# Necessary conditions for existence of $\Gamma_{n}$-contractions and examples
of $\Gamma_{3}$-contractions

Shubhankar Mandal and Avijit Pal  Department of Mathematics , Indian Institute
of Technology, Bhilai  S<EMAIL_ADDRESS>Department of
Mathematics , Indian Institute of Technology, Bhilai  A.
<EMAIL_ADDRESS>
###### Abstract.

The fundamental result of B. Sz. Nazy states that every contraction has a
coisometric extension and a unitary dilation. The isometric dilation of a
contraction on a Hilbert space motivated whether this theory can be extended
sensibly to families of operators. It is natural to ask whether this idea can
be generalized, where the contraction $T$ is substituted by a commuting
$n$-tuples of operators $(S_{1},\cdots,S_{n})$ acting on some Hilbert space
having $\Gamma_{n}$ as a spectral set. We derive the necessary conditions for
the existence of a $\Gamma_{n}$-isometric dilation for
$\Gamma_{n}$-contractions. Also we discuss an example of a
$\Gamma_{3}$-contraction $(S_{1},S_{2},S_{3})$ acting on some Hilbert space
$\mathcal{H},$ which has a $\Gamma_{3}$-isometric dilation, but it fails to
satisfy the following condition:

$E_{1}^{*}E_{1}-E_{1}E_{1}^{*}=E_{2}^{*}E_{2}-E_{2}E_{2}^{*},$

where $E_{1}$ and $E_{2}$ are the fundamental operators of
$(S_{1},S_{2},S_{3}),$ $(S_{1},S_{2})$ is a pair of commuting contractions and
$S_{3}$ is a partial isometry. Thus, the set of sufficient conditions for the
existence of a $\Gamma_{3}$-isometric dilation breaks down, in general, to be
necessary, even when the $\Gamma_{3}$-contraction $(S_{1},S_{2},S_{3})$ has
the special structure as described above.

###### Key words and phrases:

Symmetrized polydisc, Spectral set, Complete spectral set, Wold decomposition,
Pure isometry, Functional model

###### 2010 Mathematics Subject Classification:

32A60, 32C15, 47A13, 47A15, 47A20, 47A25, 47A45.

##  1\. Introduction

For $n\geq 2,$ let us consider the symmetrization map
$\mathbb{s}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$ defined by

$\mathbb{s}(\mathbb{z})=(s_{1}(\mathbb{z}),\ldots,s_{n}(\mathbb{z})),~{}~{}\mathbb{z}=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n},$

where $s_{i}(\mathbb{z})=\sum_{1\leq k_{1}<\ldots<k_{i}\leq n}z_{k_{1}}\ldots
z_{k_{i}}$ and $s_{0}=1.$ The image
$\Gamma_{n}:=\mathbb{s}(\bar{\mathbb{D}}^{n})$ is called a closed symmetrized
polydisc. We can manifest that the set $\Gamma_{n}$ is not convex but it is
polynomially convex [39]. The open symmetrized polydisc is the set
$\mathbb{G}_{n}:=\mathbb{s}(\mathbb{D}^{n})$ and it’s distinguished boundary
$b\Gamma_{n}$ can be expressed as $\mathbb{s}(\mathbb{T}^{n}),$ the image of
the $n$-torus $\mathbb{T}^{n}$ under the map $\mathbb{s}$ [23].

Let $\mathcal{O}(\Omega)$ denote the algebra of holomorphic functions on some
neighbourhood of the compact set $\Omega$ and $\mathcal{B}(\mathcal{H})$ be
the set of all bounded operators acting on some Hilbert space $\mathcal{H}.$ A
compact set $\Omega\subset\mathbb{C}^{m}$ is said to be a spectral set for a
commuting $m$-tuples of operators $\mathbf{T}=(T_{1},\ldots,T_{m})$ if
$\sigma(\mathbf{T})\subseteq\Omega$ and the homomorphism
$\rho_{\mathbf{T}}:\mathcal{O}(\Omega)\rightarrow\mathcal{B}(\mathcal{H})$ is
contractive. The concepts of spectral set for an operator on an underlying
compact subset of $\mathbb{C}$ was discovered by Von Neumann in a intricate
way. The following theorem says that for any contraction, the closed unit disc
is a spectral set.

######  Theorem 1.1.

[34, Chapter 1, Corollary 1.2] Suppose $T\in\mathcal{B}(\mathcal{H}),$ where
$\mathcal{H}$ is a separable complex Hilbert space. Then

$\|p(T)\|\leq\|p\|_{\infty,\mathbb{D}}:=\sup\\{|p(z)|:|z|<1\\}$

if and only if $\|T\|\leq 1.$

Dilations in operator theory are a technique of characterizing a given
operator as the restriction of well comprehended operator, action on a bigger
Hilbert space, to the original space. Hence the operator on a bigger Hilbert
space is named as a dilation of the original operator. The subsequent theorem
is a slight refined version of Sz.-Nagy dilation theorem [34, Chapter 1,
Theorem 1.1] which says that every contractions has a co-isometric extension
and an unitary dilation.

######  Theorem 1.2.

[34, Chapter 4 , Theorem 4.3] Suppose $T\in\mathcal{B}(\mathcal{H}).$ Then $T$
has a unitary (power) dilation if and only if there exists a unitary operator
$U$ acting on a Hilbert space $\mathcal{K}\supseteq\mathcal{H}$ such that

$P_{\mathcal{H}}\,p(U)_{|\mathcal{H}}=p(T),$

for all polynomials $p.$

Schaffer had constructed the existence of such unitary dilation for a given
contraction $T.$ Obviously, the von Neumann inequality ensued from the
existence of a power dilation via the spectral theorem for unitary operators.

Let $g=\left(\\!(g_{ij})\\!\right)$ be a matrix valued polynomial on $\Omega,$
we call $\Omega$ as a complete spectral set (complete $\Omega$-contraction)
for $\mathbf{T}$, if $\|g(\mathbf{T})\|\leq\|g\|_{\infty,\Omega}$ for all
$g\in\mathcal{A}\otimes\mathcal{M}_{k}(\mathbb{C}),k\geq 1$. We will say that
a domain $\Omega$ has the property $P$ if the following holds: if $\Omega$ is
a spectral set for a commuting $m$-tuples of operators $\mathbf{T},$ then it
is a complete spectral set for $\mathbf{T}.$ A commuting $m$-tuples of
operators $\mathbf{T}$ with $\Omega$ as a spectral set, have a
$\partial\Omega$ normal dilation if there exists a Hilbert space $\mathcal{K}$
containing $\mathcal{H}$ as a subspace and a commuting $m$-tuples of normal
operators $\mathbf{N}=(N_{1},\ldots,N_{m})$ on $\mathcal{K}$ with
$\sigma(N)\subseteq\partial\Omega$ such that
$P_{\mathcal{H}}g(\mathbf{N})\mid_{\mathcal{H}}=g(\mathbf{T})~{}{\rm{for~{}all~{}}}g\in\mathcal{O}(\Omega).$
In 1969, Arveson [1] showed that a commuting $m$-tuples of operators
$\mathbf{T}$ having $\Omega$ as a spectral set for $\mathbf{T},$ admits a
$\partial\Omega$ normal dilation if and only if it satisfies the property $P.$

In $1984,$ J. Agler [3] showed that annulus has the property $P.$ Recently, M.
Dristchell and S. McCullough [22] proved that for a domain of connectivity
$n\geq 2$ does not satisfy the property $P.$ In the multi-variable context
symmetrized bi-disc and bi-disc due to Agler and Young [7] and Ando [34]
respectively, have the property $P$. The first counter example in the multi-
variable contest was given by Parrott[35] which is for $\mathbb{D}^{n}$ for
$n>2.$ As a result, it is natural to ask if contractive linear maps on a
(finite dimensional) normed linear space are necessarily completely
contractive. Using Parrott like homomorphisms G.Misra[27, 28], V. Paulsen
[35], E. Ricard [33] proved that if $\Omega$ is any ball in $\mathbb{C}^{m},$
$m\geq 3,$ then $\Omega$ can not have this property $P.$ It was shown in [26]
that if $B_{1}$ and $B_{2}$ are not simultaneously diagonalizable via unitary,
then $\Omega_{\mathbf{B}}:=\\{(z_{1},z_{2}):\|z_{1}B_{1}+z_{2}B_{2}\|_{\rm
op}<1\\},$ does not have property $P,$ where $\mathbf{B}=(B_{1},B_{2})$ in
$\mathbb{C}^{2}\otimes\mathcal{M}_{2}(\mathbb{C})$ and $B_{1}$ and $B_{2}$ are
independent.

We will proclaim a commuting $n$-tuple of bounded operators
$(S_{1},\ldots,S_{n})$ as a $\Gamma_{n}$-contraction if $\Gamma_{n}$ is the
spectral set. The closed symmetrized polydisc $\Gamma_{n}$ is called a
complete spectral set (complete $\Gamma_{n}-$contraction) for
$(S_{1},...,S_{n})$ if

$\|g(S_{1},\ldots,S_{n})\|\leq\|g\|_{\infty,\Gamma_{n}}~{}~{}{\rm{for~{}~{}all}}~{}~{}g\in\mathcal{A}\otimes\mathcal{M}_{k}(\mathbb{C}),k\geq
1,$

where $g=\left(\\!(g_{ij})\\!\right)$ is a matrix valued polynomial defined on
$\Gamma_{n}$ and

$\|g\|_{\infty,\Gamma_{n}}=\sup\\{\|\left(\\!(g_{ij}(z))\\!\right)\|_{\rm
op}:z\in\Gamma_{n}\\}.$

Now we recollect the definitions of $\Gamma_{n}$-unitary,
$\Gamma_{n}$-isometry and pure $\Gamma_{n}$-isometry from [17].

