongo, Cosmic acceleration from a single fluid description, Phys. Dark Univ. 20 1 (2018); K. Boshkayev, R. D’Agostino and O. Luongo, Extended logotropic fluids as unified dark energy models, Eur. Phys. J. C 79 332 (2019); S. Capozziello, R. D’Agostino, R. Giambò, O. Luongo, Effective field description of the Anton-Schmidt cosmic fluid, Phys. Rev. D 99 023532 (2019).
  * [14] F. Bajardi, D. Vernieri, S. Capozziello, Bouncing cosmology in $f(Q)$ symmetric teleparallel gravity, Eur. Phys. J. Plus 135 912 (2020); B. J. Barros, T. Barreiro, T. Koivisto, N. J. Nunes, Testing $F(Q)$ gravity with redshift space distortions, Phys. Dark Univ. 30 100616 (2020); A. Narawade, L. Pati, B. Mishra, S. K. Tripathy, Dynamical system analysis for accelerating models in non-metricity f (Q) gravity, arXiv:2203.14121.
  * [15] I. Ayuso, R. Lazkoz, V. Salzano, Observational constraints on cosmological solutions of $f(Q)$ theories, Phys. Rev. D 103 063505 (2021); N. Frusciante, Signatures of $f(Q)$-gravity in cosmology, Phys. Rev. D 103, 044021 (2021); F. K. Anagnostopoulos, S. Basilakos, E. N. Saridakis, First evidence that non-metricity $f(Q)$ gravity could challenge $\Lambda$CDM, Phys. Lett. B 822 136634 (2021).
  * [16] S. Mandal, D. Wang, P. K. Sahoo, Cosmography in $f(Q)$ gravity, Phys. Rev. D 102, 124029 (2020); G. Mustafa, Z. Hassan, P. H. R. S. Moraes, P. K. Sahoo, Wormhole solutions in symmetric teleparallel gravity, Phys. Lett. B 821, 136612 (2021); R. Solanki, A. De, P. K. Sahoo, Complete dark energy scenario in $f(Q)$ gravity, Phys. Dark Univ. 36 100996 (2022); A. Pradhan, D. C. Maurya and A. Dixit, Dark energy nature of viscus universe in $f(Q)$-gravity with observational constraints, Int. J. Geom. Meth. Mod. Phys. 18, 2150124 (2021); A. Dixit, D. C. Maurya and A. Pradhan, Phantom dark energy nature of bulk-viscosity universe in modified $f(Q)$-gravity, Inter. J. Geom. Meth. Mod. Phys. 19 (2022) 2250198-581; A. Pradhan, A. Dixit and D. C. Maurya, Quintessence Behavior of an Anisotropic Bulk Viscous Cosmological Model in Modified $f(Q)$-Gravity, Symmetry 14 (2022) 2630; M. Koussour, S. H. Shekh, M. Bennai, Anisotropic $f(Q)$ gravity model with bulk viscosity, https://doi.org/10.48550/arXiv.2203.10954.
  * [17] T. Harko et al., Coupling matter in modified $f(Q)$ gravity, Phys. Rev. D 98 (2018) 084043. arXiv:1806.10437 [gr-qc].
  * [18] S. Capozziello and R. D’Agostino, Model-independent reconstruction of $f(Q)$ non-metric gravity, Phys. Lett. B 832 (2022) 137229.
  * [19] L. Järv, M. Rünkla, M. Saal, O. Vilson, Nonmetricity formulation of general relativity and its scalar-tensor extension, Phys. Rev. D 97, 124025 (2018).
  * [20] F. W. Hehl, G. D. Kerlick, P. van der Heyde, General relativity with spin and torsion: Foundations and prospects, Zeitschrift für Naturforschung A 31 111 (1976)
  * [21] D. Zhao, Covariant formulation of $f(Q)$ theory, Eur. Phys. J. C 82 303 (2022).
  * [22] Y. Xu, G. Li, T. Harko and S. D. Liang, $f(Q,T)$ gravity, Eur. Phys. J. C 79 708 (2019).
  * [23] Avik De, Tee-How Loo, and E. N. Saridakis, Non-metricity with boundary terms: $f(Q,C)$ gravity and cosmology, (2023) arxiv:2308.00652v1v1 [gr-qc].
  * [24] S. Capozziello, V. D. Falco, and C. Ferrara, The role of the boundary term in $f(Q,B)$ symmetric teleparallel gravity, (2023) arxiv:2307.13280v1 [gr-qc].
  * [25] D.C. Maurya, Transit cosmological model with specific Hubble parameter in $F(R,T)$ gravity, New Astronomy 77 (2020) 101355.
  * [26] E. J. Copeland, et al., Dynamics of dark energy, Int. J. Mod. Phys. D 15 (2006) 1753.
  * [27] S. Agarwal, R. K. Pandey, A. Pradhan, LRS Bianchi type II perfect fluid cosmological models in normal gauge for Lyra’s manifold, Int. J. Theor. Phys. 50 (2011) 296-307.
  * [28] A. Pradhan, S. Agarwal, G. P. Singh, LRS Bianchi type-I universe in Barber’s second self creation theory, Int. J. Theor. Phys. 48 (2009) 158-166.
  * [29] E. Macaulay, et al., First cosmological results using Type Ia supernovae from the dark energy survey: measurement of the Hubble constant, Mon. Not. R. Astro. Soc. 486 (2019) 2184-2196.