######  Definition 1.3.

Suppose $(S_{1},\ldots,S_{n})$ is a commuting $n$-tuple of operators acting on
some Hilbert space $\mathcal{H}.$ We call $(S_{1},\ldots,S_{n})$ is

  1. (1)

a $\Gamma_{n}$-unitary if the joint spectrum
$\sigma(S_{1},\ldots,S_{n})\subseteq b\Gamma_{n}$ and $S_{1},\ldots,S_{n}$ are
normal operators on $\mathcal{H}.$

  2. (2)

a $\Gamma_{n}$-isometry if there exists a Hilbert space
$\mathcal{K}\supseteq\mathcal{H}$ and a $\Gamma_{n}$-unitary
$(\tilde{S}_{1},\ldots,\tilde{S}_{n})$ acting on $\mathcal{K}$ such that
$\mathcal{H}$ is a common invariant subspace for
$\tilde{S}_{1},\ldots,\tilde{S}_{n}$ and
$S_{i}=\tilde{S}_{i}\mid_{\mathcal{H}}$ for $i=1,\ldots,n.$

  3. (3)

a $\Gamma_{n}$-co-isometry if $(S_{1}^{*},\ldots,S_{n}^{*})$ is a
$\Gamma_{n}$-isometry.

  4. (4)

a pure $\Gamma_{n}$-isometry if $(S_{1},\ldots,S_{n})$ is a
$\Gamma_{n}$-isometry and $S_{n}$ is a pure isometry.

In section $2,$ we prove the necessary conditions for existence of a
$\Gamma_{n}$-isometric dilation of $\Gamma_{n}$-contractions. In section $3,$
we consider the $\Gamma_{3}$-contractions with some special structure. We also
construct an example of $\Gamma_{3}$-contraction which does have
$\Gamma_{3}$-isometric dilation, but at the same time it would not satisfy the
following condition:

$E_{1}^{*}E_{1}-E_{1}E_{1}^{*}=E_{2}^{*}E_{2}-E_{2}E_{2}^{*},$

where $E_{1}$ and $E_{2}$ are the fundamental operators of
$(S_{1},S_{2},S_{3}).$ Therefore, the set of sufficient conditions which is
described in Theorem (2.8) fail, in general, to be necessary. Still, now, we
are not able to produce an example of $\Gamma_{n}$-contraction, which fails to
satisfy one of these necessary conditions. However, the existence of rational
dilation for a $\Gamma_{n}$-contraction for $n\geq 3,$ is still an open
question.

##  2\. $\Gamma_{n}$-isometric dilation for $\Gamma_{n}$-contractions:
necessary and sufficient conditions:

We begin this section with some basic terminologies that we need further. In
this way we are going to recollect few definitions of spectrum, spectral
radius and numerical radius of an operator. Assume that $T$ is a bounded
linear operator acting on a Hilbert space $\mathcal{H}$. Then spectrum of $T$
is denoted by $\sigma(T)$ and has the succeeding identification

$\sigma(T)=\\{\lambda\in\mathbb{C}|\,\,T-\lambda
I\,\,\text{is}\,\,\text{not}\,\,\text{invertible}\\}.$

Also the numerical radius of a bounded operator $T$ on some Hilbert space
$\mathcal{H}$ explicated as

$\omega(A)=\sup\\{|\langle Ax,x\rangle|:\|x\|=1\\}.$

An easy calculation boils up an inequality: $r(A)\leq\omega(A)$, where
$r(A)=\sup_{\lambda\in\sigma(A)}|\lambda|$ is popularly known as spectral
radius. Let $(S_{1},\ldots,S_{n})$ with $\|S_{n}\|\leq 1,$ be a commuting
$n$-tuple of operators and $D_{S_{n}}=(I-S_{n}^{*}S_{n})^{\frac{1}{2}}$ be the
defect operator of $S_{n}$ and
$\mathcal{D}_{S_{n}}=\overline{\operatorname{Ran}}D_{S_{n}}$ be the closure of
the range of $D_{S_{n}}.$ We now look at the succeeding operator equations

$S_{i}-S_{n-i}^{*}S_{n}=D_{S_{n}}E_{i}D_{S_{n}}~{}~{}{\rm{and}}~{}~{}S_{n-i}-S_{i}^{*}S_{n}=D_{S_{n}}E_{n-i}D_{S_{n}}~{}{\rm~{}for}~{}i=1,\ldots,(n-1).$
(2.1)

These equations are called the fundamental equations and the operators
$E_{i}{,}{\rm{s}}$ for $i=1,\ldots,(n-1)$ are said to be the fundamental
operators for $(S_{1},\ldots,S_{n})$. The following theorem tells the
existence and uniqueness of the fundamental operators $E_{i}^{,}{\rm{s}}$ for
$\Gamma_{n}$-contractions of $(S_{1},\ldots,S_{n}).$

######  Theorem 2.1.

[29, Theorem $4.4$] For $n\geq 2,$ suppose $(S_{1},\ldots,S_{n})$ is a
$\Gamma_{n}$-contraction acting on a Hilbert space $\mathcal{H}.$ Then there
are operators $E_{1},\ldots,E_{n-1}\in\mathcal{B}(\mathcal{D}_{S_{n}})$ which
are unique and satisfies (2.1). Furthermore,
$\omega(E_{i}+E_{n-i}z)\leq\binom{n-1}{i}+\binom{n-1}{n-i}$ for all
$z\in\mathbb{T}.$

For $i=1,2,...,(n-1)$ we define $k(i)=\binom{n-1}{i}+\binom{n-1}{n-i}.$ and
operator pencil $\Phi^{(i)}_{1}$ and $\Phi^{(i)}_{2}$ for a commuting
$n-$tuples of bounded operator $(S_{1},...,S_{n})$ as

$\displaystyle\Phi_{1}^{(i)}\left(\alpha^{i}S_{i},\alpha^{n-i}S_{n-i},\alpha^{n}S_{n}\right)$
$\displaystyle=$ $\displaystyle
k(i)^{2}\left(1-\left.|\alpha^{n}S_{n}\right|^{2}\right)+\left(|\alpha^{i}S_{i}|^{2}-\left.|\alpha^{n-i}S_{n-i}\right|^{2}\right)$
$\displaystyle-k(i)\alpha^{i}\left(S_{i}-|\alpha|^{2n-2i}S^{*}_{n-i}S_{n}\right)-k(i)\bar{\alpha}^{i}\left(S_{i}^{*}-|\alpha|^{2n-2i}S_{n-i}S_{n}^{*}\right)$

and

$\displaystyle\Phi_{2}^{(i)}\left(\alpha^{i}S_{i},\alpha^{n-i}S_{n-i},\alpha^{n}S_{n}\right)$
$\displaystyle=$ $\displaystyle
k(i)^{2}\left(1-\left.|\alpha^{n}S_{n}\right|^{2}\right)+\left(|\alpha^{n-i}S_{n-i}|^{2}-\left.|\alpha^{i}S_{i}\right|^{2}\right)$
$\displaystyle-k(i)\alpha^{n-i}\left(S_{n-i}-|\alpha|^{2i}S^{*}_{i}S_{n}\right)-k(i)\bar{\alpha}^{n-i}\left(S_{n-i}^{*}-|\alpha|^{2i}S_{i}S_{n}^{*}\right).$

These operator pencils are important tool, used for figuring out the structure
of $\Gamma_{n}-$ contraction. The succeeding theorem will give a new
description of $\Gamma_{n}$-isometry in terms of these operator pencils, which
were obtained in [17, Theorem 4.12] and [29, Theorem $5.2$ ].

######  Theorem 2.2.

Suppose $S_{1},\ldots,S_{n}$ are commuting operators acting on a Hilbert space
$\mathcal{H}$ and $\gamma_{i}=\frac{n-i}{n},$ for $i=1,\ldots,n-1.$ Then the
following are equivalent:

  1. (1)

$(S_{1},\ldots,S_{n})$ is a $\Gamma_{n}$-isometry ;

  2. (2)

$(\gamma_{1}S_{1},\ldots,\gamma_{n-1}S_{n-1})$ is $\Gamma_{n-1}$-contraction,
$S_{i}=S_{n-i}^{*}S_{n}$ and $S_{n}$ is a isometry;

  3. (3)

( Wold-Decomposition ): there exist an orthogonal decomposition
$\mathcal{H}=\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ into common invariant
subspaces of $S_{1},\ldots,S_{n}$ such that
$(S_{1}\mid\mathcal{H}_{2},\ldots,S_{n}\mid\mathcal{H}_{2})$ is a pure
$\Gamma_{n}$-isometry and
$(S_{1}\mid\mathcal{H}_{1},\ldots,S_{n}\mid\mathcal{H}_{1})$ is a
$\Gamma_{n}$-unitary;

  4. (4)

$S_{n}$ is a isometry and $(S_{1},\ldots,S_{n})$ is a
$\Gamma_{n}$-contraction;

  5. (5)

$(\gamma_{1}S_{1},\ldots,\gamma_{n-1}S_{n-1})$ is a $\Gamma_{n-1}$-contraction
and

$\Phi^{(i)}_{1}(\beta^{i}S_{i},\beta^{n-i}S_{n-i},\beta^{n}S_{n})=0~{}~{}{\rm
and}~{}~{}\Phi^{(i)}_{2}(\beta^{i}S_{i},\beta^{n-i}S_{n-i},\beta^{n}S_{n})=0~{}~{}{\rm{for~{}~{}all}}~{}~{}\beta\in\mathbb{T}~{}~{}{\rm{and}}~{}~{}i=1,\ldots,(n-1);$

Furthermore, if $r(S_{i})<\binom{n-1}{i}+\binom{n-1}{n-i}$ for
$i=1,\ldots,(n-1),$ then all of the above are equivalent to :

  6. (6)

$(\gamma_{1}S_{1},\ldots,\gamma_{n-1}S_{n-1})$ is a $\Gamma_{n-1}$-contraction
along with $(k(i)\beta^{n}S_{n}-S_{n-i})(k(i)I-\beta^{i}S_{i})^{-1}$ and
$(k(i)\beta^{n}S_{n}-S_{i})(k(i)I-\beta^{n-i}S_{n-i})^{-1}$ are isometries for
$i=1,\ldots,(n-1)$ with $k(i)=\binom{n-1}{i}+\binom{n-1}{n-i}$ and for all
$\beta\in\mathbb{T}$ .

###  2.1. Necessary conditions for existence of a $\Gamma_{n}$-isometry:

#### We now define $\Gamma_{n}$-isometric dilation of
$\Gamma_{n}$-contraction.

######  Definition 2.3.

A commuting $n$-tuple of operators $(V_{1},\ldots,V_{n})$ acting on a Hilbert
space $\mathcal{K}\supseteq\mathcal{H}$ is called a $\Gamma_{n}$-isometric
dilation of a $\Gamma_{n}$-contraction $(S_{1},\ldots,S_{n})$ acting on a
Hilbert space $\mathcal{H}$, if it has the following properties:

  * •

$(V_{1},\ldots,V_{n})$ is $\Gamma_{n}$-isometric;

  * •

$P_{\mathcal{H}}V_{1}^{m_{1}}\ldots
V_{n}^{m_{n}}\mid_{\mathcal{H}}=S_{1}^{m_{1}}\ldots S_{n}^{m_{n}},$ for all
$m_{1},\ldots,m_{n-1},m_{n}\in\mathbb{N}\cup\\{0\\}.$

In the same way, we can define $\Gamma_{n}$-unitary dilation of a
$\Gamma_{n}$-contraction.

######  Proposition 2.4.

Let $(V_{1},\ldots,V_{n})$ be a $\Gamma_{n}$-isometric dilation of a
$\Gamma_{n}$-contraction $(S_{1},\ldots,S_{n})$ acting on a Hilbert space
$\mathcal{H}$. Then $(V_{1},\ldots,V_{n})$ is minimal $\Gamma_{n}$-isometric
dilation of $(S_{1},\ldots,S_{n}).$

###### Proof.

Since $(V_{1},\ldots,V_{n})$ is a $\Gamma_{n}$-isometric dilation of
$(S_{1},\ldots,S_{n})$ and

$\mathcal{K}_{0}=\overline{{\rm{span}}}\\{V_{1}^{m_{1}}\ldots
V_{n}^{m_{n}}h:h\in\mathcal{H}~{}{\rm{and}}~{}m_{1},\ldots,m_{n-1},m_{n}\in\mathbb{N}\cup\\{0\\}\\}.$

Clearly, $\mathcal{K}_{0}$ is an invariant subspace of
$V_{1}^{m_{1}},\ldots,V_{n}^{m_{n}}$ for any non-negative integers
$m_{1},\ldots,m_{n}.$ For $i=1,\dots n,$ set
$V_{1i}=V_{i}\mid_{\mathcal{K}_{0}}.$ Then one can easily verify that

$\mathcal{K}_{0}=\overline{{\rm{span}}}\\{V_{11}^{m_{1}}\ldots
V_{1n}^{m_{n}}h:h\in\mathcal{H}~{}{\rm{and}}~{}m_{1},\ldots,m_{n-1},m_{n}\in\mathbb{N}\cup\\{0\\}\\},$

which implies that for all
$m_{1},\ldots,m_{n-1},m_{n}\in\mathbb{N}\cup\\{0\\},$ we get

$P_{\mathcal{H}}(V_{1}^{m_{1}}\ldots
V_{n}^{m_{n}})h=P_{\mathcal{H}}(V_{11}^{m_{1}}\ldots
V_{1n}^{m_{n}})h~{}{\rm{for~{}all~{}h\in\mathcal{H}}}.$

$(V_{11},\ldots,V_{1n})$ is obtained from the restriction of a
$\Gamma_{n}$-contraction $(V_{1},\ldots,V_{n})$ to a common invariant subspace
$\mathcal{K}_{0},$ therefore, $(V_{11},\ldots,V_{1n})$ is a
$\Gamma_{n}$-contraction. Since $V_{1n}=V_{n}\mid_{\mathcal{K}_{0}}$ and
$V_{n}$ is isometry, which implies that $V_{1n}$ is isometry. Hence, by
Theorem 2.2, $(V_{11},\ldots,V_{1n})$ is a $\Gamma_{n}$-isometry. Thus,
$(V_{11},\ldots,V_{1n})$ is a minimal $\Gamma_{n}$-isometry of
$(S_{1},\ldots,S_{n}).$ ∎

######  Proposition 2.5.

A commuting $n$-tuple of operators $(V_{1},\ldots,V_{n})$ acting on Hilbert
space $\mathcal{K}$ is minimal $\Gamma_{n}$-isometric dilation of a
$\Gamma_{n}$-contraction $(S_{1},\ldots,S_{n})$ if and only if
$(V_{1}^{*},\ldots,V_{n}^{*})$ is a $\Gamma_{n}$-coisometric extension of
$(S_{1}^{*},\ldots,S_{n}^{*}).$

###### Proof.

Since $(V_{1},...,V_{n})$ is minimal $\Gamma_{n}-$isometric dilation of a
$\Gamma_{n}$-contraction $(S_{1},\ldots,S_{n})$ therefore

$\mathcal{K}={\overline{\rm{span}}}\\{V_{1}^{m_{1}}\ldots
V_{n}^{m_{n}}h:h\in\mathcal{H}~{}{\rm{and}}~{}m_{1},\ldots,m_{n}\in\mathbb{N}\cup\\{0\\}\\}.$

Now, we will prove that $S_{i}P_{\mathcal{H}}=P_{\mathcal{H}}V_{i}$ for
$i=1,\ldots,n,$ where $P_{\mathcal{H}}:\mathcal{K}\rightarrow\mathcal{H}$ is
orthogonal projection onto $\mathcal{H}.$ Hence, for $h\in\mathcal{H},$ we
have

$\displaystyle S_{i}P_{\mathcal{H}}(V_{1}^{m_{1}}\ldots
V_{n}^{m_{n}}h)=S_{i}(S_{1}^{m_{1}}\ldots S_{n}^{m_{n}}h)$
$\displaystyle=(S_{1}^{m_{1}}\ldots S_{i}^{m_{i}+1}\ldots
S_{n-1}^{m_{n-1}}S_{n}^{m_{n}}h)$
$\displaystyle=P_{\mathcal{H}}(V_{1}^{m_{1}}\ldots V_{i}^{m_{i}+1}\ldots
V_{n}^{m_{n}}h)$ $\displaystyle=P_{\mathcal{H}}V_{i}(V_{1}^{m_{1}}\ldots
V_{n}^{m_{n}}h)$

which gives $S_{i}P_{\mathcal{H}}=P_{\mathcal{H}}V_{i}$ for $i=1,\ldots,n.$
Also, for any $h\in\mathcal{H}$ and $k\in\mathcal{K},$ we obtain

$\langle S_{i}^{*}h,k\rangle=\langle
P_{\mathcal{H}}S_{i}^{*}h,k\rangle=\langle
S_{i}^{*}h,P_{\mathcal{H}}k\rangle=\langle
h,S_{i}P_{\mathcal{H}}k\rangle=\langle h,P_{\mathcal{H}}V_{i}k\rangle=\langle
V_{i}^{*}h,k\rangle,$

which implies that $S_{i}^{*}=V_{i}^{*}\mid_{\mathcal{H}}$ for $i=1,\ldots,n.$

It is also easy to verify the converse part. This completes the proof. ∎

The next Proposition gives the different characterization of the fundamental
operators $E_{i}$’s of a $\Gamma_{n}$-contraction $(S_{1},\ldots,S_{n})$ which
are described in [29, Theorem $2.1$] and [15, Lemma $4.1$].

######  Proposition 2.6.

The fundamental operators $E_{i}$ of a $\Gamma_{n}$-contraction
$(S_{1},\ldots,S_{n})$ is the unique bounded linear operator $X_{i}$ on
$D_{S_{n}}$ that satisfies the following operator equation:

$D_{S_{n}}S_{i}=X_{i}D_{S_{n}}+X_{n-i}^{*}D_{S_{n}}S_{n}~{}~{}{\rm{for}}~{}~{}i=1,\ldots,(n-1).\
$ (2.2)

We will now prove the necessary conditions for existence of a
$\Gamma_{n}$-isometry of $\Gamma_{n}$-contractions $(S_{1},\ldots,S_{n}).$

######  Theorem 2.7.

Let $(S_{1},\ldots,S_{n})$ be a $\Gamma_{n}$-contractions with fundamental
operators $E_{i}\in\mathcal{B}(D_{S_{n}})$ and $\gamma_{i}=\frac{n-i}{n},$ for
$i=1,\ldots,n-1.$ Each of the following conditions is necessary condition for
existence of a $\Gamma_{n}$-isometry for $(S_{1},\ldots,S_{n}):$

  1. (1)

The $(n-1)$-tuple $(\gamma_{1}E_{1},\ldots,\gamma_{n-1}E_{n-1})$ has joint
dilation to a commuting $(n-1)$-tuple of subnormal operators
$(\gamma_{1}\tilde{S}_{1},\ldots,\gamma_{n-1}\tilde{S}_{n-1}),$ that is, there
exists an isometry $\Theta$ of $\mathcal{D}_{S_{n}}$ into a large Hilbert
space $\mathcal{\tilde{H}}$ such that
$\gamma_{i}E_{i}=\Theta^{*}\gamma_{i}\tilde{S}_{i}\Theta$ for
$i=1,\ldots,(n-1),$ where
$(\gamma_{1}\tilde{S}_{1},\ldots,\gamma_{n-1}\tilde{S}_{n-1})$ can be extended
to a commuting $(n-1)$-tuple of normal operators
$(\gamma_{1}U_{1},\ldots,\gamma_{n-1}U_{n-1})$ whose joint spectrum
$\sigma(\gamma_{1}U_{1},\ldots,\gamma_{n-1}U_{n-1})$ lies in $\Gamma_{n-1}.$

  2. (2)

For $i=1,\ldots,(n-1),$
$(E_{i}^{*}D_{S_{n}}S_{i}-E_{n-i}^{*}D_{S_{n}}S_{n-i})_{|_{\ker
D_{S_{n}}}}=0.$

  3. (3)

For $i=1,\ldots,(n-1),$
$(E_{i}^{*}E_{n-i}^{*}-E_{n-i}^{*}E_{i}^{*})D_{S_{n}}{S_{n}}_{|_{\ker
D_{S_{n}}}}=0.$

###### Proof.

Suppose $(V_{1},\ldots,V_{n})$ on $\mathcal{K}$ is a $\Gamma_{n}$-isometry of
a $\Gamma_{n}$ contraction $(S_{1},\ldots,S_{n}),$ therefore by Proposition
$2.4$ we have

$\mathcal{K}={\overline{\rm{span}}}\\{V_{1}^{m_{1}}\ldots
V_{n}^{m_{n}}h:h\in\mathcal{H}~{}{\rm{and}}~{}m_{1},\ldots,m_{n}\in\mathbb{N}\cup\\{0\\}\\}.$