  * [30] C. Zhang, et al., Four new observational $H(z)$ data from luminous red galaxies in the sloan digital sky survey data release seven, Res. Astron. Astrophys. 14 (2014) 1221-1233.
  * [31] D. Stern, et al., Cosmic chronometers: constraining the equation of state of dark energy.I: $H(z)$ measurements, J. Cosmol. Astropart. Phys. 1002 (2010) 008.
  * [32] E. G. Naga, et al., Clustering of luminous red galaxies-IV: Baryon acoustic peak in the line-of-sight direction and a direct measurement of $H(z)$, Mon. Not. R. Astro. Soc. 399 (2009) 1663-1680.
  * [33] D. H Chauang, Y. Wang, Modelling the anisotropic two-point galaxy correlation function on small scales and single-probe measurements of $H(z)$, $D_{A}(z)$ and $f(z)$, $\sigma_{8}(z)$ from the sloan digital sky survey DR7 luminous red galaxies, Mon. Not. R. Astro. Soc. 435 (2013) 255-262.
  * [34] S. Alam, et al., The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample, Mon. Not. R. Astron. Soc. 470 (2017) 2617.
  * [35] A. L. Ratsimbazafy, et al., Age-dating luminous red galaxies observed with the Southern African Large Telescope, Mon. Not. R. Astron. Soc. 467 (2017) 3239.
  * [36] L. Anderson, et al., The clustering of galaxies in the SDSS-III Baryon oscillation Spectro-scopic Survey: baryon acoustic oscillations in the data releases 10 and 11 galaxy samples, Mon. Not. R. Astron. Soc. 441 (2014) 24.
  * [37] M. Moresco, Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z $\equiv$ 2, Mon. Not. R. Astron. Soc. 450 (2015) L16.
  * [38] N. G. Busa, et al., Baryon acoustic oscillations in the Ly$\alpha$ forest of BOSS quasars, Astron. $\&$ Astrophys 552 (2013) A96.
  * [39] M. Moresco, et al., Improved constraints on the expansion rate of the Universe up to $z\sim 1.1$ from the spectroscopic evolution of cosmic chronometers, J. Cosmol. Astropart. Phys. 2012 (2012) 006.
  * [40] J. Simon, L. Verde, R. Jimenez, Constraints on the redshift dependence of the dark energy potential, Phys. Rev. D 71 (2005) 123001.
  * [41] M. Moresco et al., A 6 $\%$ measurement of the Hubble parameter at z $\sim 0.45$ direct evidence of the epoch of cosmic re-acceleration, J. Cosmol. Astropart. Phys. 05 (2016) 014.
  * [42] G. F. R. Ellis, M. A. H. MacCallum, A class of homogeneous cosmological models, Commun. Math. Phys. 12 (1969) 108.
  * [43] Planck Collaboration, Aghanim N, Akrami Y et al (2020) Planck 2018 results. VI. Cosmological parameters. A & A 641:A6. https://doi.org/10.1051/0004-6361/201833910. arXiv:1807.06209 [astroph.CO]
  * [44] Riess AG, Casertano S, Yuan W et al (2021) Cosmic distances calibrated to $1\%$ precision with Gaia EDR3 parallaxes and Hubble Space Telescope photometry of 75 Milky Way Cepheids Confirm Tension with KCDM. ApJ 908(1):L6. https://doi.org/10.3847/20418213/abdbaf. arXiv:2012.08534 [astro-ph.CO]
  * [45] G. K. Goswami, R. N. Dewangan and A. K. Yadav, Anisotropic universe with magnetized dark energy, Astrophys. Space Sci. 361, 119 (2016).
  * [46] G. K. Goswami, R. N. Dewangan, A. K. Yadav and A. Pradhan, Anisotropic string cosmological models in Heckmann-Schucking space-time, Astrophys. Space Sci. 361, 47 (2016).
  * [47] A. K. Camlibel, I. Semiz and M. A. Feyizoglu, Pantheon update on a model-independent analysis of cosmological supernova data, Class. Quantum Grav. 37 (2020) 235001.
  * [48] D. M. Scolnic et al., The complete light-curve sample of spectroscopically confirmed SNe Ia from Pan$-$STARRS1 and cosmological constraints from the combined pantheon sample, Astrophys. J. 859 (2018) 101.
  * [49] Suzuki, N.; Rubin, D.; Lidman, C.; Aldering, G.; Amanullah, R.; Barbary, K.; Barrientos, L. F.; Botyanszki, J.; Brodwin, M.; Connolly, N.; Dawson, K. S. The Hubble space telescope cluster supernova survey. V. Improving the dark-energy constraints above $z>1$ and building an early-type-hosted supernova sample, Astrophys. J. 2012, 746, 85.
  * [50] Spurnova Serach Team collaboration (A. G. Riess et al.), New hubble space telescope discoveries of type Ia supernovae at $z>1$: narrowing constraints on the early behavior of dark energy, Astrophys. J. 659 (2007) 98.
  * [51] P. Astier et al., The Supernova Legacy Survey: measurement of, and w from the first year data set, Astron. Astrophys. 447 (2006) 31-48.
  * [52] J.A.S. Lima, J.F. Jesus, R.C. Santos, M.S.S. Gill, Is the transition redshift a new cosmological number?, arXiv:1205.4688v2[astro-ph.CO], (2012).
  * [53] V. Sahni, A. Shafieloo, A. A. Starobinsky, Two new diagnostics of dark energy, Phys. Rev. D 78 (2008) 103502.