Also, from Proposition (2.5), we observe that

$(V_{1}^{*},\ldots,V_{n}^{*})\mid_{\mathcal{H}}=(S_{1}^{*},\ldots,S_{n}^{*}).$

The $2\times 2$ block operator matrix representation corresponding to the
decomposition $\mathcal{K}=\mathcal{H}\oplus(\mathcal{K}\ominus\mathcal{H})$
are of the form

$V_{i}=\begin{pmatrix}S_{i}&0\\\
C_{i}&\tilde{S}_{i}\end{pmatrix}~{}~{}{\rm{for}}~{}~{}i=1,\ldots,n.$ (2.3)

Since $V_{n}$ is isometry, the entries $S_{n},C_{n}$ and $\tilde{S}_{n}$ of
its $2\times 2$ block operator matrix satisfy the following operator
identities:

$S_{n}^{*}S_{n}+C_{n}^{*}C_{n}=I_{\mathcal{H}}~{}~{}{\rm{and}}~{}~{}\tilde{S}_{n}^{*}\tilde{S}_{n}=I_{\mathcal{H}}.$
(2.4)

From (2.4) we conclude that there exist an isometry
$\Theta:\mathcal{D}_{S_{n}}\to\mathcal{K}\ominus\mathcal{H}$ such that

$\Theta D_{S_{n}}=C_{n}.$ (2.5)

As $(V_{1},\ldots,V_{n})$ is a $\Gamma_{n}$-isometry, from part $(2)$ of
Theorem (2.2) we have $V_{i}=V_{n-i}^{*}V_{n}$ for $i=1,\ldots,(n-1).$ Thus,
from (2.3) we get

$\begin{pmatrix}S_{i}&0\\\
C_{i}&\tilde{S}_{i}\end{pmatrix}=\begin{pmatrix}S_{n-i}^{*}&C_{n-i}^{*}\\\
0&\tilde{S}_{n-i}^{*}\end{pmatrix}\begin{pmatrix}S_{n}&0\\\
C_{n}&\tilde{S}_{n}\end{pmatrix}=\begin{pmatrix}S_{n-i}^{*}S_{n}+C_{n-i}^{*}C_{n}&C_{n-i}^{*}\tilde{S}_{n}\\\
\tilde{S}_{n-i}^{*}C_{n}&\tilde{S}_{n-i}^{*}\tilde{S}_{n}\end{pmatrix},$

which implies that

$S_{i}-S_{n-i}^{*}S_{n}=C_{n-i}^{*}C_{n},~{}~{}C_{n-i}^{*}\tilde{S}_{n}=0,~{}~{}C_{i}=\tilde{S}_{n-i}^{*}C_{n}~{}~{}{\rm{and}}~{}~{}\tilde{S}_{i}=\tilde{S}_{n-i}^{*}\tilde{S}_{n}~{}~{}{\rm{for}}~{}~{}i=1,\ldots,(n-1).$
(2.6)

From (2.6) and (2.5), it follows immediately that

$D_{S_{n}}E_{i}D_{S_{n}}=S_{i}-S_{n-i}^{*}S_{n}=C_{n-i}^{*}C_{n}=C_{n}^{*}\tilde{S}_{i}C_{n}=D_{S_{n}}\Theta^{*}\tilde{S}_{i}\Theta
D_{S_{n}}~{}~{}{\rm{for}}~{}~{}i=1,\ldots,(n-1).$

By uniqueness of the fundamental operators from, we obtain

$E_{i}=\Theta^{*}\tilde{S}_{i}\Theta~{}~{}~{}~{}{\rm{for}}~{}~{}i=1,\ldots,(n-1).$

We will use Theorem $2.2$ to show that $(\tilde{S}_{1},\ldots,\tilde{S}_{n})$
is a $\Gamma_{n}$-isometry. From (2.4) and (2.6), we get

$\tilde{S}_{i}=\tilde{S}_{n-i}^{*}\tilde{S}_{n}~{}~{}{\rm{and}}~{}~{}\tilde{S}_{n}^{*}\tilde{S}_{n}=I_{\mathcal{H}}.$

Hence it is enough to show that
$(\gamma_{1}\tilde{S}_{1},\ldots,\gamma_{n-1}\tilde{S}_{n-1})$ is a
$\Gamma_{n-1}$-contraction. Since $\Gamma_{n-1}$ is a polynomially convex,
therefore to show
$(\gamma_{1}\tilde{S}_{1},\ldots,\gamma_{n-1}\tilde{S}_{n-1})$ is a
$\Gamma_{n-1}$-contraction, it suffices to work with polynomials rather than
the full algebra $\mathcal{O}(\Gamma_{n-1})$(See Oka-Weil Approximation
Theorem [24, Page 84]). As $(V_{1},\ldots,V_{n})$ is a $\Gamma_{n}$-isometry,
from part $(2)$ of Theorem (2.2), we have
$(\gamma_{1}V_{1},\ldots,\gamma_{n-1}V_{n-1})$ is a
$\Gamma_{n-1}$-contraction. Therefore, for any polynomial $p$, we get

$\displaystyle\|p(\gamma_{1}\tilde{S}_{1},\ldots,\gamma_{n-1}\tilde{S}_{n-1})\|$
$\displaystyle=\|P_{\mathcal{K}\ominus\mathcal{H}}p(\gamma_{1}V_{1},\ldots,\gamma_{n-1}V_{n-1})_{\mid_{\mathcal{K}\ominus\mathcal{H}}}\|$
$\displaystyle\leq\|p(\gamma_{1}V_{1},\ldots,\gamma_{n-1}V_{n-1})\|$
$\displaystyle\leq\|p\|_{\infty,\Gamma_{n-1}},$

which implies that
$(\gamma_{1}\tilde{S}_{1},\ldots,\gamma_{n-1}\tilde{S}_{n-1})$ is a
$\Gamma_{n-1}$-contraction. This shows that
$(\tilde{S}_{1},\ldots,\tilde{S}_{n})$ is a $\Gamma_{n}$-isometry. Therefore,
$(\tilde{S}_{1},\ldots,\tilde{S}_{n})$ has a $\Gamma_{n}$-unitary extension,
say $(U_{1},\ldots,U_{n}).$ By definition of $\Gamma_{n}$-unitary, the joint
spectrum $\sigma(U_{1},\ldots,U_{n})$ lies in the distinguish boundary of
$\Gamma_{n}$ and $U_{1},\ldots,U_{n}$ are normal operators. Note that
$(\gamma_{1}U_{1},\ldots,\gamma_{n-1}U_{n-1})$ is a
$\Gamma_{n-1}$-contraction, because $(U_{1},\ldots,U_{n})$ is a
$\Gamma_{n}$-unitary. Therefore, the joint spectrum of commuting $(n-1)$
normal operators $(\gamma_{1}U_{1},\ldots,\gamma_{n-1}U_{n-1})$ is contained
in $\Gamma_{n-1},$ and item $(1)$ follows.

Since $V_{i}$ and $V_{n-i}$ commute, by equating $(2,1)$\- entry of
$V_{i}V_{n-i}$ with the $(2,1)$\- entry of $V_{n-i}V_{i},$ we observe that

$C_{i}S_{n-i}+\tilde{S}_{i}C_{n-i}=C_{n-i}S_{i}+\tilde{S}_{n-i}C_{i}~{}~{}{\rm{for}}~{}~{}i=1,\ldots,(n-1).$
(2.7)

Hence from (2.6) and (2.7), we have

${S}_{i}^{*}C_{n}S_{i}-\tilde{S}_{n-i}^{*}C_{n}S_{n-i}=(\tilde{S}_{i}\tilde{S}_{i}^{*}-\tilde{S}_{n-i}\tilde{S}_{n-i}^{*})C_{n}~{}~{}{\rm{for}}~{}~{}i=1,\ldots,(n-1).$
(2.8)

By multiplying $\Theta^{*}$ on the left side of (2.8) and using (2.5) and the
formulas for $E_{i}$ for $i=1,\ldots,(n-1),$ we get

$E_{i}^{*}D_{S_{n}}S_{i}-E_{n-i}^{*}D_{S_{n}}S_{n-i}=\Theta^{*}(\tilde{S}_{i}\tilde{S}_{i}^{*}-\tilde{S}_{n-i}\tilde{S}_{n-i}^{*})\Theta
D_{S_{n}}~{}~{}{\rm{for}}~{}~{}i=1,\ldots,(n-1).$

This completes the proof of part $(2).$

We use Proposition (2.6) to prove the part $(3).$ Note that for
$i=1,\ldots,(n-1)$

$\displaystyle\Theta^{*}(\tilde{S}_{i}\tilde{S}_{i}^{*}-\tilde{S}_{n-i}\tilde{S}_{n-i}^{*})\Theta
D_{S_{n}}$ $\displaystyle=E_{i}^{*}D_{S_{n}}S_{i}-E_{n-i}^{*}D_{S_{n}}S_{n-i}$
$\displaystyle=E_{i}^{*}(E_{i}D_{S_{n}}+E_{n-i}^{*}D_{S_{n}}S_{n})-E_{n-i}^{*}(E_{n-i}D_{S_{n}}+E_{i}^{*}D_{S_{n}}S_{n})$
$\displaystyle=(E_{i}^{*}E_{i}-E_{n-i}^{*}E_{n-i})D_{S_{n}}+(E_{i}^{*}E_{n-i}^{*}-E_{n-i}^{*}E_{i}^{*})D_{S_{n}}S_{n}.$

This completes the proof of part $(3).$ Also, from above observations, we
conclude that the part $(2)$ and part $(3)$ are equivalent.

∎

###  2.2. Sufficient conditions for existence of a $\Gamma_{n}$-isometry:

#### The following theorem gives the sufficient conditions for unitary
dilation of $\Gamma_{n}$-contractions.

######  Theorem 2.8.

[29, Theorem $6.6$] Suppose $(S_{1},\ldots,S_{n})$ is a
$\Gamma_{n}$-contraction acting on a Hilbert space $\mathcal{H}.$ The
commuting $(n-1)$-tuple of fundamental operators $(E_{1},\ldots,E_{n-1})$ and
$(F_{1},\ldots,F_{n-1})$ of $(S_{1},\ldots,S_{n})$ and
$(S_{1}^{*},\ldots,S_{n}^{*})$ respectively obey the conditions
$E_{l}E_{n-k}^{*}-E_{k}E_{n-l}^{*}=E_{n-k}^{*}E_{l}-E_{n-l}^{*}E_{k}$ and
$F_{l}^{*}F_{n-k}-F_{k}^{*}F_{n-l}=F_{n-k}F_{l}^{*}-F_{n-l}F_{k}^{*}.$ Let
$\mathcal{K}=\ldots\oplus\mathcal{D}_{S_{n}}\oplus\mathcal{D}_{S_{n}}\oplus\mathcal{D}_{S_{n}}\oplus\mathcal{H}\oplus\mathcal{D}_{S_{n}^{*}}\oplus\mathcal{D}_{S_{n}^{*}}\oplus\mathcal{D}_{S_{n}^{*}}\oplus\ldots$
and let $(R_{1},\ldots,R_{n-1},U)$ be a $\Gamma_{n}$-contraction defined on
$\mathcal{K}$ by

$R_{i}=\left[\begin{array}[]{cccc|c|cccc}\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\iddots\\\
\ldots&E_{i}&E_{n-i}^{*}&0&0&0&0&0&\ldots\\\
\ldots&0&E_{i}&E_{n-i}^{*}&0&0&0&0&\ldots\\\
\ldots&0&0&E_{i}&E_{n-i}^{*}D_{S_{n}}&-E_{n-i}^{*}S_{n}^{*}&0&0&\ldots\\\
\hline\cr\ldots&0&0&0&S_{i}&D_{S_{n}^{*}}F_{n-i}&0&0&\ldots\\\
\hline\cr\ldots&0&0&0&0&F_{i}^{*}&F_{n-i}&0&\ldots\\\
\ldots&0&0&0&0&0&F_{i}^{*}&F_{n-i}&\ldots\\\
\iddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right]$
(2.9)

and

$U=\left[\begin{array}[]{cccc|c|cccc}\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\iddots\\\
\ldots&0&I&0&0&0&0&0&\ldots\\\ \ldots&0&0&I&0&0&0&0&\ldots\\\
\ldots&0&0&0&D_{S_{n}}&-S_{n}^{*}&0&0&\ldots\\\
\hline\cr\ldots&0&0&0&P&D_{S_{n}^{*}}&0&0&\ldots\\\
\hline\cr\ldots&0&0&0&0&0&I&0&\ldots\\\ \ldots&0&0&0&0&0&0&I&\ldots\\\
\iddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right].$
(2.10)

Then $(R_{1},\ldots,R_{n-1},U)$ is a minimal $\Gamma_{n}$-unitary dilation of
$(S_{1},\ldots,S_{n}).$

##  3\.  $\Gamma_{3}$-contractions with some special forms:

Consider $(T_{1},T_{2},V_{3})$ is a commuting $3$-tuple of bounded operators
on some Hilbert space $\mathcal{H},$ where $T_{1}$ and $T_{2}$ are
contractions on $\mathcal{H}$ and $V_{3}$ is an isometry on $\mathcal{H}$.
Suppose $T_{1},T_{2}$ and $V_{3}$ are of the above form, then we will show
that there is a $\Gamma_{3}$-contraction of the form
$\left(\frac{T_{1}+T_{2}+V_{3}}{3},\frac{T_{1}T_{2}+T_{2}V_{3}+V_{3}T_{1}}{3},T_{1}T_{2}V_{3}\right)$
which always dilate.

######  Lemma 3.1.

Suppose $T_{1},T_{2}$ and $V_{3}$ as above. Then $(V_{1},V_{2},V_{3}\oplus
I_{\mathcal{K}\ominus\mathcal{H}})$ is an isometric dilation of
$(T_{1},T_{2},V_{3}),$ where $(V_{1},V_{2})$ is the isometric dilation of
$(T_{1},T_{2})$ on $\mathcal{K}$ and $I_{\mathcal{K}\ominus\mathcal{H}}$ is
the identity operator on $\mathcal{K}\ominus\mathcal{H}.$

###### Proof.

Since $(T_{1},T_{2})$ are commuting contractions on $\mathcal{H},$ by Ando’s
theorem, there exist a Hilbert space $\mathcal{K}$ containing $\mathcal{H}$ as
a subspace such that $T_{i}=P_{\mathcal{H}}{V_{i}}_{|_{\mathcal{H}}}$ for
$i=1,2.$ As $V_{1},V_{2}$ and $V_{3}\oplus I_{\mathcal{K}\ominus\mathcal{H}}$
are isometries, we conclude that $(V_{1},V_{2},V_{3}\oplus
I_{\mathcal{K}\ominus\mathcal{H}})$ is an isometric dilation of
$(T_{1},T_{2},V_{3}).$ This completes the proof. ∎

######  Lemma 3.2.

Let $(V_{1},V_{2},V_{3})$ be a commuting $3$-tuple of isometries on some
Hilbert space $\mathcal{H}.$ Then

$\left(S_{1}=\frac{V_{1}+V_{2}+V_{3}}{3},S_{2}=\frac{V_{1}V_{2}+V_{2}V_{3}+V_{3}V_{1}}{3},S_{3}=V_{1}V_{2}V_{3}\right)$

is $\Gamma_{3}$-isometry.

###### Proof.

Clearly, $S_{3}$ is an isometry as $V_{1},V_{2}$ and $V_{3}$ are isometries.
It is easy to check that $S_{1}=S_{2}^{*}S_{3}$ and $S_{2}=S_{1}^{*}S_{3}.$
Now all we require to show that $(\frac{2}{3}S_{1},\frac{1}{3}S_{2})$ is a
$\Gamma_{2}$-contraction which is equivalent to

$(2-\frac{2}{3}S_{1})^{*}(2-\frac{2}{3}S_{1})-(2\frac{1}{3}S_{2}-\frac{2}{3}S_{1})^{*}(2\frac{1}{3}S_{2}-\frac{2}{3}S_{1})\geq
0.$

Note that

$\displaystyle A$ $\displaystyle=(2-\frac{2}{3}S_{1})^{*}(2-\frac{2}{3}S_{1})$
$\displaystyle=\frac{4}{81}\left(84I-9V_{1}-9V_{2}-9V_{3}-9V_{1}^{*}-9V_{2}^{*}-9V_{3}^{*}+V_{2}^{*}V_{1}+V_{3}^{*}V_{1}+V_{1}^{*}V_{2}+V_{3}^{*}V_{2}+V_{1}^{*}V_{3}+V_{2}^{*}V_{3}\right)$
(3.1)

and

$\displaystyle B$
$\displaystyle=(2\frac{1}{3}S_{2}-\frac{2}{3}S_{1})^{*}(2\frac{1}{3}S_{2}-\frac{2}{3}S_{1})$
$\displaystyle=\frac{4}{81}\left(6I-2V_{1}-2V_{2}-2V_{3}-2V_{1}^{*}-2V_{2}^{*}-2V_{3}^{*}+2V_{2}^{*}V_{1}+2V_{3}^{*}V_{1}+2V_{1}^{*}V_{2}+2V_{3}^{*}V_{2}+2V_{1}^{*}V_{3}+2V_{2}^{*}V_{3}-C\right),$
(3.2)

where
$C=V_{2}^{*}V_{1}V_{3}+V_{3}^{*}V_{1}^{*}V_{2}+V_{1}^{*}V_{2}V_{3}+V_{3}^{*}V_{2}^{*}V_{1}+V_{3}^{*}V_{1}V_{2}+V_{2}^{*}V_{1}^{*}V_{3}.$
From (3) and (3), we have

$\displaystyle A-B$
$\displaystyle=\frac{4}{81}\\{78I-7(V_{1}+V_{2}+V_{3}+V_{1}^{*}+V_{2}^{*}+V_{3}^{*})-V_{2}^{*}V_{1}-V_{3}^{*}V_{1}-V_{1}^{*}V_{2}-V_{3}^{*}V_{2}-V_{1}^{*}V_{3}-V_{2}^{*}V_{3}$
$\displaystyle+V_{2}^{*}V_{1}V_{3}+V_{3}^{*}V_{1}^{*}V_{2}+V_{1}^{*}V_{2}V_{3}+V_{3}^{*}V_{2}^{*}V_{1}+V_{3}^{*}V_{1}V_{2}+V_{2}^{*}V_{1}^{*}V_{3}\\}$
$\displaystyle=\frac{4}{81}\\{24I+7(I-V_{1})^{*}(I-V_{1})+7(I-V_{2})^{*}(I-V_{2})+7(I-V_{3})^{*}(I-V_{3})+(I-V_{2}^{*}V_{1})^{*}(I-V_{2}^{*}V_{1})$
$\displaystyle+(I-V_{3}^{*}V_{1})^{*}(I-V_{3}^{*}V_{1})+(I-V_{2}^{*}V_{3})^{*}(I-V_{2}^{*}V_{3})+(I+V_{2}^{*}V_{1}V_{3})^{*}(I+V_{2}^{*}V_{1}V_{3})$
$\displaystyle+(I+V_{1}^{*}V_{2}V_{3})^{*}(I+V_{1}^{*}V_{2}V_{3})+(I+V_{3}^{*}V_{1}V_{2})^{*}(I+V_{3}^{*}V_{1}V_{2})\\}\geq
0,$

Thus, from item $2$ of the Theorem $2.2$, we conclude that
$(S_{1},S_{2},S_{3})$ is a $\Gamma_{3}$-isometry.

∎

######  Theorem 3.3.

Suppose $(T_{1},T_{2},V_{3})$ is a commuting $3$-tuple of operators acting on
some Hilbert space $\mathcal{H}$ with $T_{1}$ and $T_{2}$ are contractions and
$V_{3}$ is an isometry. Then
$\left(\frac{T_{1}+T_{2}+V_{3}}{3},\frac{T_{1}T_{2}+T_{2}V_{3}+V_{3}T_{1}}{3},T_{1}T_{2}V_{3}\right)$
is $\Gamma_{3}$-contraction and
$\left(\frac{T_{1}+T_{2}+V_{3}}{3},\frac{T_{1}T_{2}+T_{2}V_{3}+V_{3}T_{1}}{3},T_{1}T_{2}V_{3}\right)$
has a $\Gamma_{3}$-isometric dilation.

###### Proof.

Let $(V_{1},V_{2})$ be an Ando isometric dilation of $(T_{1},T_{2}).$ Then
there exist a Hilbert space $\mathcal{K}\supseteq\mathcal{H}$ as a subspace
such that $T_{i}=P_{\mathcal{H}}{V_{i}}_{|_{\mathcal{H}}}$ for $i=1,2.$ Now
from Lemma 3.1, it implies that $(V_{1},V_{2},V_{3}\oplus
I_{\mathcal{K}\ominus\mathcal{H}})$ is an isometric dilation of
$(T_{1},T_{2},V_{3}).$ Set

$\left(S_{1}=\frac{V_{1}+V_{2}+V_{3}\oplus
I_{\mathcal{K}\ominus\mathcal{H}}}{3},S_{2}=\frac{V_{1}V_{2}+V_{2}(V_{3}\oplus
I_{\mathcal{K}\ominus\mathcal{H}})+(V_{3}\oplus
I_{\mathcal{K}\ominus\mathcal{H}})V_{1}}{3},S_{3}=V_{1}V_{2}(V_{3}\oplus
I_{\mathcal{K}\ominus\mathcal{H}})\right).$

Since $\left(S_{1},S_{2},S_{3}\right)$ is $\Gamma_{3}$-isometry, from item $4$
of Theorem $2.2$ it implies that $\left(S_{1},S_{2},S_{3}\right)$ is
$\Gamma_{3}$-contraction and $S_{3}$ is an isometry. We all knows that
$\Gamma_{3}$ is polynomially convex therefore we can use Oka-Weil
Approximation Theorem (See [24, Pg. 84 Theorem ]). So, it is enough to work
with polynomials rather than the full algebra $\mathcal{O}(\Gamma_{3}).$ Let
$p$ be any polynomial we have

$\displaystyle\|p(\frac{T_{1}+T_{2}+V_{3}}{3},\frac{T_{1}T_{2}+T_{2}V_{3}+V_{3}T_{1}}{3},T_{1}T_{2}V_{3})\|$
$\displaystyle=\|P_{\mathcal{H}}p(S_{1},S_{2},S_{3})_{|_{\mathcal{H}}}\|$
$\displaystyle\leq\|p(S_{1},S_{2},S_{3})\|$
$\displaystyle\leq\|p\|_{\infty,\Gamma_{3}}.$

This shows that
$\left(\frac{T_{1}+T_{2}+V_{3}}{3},\frac{T_{1}T_{2}+T_{2}V_{3}+V_{3}T_{1}}{3},T_{1}T_{2}V_{3}\right)$
is $\Gamma_{3}$-contraction. This completes the proof.

∎

To work with more tractable examples, in this section we will consider
$\Gamma_{3}$-contractions $(S_{1},S_{2},S_{3})$, where $S_{3}$ is a partial
isometry. We begin with the following result which appears in [18, Proposition
3.2].

######  Proposition 3.4.

Let $(T_{1},T_{2})$ be a pair of contractions on $\mathcal{H}$ and $T$ be a
partial isometry on $\mathcal{H}.$ Suppose there exist two operators
$E_{1},E_{2}$ in $\mathcal{B}(\mathcal{D}_{T})$ such that

$T_{i}-T_{3-i}^{*}T=D_{T}E_{i}D_{T}~{}~{}{\rm{for}~{}~{}}i=1,2.$

Then

  1. (1)

$\ker T$ is jointly invariant under $(T_{1},T_{2})$ and,

  2. (2)

if we represent the restriction $(T_{1},T_{2})_{|_{\ker T}}$ by
$(D_{1},D_{2}),$ then

    1. (a)

$E_{1}^{*}E_{1}-E_{1}E_{1}^{*}=E_{2}^{*}E_{2}-E_{2}E_{2}^{*}$ if and only if
$D_{1}^{*}D_{1}-D_{1}D_{1}^{*}=D_{2}^{*}D_{2}-D_{2}D_{2}^{*},$ and

    2. (b)

$E_{1}E_{2}=E_{2}E_{1}$ if and only if $D_{1}D_{2}=D_{2}D_{1}.$

The following corollary is a simple outcome of the above Proposition.

######  Corollary 3.5.

Let $(S_{1},S_{2},S_{3})$ be a $\Gamma_{3}$-contraction acting on a Hilbert
space $\mathcal{H}$ having fundamental operators $E_{1}$ and $E_{2}$ with
$(S_{1},S_{2})$ a $2$-tuple of commuting contractions on $\mathcal{H}.$ If
$S_{3}$ is a partial isometry, then

  1. (1)

$\ker S_{3}$ is jointly invariant under $(S_{1},S_{2})$ and,

  2. (2)

$E_{1}E_{2}=E_{2}E_{1}$ and

  3. (3)

if we denote the restriction $(S_{1},S_{2})_{|_{\ker S_{3}}}$ by
$(D_{1},D_{2}),$ then
$E_{1}^{*}E_{1}-E_{1}E_{1}^{*}=E_{2}^{*}E_{2}-E_{2}E_{2}^{*}$ if and only if
$D_{1}^{*}D_{1}-D_{1}D_{1}^{*}=D_{2}^{*}D_{2}-D_{2}D_{2}^{*}.$

We will now produce an example of $\Gamma_{3}$ contraction
$(S_{1},S_{2},S_{3})$ which satisfies all conditions in Corollary (3.5).

######  Example 1.

Consider the following triple of commuting operators on $H^{2}\oplus H^{2}:$

$(T_{1},T_{2},T_{3})=\left(\left(\begin{smallmatrix}0&0\\\
I_{H^{2}}&0\end{smallmatrix}\right),\left(\begin{smallmatrix}T_{z}&0\\\
0&T_{z}\end{smallmatrix}\right),\left(\begin{smallmatrix}I_{H^{2}}&0\\\
0&I_{H^{2}}\end{smallmatrix}\right)\right),$

where $T_{z}$ is a multiplication operator on $H^{2}.$ Let

$S_{1}=\frac{1}{3}(T_{1}+T_{2}+V_{3})=\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0\\\
I_{H^{2}}&I_{H^{2}}+T_{z}\end{smallmatrix}\right),S_{2}=\frac{1}{3}(T_{1}T_{2}+T_{2}T_{3}+T_{1}T_{3})=\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0\\\
I_{H^{2}}+T_{z}&T_{z}\end{smallmatrix}\right)$

and $S_{3}=T_{1}T_{2}V_{3}=\left(\begin{smallmatrix}0&0\\\
T_{z}&0\end{smallmatrix}\right).$ Therefore by using Theorem (3.3) we conclude
that the triple $\left(S_{1},S_{2},S_{3}\right)$ is $\Gamma_{3}$-contraction
and has a $\Gamma_{3}$-isometric dilation. Since $T_{z}$ is an isometry, one
can easily verify that $S_{3}$ is a partial isometry. First we will compute
the defect operator for $S_{3}:$

$D^{2}_{S_{3}}=\left(\begin{smallmatrix}I_{H^{2}}&0\\\
0&I_{H^{2}}\end{smallmatrix}\right)-\left(\begin{smallmatrix}0&0\\\
T_{z}&0\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}0&0\\\
T_{z}&0\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&0\\\
0&I_{H^{2}}\end{smallmatrix}\right)=D_{S_{3}}.$

Let us consider

$(A_{1},A_{2})=(\frac{I_{H^{2}}+T_{z}}{3},\frac{T_{z}}{3}).$

Notice that

$\displaystyle S_{1}-S_{2}^{*}S_{3}$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0\\\
I_{H^{2}}&I_{H^{2}}+T_{z}\end{smallmatrix}\right)-\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0\\\
I_{H^{2}}+T_{z}&T_{z}\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}0&0\\\
T_{z}&0\end{smallmatrix}\right)$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0\\\
I_{H^{2}}&I_{H^{2}}+T_{z}\end{smallmatrix}\right)-\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0\\\
I_{H^{2}}&0\end{smallmatrix}\right)$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}0&0\\\
0&I_{H^{2}}+T_{z}\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&0\\\
0&I_{H^{2}}\end{smallmatrix}\right)\frac{1}{3}\left(\begin{smallmatrix}0&0\\\
0&I_{H^{2}}+T_{z}\end{smallmatrix}\right)\left(\begin{smallmatrix}0&0\\\
0&I_{H^{2}}\end{smallmatrix}\right)=D_{S_{3}}A_{1}D_{S_{3}}$

and

$\displaystyle S_{2}-S_{1}^{*}S_{3}$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0\\\
I_{H^{2}}+T_{z}&T_{z}\end{smallmatrix}\right)-\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0\\\
I_{H^{2}}&I_{H^{2}}+T_{z}\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}0&0\\\
T_{z}&0\end{smallmatrix}\right)$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0\\\
I_{H^{2}}+T_{z}&T_{z}\end{smallmatrix}\right)-\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0\\\
I_{H^{2}}+T_{z}&0\end{smallmatrix}\right)$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}0&0\\\
0&T_{z}\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&0\\\
0&I_{H^{2}}\end{smallmatrix}\right)\frac{1}{3}\left(\begin{smallmatrix}0&0\\\
0&T_{z}\end{smallmatrix}\right)\left(\begin{smallmatrix}0&0\\\
0&I_{H^{2}}\end{smallmatrix}\right)=D_{S_{3}}A_{2}D_{S_{3}}$

which implies that $(A_{1},A_{2})$ are same as the fundamental operators
$(E_{1},E_{2})$ of $\left(S_{1},S_{2},S_{3}\right).$ Notice that

$A_{1}^{*}A_{1}-A_{1}A_{1}^{*}=\frac{1}{9}(I-T_{z}T_{z}^{*})=A_{2}^{*}A_{2}-A_{2}A_{2}^{*}.$

One can easily verify that the conditions $(1)$ and $(2)$ of Corollary (3.5)
are satisfied. To verify the condition $(3)$ of Corollary (3.5), we need to
show that $D_{1}^{*}D_{1}-D_{1}D_{1}^{*}=D_{2}^{*}D_{2}-D_{2}D_{2}^{*}.$ It is
also easy to check that $\ker S_{3}=\\{0\\}\oplus H^{2}$ and
$(S_{1},S_{2})_{|_{\ker
S_{3}}}=(D_{1},D_{2})=(\frac{I_{H^{2}}+T_{z}}{3},\frac{T_{z}}{3}),$ which
implies

$D_{1}^{*}D_{1}-D_{1}D_{1}^{*}=D_{2}^{*}D_{2}-D_{2}D_{2}^{*}=\frac{1}{9}(I-T_{z}T_{z}^{*}).$

Thus, the condition $(3)$ of Corollary (3.5) is satisfied.

The subsequent example of a $\Gamma_{3}$-contraction has a
$\Gamma_{3}$-isometric dilation for which condition $(3)$ in Corollary (3.5)
fails, that is,

$E_{1}^{*}E_{1}-E_{1}E_{1}^{*}\neq E_{2}^{*}E_{2}-E_{2}E_{2}^{*}.$

In other words, we can conclude that the set of sufficient conditions for
existence of a $\Gamma_{3}$-isometric dilation in Theorem (2.8) breaks down in
general to be necessary, even when the $\Gamma_{3}$-contraction
$(S_{1},S_{2},S_{3})$ has the following forms:

  1. (1)

$(S_{1},S_{2})$ is a $2$-tuple of commuting contractions on some Hilbert space
$\mathcal{H},$ and

  2. (2)

$S_{3}$ is a partial isometry on $\mathcal{H}.$

######  Example 2.

Let us consider the following pair of contractions on $H^{2}\oplus H^{2}\oplus
H^{2}$:

$(T_{1},T_{2})=\left(\left(\begin{smallmatrix}0&0&I_{H^{2}}\\\ 0&0&0\\\
I_{H^{2}}&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}T_{z}&0&0\\\
0&0&0\\\ 0&0&T_{z}\end{smallmatrix}\right)\right),$

where $T_{z}$ is a multiplication operator on $H^{2}.$ Let

$S_{1}=\frac{1}{3}(T_{1}+T_{2}+V_{3})=\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0&I_{H^{2}}\\\
0&T_{z}&0\\\
I_{H^{2}}&0&I_{H^{2}}+T_{z}\end{smallmatrix}\right),S_{2}=\frac{1}{3}(T_{1}T_{2}+T_{2}V_{3}+T_{1}V_{3})=\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0&I_{H^{2}}+T_{z}\\\
0&0&0\\\ I_{H^{2}}+T_{z}&0&T_{z}\end{smallmatrix}\right)$

and $S_{3}=T_{1}T_{2}V_{3}=\left(\begin{smallmatrix}0&0&T_{z}\\\ 0&0&0\\\
T_{z}&0&0\end{smallmatrix}\right),$ where
$V_{3}=\left(\begin{smallmatrix}I_{H^{2}}&0&0\\\ 0&T_{z}&0\\\
0&0&I_{H^{2}}\end{smallmatrix}\right).$ Therefore by Theorem (3.3) the triple
$\left(S_{1},S_{2},S_{3}\right)$ is $\Gamma_{3}$-contraction which has a
$\Gamma_{3}$-isometric dilation. As $T_{z}$ is an isometry, one can easily
check that $S_{3}$ is a partial isometry. Therefore we can apply the
conclusion of the Proposition (3.4) for this triple. The first step for
computing the fundamental operators $E_{1}$ and $E_{2},$ is to calculate the
defect operator for $S_{3}:$

$D^{2}_{S_{3}}=\left(\begin{smallmatrix}I_{H^{2}}&0&0\\\ 0&I_{H^{2}}&0\\\
0&0&I_{H^{2}}\end{smallmatrix}\right)-\left(\begin{smallmatrix}0&0&T_{z}\\\
0&0&0\\\
T_{z}&0&0\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}0&0&T_{z}\\\
0&0&0\\\ T_{z}&0&0\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&0&0\\\
0&I_{H^{2}}&0\\\ 0&0&0\end{smallmatrix}\right)=D_{S_{3}}.$

Set

$(A_{1},A_{2})=(\frac{T_{z}}{3},0).$

The following computations

$\displaystyle S_{1}-S_{2}^{*}S_{3}$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0&I_{H^{2}}\\\
0&T_{z}&0\\\
I_{H^{2}}&0&I_{H^{2}}+T_{z}\end{smallmatrix}\right)-\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0&I_{H^{2}}+T_{z}\\\
0&0&0\\\
I_{H^{2}}+T_{z}&0&T_{z}\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}0&0&T_{z}\\\
0&0&0\\\ T_{z}&0&0\end{smallmatrix}\right)$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0&I_{H^{2}}\\\
0&T_{z}&0\\\
I_{H^{2}}&0&I_{H^{2}}+T_{z}\end{smallmatrix}\right)-\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0&I_{H^{2}}\\\
0&0&0\\\ I_{H^{2}}&0&I_{H^{2}}+T_{z}\end{smallmatrix}\right)$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}0&0&0\\\ 0&T_{z}&0\\\
0&0&0\end{smallmatrix}\right)$
$\displaystyle=\left(\begin{smallmatrix}0&0&0\\\ 0&I_{H^{2}}&0\\\
0&0&0\end{smallmatrix}\right)\frac{1}{3}\left(\begin{smallmatrix}0&0&0\\\
0&T_{z}&0\\\ 0&0&0\end{smallmatrix}\right)\left(\begin{smallmatrix}0&0&0\\\
0&I_{H^{2}}&0\\\ 0&0&0\end{smallmatrix}\right)=D_{S_{3}}A_{1}D_{S_{3}}$

and

$\displaystyle S_{2}-S_{1}^{*}S_{3}$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0&I_{H^{2}}+T_{z}\\\
0&0&0\\\
I_{H^{2}}+T_{z}&0&T_{z}\end{smallmatrix}\right)-\frac{1}{3}\left(\begin{smallmatrix}I_{H^{2}}+T_{z}&0&I_{H^{2}}\\\
0&T_{z}&0\\\
I_{H^{2}}&0&I_{H^{2}}+T_{z}\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}0&0&T_{z}\\\
0&0&0\\\ T_{z}&0&0\end{smallmatrix}\right)$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0&I_{H^{2}}+T_{z}\\\
0&0&0\\\
I_{H^{2}}+T_{z}&0&T_{z}\end{smallmatrix}\right)-\frac{1}{3}\left(\begin{smallmatrix}T_{z}&0&I_{H^{2}}+T_{z}\\\
0&0&0\\\ I_{H^{2}}+T_{z}&0&T_{z}\end{smallmatrix}\right)$
$\displaystyle=\frac{1}{3}\left(\begin{smallmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{smallmatrix}\right)=D_{S_{3}}A_{2}D_{S_{3}}$

implies that $(A_{1},A_{2})$ are the same as the fundamental operators
$(E_{1},E_{2})$ of $\left(S_{1},S_{2},S_{3}\right).$ Note that

$A_{1}^{*}A_{1}-A_{1}A_{1}^{*}=\frac{1}{9}(I-T_{z}T_{z}^{*})\neq
0~{}~{}{\rm{while~{}~{}}}A_{2}^{*}A_{2}-A_{2}A_{2}^{*}=0.$

One can easily verify that $\ker S_{3}=\\{0\\}\oplus H^{2}\oplus\\{0\\}$ and
$(S_{1},S_{2})_{|_{\ker S_{3}}}=(D_{1},D_{2})=(\frac{T_{z}}{3},0).$ Also, we
have

$D_{1}^{*}D_{1}-D_{1}D_{1}^{*}=\frac{1}{9}(I-T_{z}T_{z}^{*})\neq
0~{}~{}{\rm{while~{}~{}}}D_{2}^{*}D_{2}-D_{2}D_{2}^{*}=0,$

which shows that the condition $(3)$ of Corollary (3.5) is disobeyed.

######  Remark 3.6.

In Example $2,$ we notice that if $(S_{1},S_{2})$ is a $2$-tuple of commuting
contractions on $H^{2},$ and $S_{3}$ is a partial isometry on $H^{2},$ then we
have

$D_{1}^{*}D_{1}-D_{1}D_{1}^{*}\neq D_{2}^{*}D_{2}-D_{2}D_{2}^{*}.$

Therefore, it follows from condition $(3)$ of Corollary (3.5) that

$E_{1}^{*}E_{1}-E_{1}E_{1}^{*}\neq E_{2}^{*}E_{2}-E_{2}E_{2}^{*}$

This shows that the $\Gamma_{3}$-contraction $(S_{1},S_{2},S_{3})$ does have
$\Gamma_{3}$-isometric dilation, but it fails to satisfy the following
condition:

$E_{1}^{*}E_{1}-E_{1}E_{1}^{*}\neq E_{2}^{*}E_{2}-E_{2}E_{2}^{*}.$

However, as has been mentioned, Example $2$ above appears that the condition
$E_{1}^{*}E_{1}-E_{1}E_{1}^{*}\neq E_{2}^{*}E_{2}-E_{2}E_{2}^{*}$ is not
necessary for existing of a $\Gamma_{3}$-isometric dilation. Still now we are
not able to locate an example of $\Gamma_{n}$-contraction for $n\geq 3,$ which
fails to satisfy one of the necessary conditions. Thus, the existence of
rational dilation for $\Gamma_{n}$-contraction, for $n\geq 3$ is still an open
question.

Acknowledgements: The first named author thanks CSIR for financial support and
the second named author thankfully acknowledges the financial support provided
by Mathe-matical Research Impact Centric Support (MATRICS) grant, File no:
MTR/2020/000493,by the Science and Engineering Research Board (SERB),
Department of Science and Tech-nology (DST), Government of India.

## References

  * [1] W. Arveson, Subalgebras of $C^{*}$-algebras,  Acta Math., 123 (1969), 141 -224.
  * [2] W. Arveson, Subalgebras of $C^{*}$-algebras II, Acta Math., 128 (1972), 271 -308.
  * [3] J. Agler, Rational dilation on an annulus, Ann. of Math., 121 (1985), 537 - 563.
  * [4] J. Agler, J. Harland, B. J. Raphael, Classical function theory, operator dilation theory, and machine computation on multiply-connected domains, Mem. Amer. Math. Soc. 191 (2008), 289 – 312.
  * [5] J. Agler, Z. A. Lykova, N. J. Young, Extremal holomorphic maps and the symmetrized bidisc, Proc. London Math. Soc., 106 (2013) 781–818.
  * [6] J. Agler, J. McCarthy, Pick Interpolation and Hilbert Function Spaces, Graduate studies in mathematics 44, Amer. Math. Soc., Providence, R.I. (2002).
  * [7] J. Agler, N. J. Young, Operators having the symmetrized bidisc as a spectral set, Proc. Edinburgh Math. Soc., 43 (2000), 195 -210.
  * [8] J. Agler, N.J. Young, A commutant lifting theorem for a domain in $\mathbb{C}^{2}$ and spectral interpolation, J. Funct. Anal. 161 (1999) 452–477.
  * [9] J. Agler, N.J. Young, A model theory for $\Gamma$-contractions, J. Operator Theory 49 (2003) 45–60.
  * [10] J. Agler, N.J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004) 375–403.
  * [11] T. Ando, On a pair of commutative contractions, Acta Sci Math 24 (1963), 88 – 90.
  * [12] T. Ando, Structure of operators with numerical radius one, Acta Sci. Math. (Szeged)34 (1973) 11–15.
  * [13] H. Bercovici, C. Foias, L. Kerchy, B. Sz.-Nagy, Harmonic analysis of operators on Hilbert space, Universitext, Springer, New York, (2010).
  * [14] T. Bhattacharyya, S. Pal, S. Shyam Roy, Dilations of $\Gamma$-contractions by solving operator equations, Adv. Math.230 (2012), 577 – 606.
  * [15] T. Bhattacharyya, The tetrablock as a spectral set,  Indiana Univ. Math. J., 63(6) (2014), 1601-1629.
  * [16] T. Bhattacharyya, Sneh Lata, H. Sau, Admissible fundamental operators, J . Math. Anal. Appl., 425 (2015), 983 –1003.
  * [17] S. Biswas, S. Shyam Roy, Functional models for $\Gamma_{n}$-contractions and characterization of $\Gamma_{n}$-isometries, J. Func. Anal., 266 (2014), 6224 –6255.
  * [18] J. A. Ball and H. Sau, Rational dilations of tetrablock contractions revisited, J.Functional Analysis,278(2020)1-15.
  * [19] C. Costara, The symmetrized bidisc and Lempert?s theorem, Bull. London Math. Soc. 36 (2004) 656–662.
  * [20] C. Costara, On the spectral Nevanlinna-Pick problem, Studia Math., 170 (2005), 23–55.
  * [21] R. E. Curto, Applications of several complex variables to multiparameter spectral theory, Surveys of Some Recent Results in Operator Theory, Vol. II, Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 192 (1988), 25-90.
  * [22] M. Dritschel, S. McCullough, The failure of rational dilation on a triply connected domain, J. Amer. Math. Soc., 18 (2005), 873- 918.
  * [23] A. Edigarian, W. Zwonek, Geometry of symmetrized polydisc, Archiv der Mathematik, 84 (2005), 364 -374.
  * [24] T. Gamelin, Uniform Algebras, Prentice Hall, New Jersey, (1969).
  * [25] S. Gorai, J. Sarkar, Characterizations of symmetrized polydisc, To appear in Indian Jour. Pure and Applied Math., available at arXiv:1503.03473v1 [math.CV] 11 Mar (2015).
  * [26] G. Misra, A. Pal and C. Varughese, Contractivity and complete contractivity for finite dimensional Banach spaces, Journal of operator theory, 1 (2019), 23-47.
  * [27] G. Misra, N. S. N. Sastry, Contractive modules, extremal problems and curvature inequalities, J. Funct. Anal., 88 (1990), 118 - 134.
  * [28] G. Misra, N. S. N. Sastry, Completely contractive modules and associated extremal problems, J. Funct. Anal., 91  (1990), 213 - 220.
  * [29] A.Pal , On $\Gamma_{n}$-contractions and their Conditional Dilations arXiv:1704.04508v2
  * [30] S. Pal The failure of rational dilation on the symmetrized $n$-disk for any $n\geq 3.$,  arXiv:1712.05707
  * [31] S. Pal Canonical decomposition of operators associated with the symmetrized polydisc  arXiv:1708.00724
  * [32] S. Pal Operator theory and distinguished varities in the symmetrized $n$-disk  arXiv:1708.06015
  * [33] G. Pisier,  Introduction to Operator Spaces Theory, Cambridge Univ. Press, (2003).
  * [34] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Univ. Press, (2002).
  * [35] V. Paulsen, Representations of Function Algebras, Abstract Operator Spaces and Banach Space Geometry,  J. Funct. Anal., 109 (1992), 113 - 129.
  * [36] V. Ptak, N. J. Young, A generalization of zero location theorem of Schur and Cohn, Trans. Inst. Electrical Electron. Automat. Control, 25 (1980), 978 -980.
  * [37] I. Schur, Über Potenzreihen die im Innern des Einheitskreises beschränkt sind, Jour. f̈ur Math., I 147  (1917), 205 –232 ; II148 (1918), 122 – 145.
  * [38] Bela Sz.-Nagy, Sur les contractions de lespace de Hilbert, Acta Sci. Math., 15 (1953), 87 -92.
  * [39] E. L. Stout, Polynomial Convexity, Progr. Math., Birkhäuser Boston, Inc., Boston, MA, (2007).
  * [40] J. L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1 –38.
  * [41] J. L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal., 6 (1970), 172 – 191.
  * [42] F. H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, Editura Academiei: Bucuresti, Romania and D. Reidel Publishing Company, (1982).


Fractional difference sequence spaces via Pascal mean

Salila Dutta †, and Diptimayee Jena‡

†Department of Mathematics, Utkal University, Vani Vihar

Bhubaneswar, India

e-mail<EMAIL_ADDRESS>
‡Department of Mathematics, Utkal University, Vanivihar, India

e-mail<EMAIL_ADDRESS>

Abstract : The main purpose of this article is to introduce Pascal difference
sequence spaces of fractional order $\tau$ over the sequence space $\ell_{p}$
and $\ell_{\infty}$. Some topological properties of these spaces are
considered here along with the Schauder basis, ${\alpha}-,\beta-$ and
$\gamma-$duals of the spaces.

Keywords : difference operator $\Delta^{\tau}$; Schauder basis, Pascal mean,
dual spaces, sequence spaces.

2010 Mathematics Subject Classification : 47A10; 40A05; 46A45.

##  1 Introduction, Preliminaries and Definitions

Let $\tau\in\mathbb{R}-\left({\mathbb{Z}^{-}\bigcup\left\\{0\right\\}}\right)$
where $\mathbb{R}=$ Set of all real numbers and $\mathbb{Z}^{-}=$ set of
negative integers. For all real number $\tau$ ,the Euler gamma function
$\Gamma\left({\tau}\right)$ is defined as

$\Gamma\left({\tau}\right)=\displaystyle\int_{0}^{\infty}e^{-t}\
t^{\tau-1}\,dt,$ (1.1)

which is an improper integral and satisfies the following properties:

1\. $\Gamma\left({n+1}\right)=n!$, $n\in\mathbb{N}$ ,( $\mathbb{N}$ is set of
natural numbers).

2\. $\Gamma\left({\tau+1}\right)=\tau\Gamma\left({\tau}\right)$ for each real
number $\tau\not\in\\{0,-1,-2,-3,....\\}$.

It is obvious by previous knowledge based on sequence spaces the space of all
real or complex valued sequences are denoted by $\omega$. Any subspace of
$\omega$ is called a sequence space. The spaces $c_{0},c$ and $l_{\infty}$ are
of null, convergent and bounded sequences respectively. Also by $\ell_{p}$ ,
we mean the space of absolutely p-summable series, for $1\leq p<\infty$ with
the norm $\parallel
x\parallel_{\ell_{p}}={\left(\sum_{k}{\|x_{k}\|^{p}}\right)}^{1/p}$, which is
a BK- space.
Let $M$ and $N$ be two sequence spaces and $A=\left({a_{nk}}\right)$ an
infinite matrix of real or comlpex elements. We write
$Ax={\left\\{A_{n}(x)\right\\}}_{n=0}^{\infty}\in N$ if
${A_{n}(x)}=\sum_{k}{a_{nk}x_{k}}$ converges for each $n$. if $x=(x_{k})\in M$
implies that $Ax\in N$. Then we say $A$ defines a matrix mapping from $M$ into
$N$ and we denote it be $A:M\rightarrow N$. And the sequence $Ax$ is called
$A-$transform of $x$. The notion of difference sequence spaces
$X\left(\Delta\right)$ for $X=\left\\{{\ell_{\infty}},{c_{o}},c\right\\}$ was
introduced by Kizmaz [[26] ], generalized by Et and Çolak [[22]] and many
others [ see [23, 27, 32, 25]]. For a proper fraction ${\tau}$, the concept of
fractional difference operator $\Delta^{\left({\tau}\right)}$ was introduced
by (Baliarsingh [[8]]), Baliarsingh and Dutta [[9, 10, 11, 20, 19]] and the
difference operator $\Delta^{\left({\tau}\right)}$ as

$\Delta^{\left({\tau}\right)}x_{k}=\sum_{i}\left(-1\right)^{i}\frac{\Gamma\left({{\tau}}+1\right)}{i!\Gamma({{\tau}}-i+1)}x_{k-i}$
(1.2)

$\Delta^{-\left({\tau}\right)}x_{k}=\sum_{i}\left(-1\right)^{i}\frac{\Gamma\left({{-\tau}}+1\right)}{i!\Gamma({-{\tau}}-i+1)}x_{k-i}$
(1.3)

The series of fractional difference operators are convergent. It is also
appropriate to express the difference operator
$\left(\Delta^{\left({\tau}\right)}\right)$ and its inverse
$\left(\Delta^{\left({-\tau}\right)}\right)$ respectively, as triangles by :

$\left(\Delta^{\left({\tau}\right)}\right)_{nk}=\left\\{\begin{array}[]{l}{\left(-1\right)^{n-k}\frac{\Gamma\left({{\tau}}+1\right)}{\left(n-k\right)!\Gamma({{\tau}}-n+k+1)}\qquad
if\,0\leq k\leq n}\\\ {0\qquad\qquad\qquad\qquad\qquad\qquad\qquad
if\,k>n}\end{array}\right.$
$\left(\Delta^{\left({-\tau}\right)}\right)_{nk}=\left\\{\begin{array}[]{l}{\left(-1\right)^{n-k}\frac{\Gamma\left({{-\tau}}+1\right)}{\left(n-k\right)!\Gamma({-{\tau}}-n+k+1)}\qquad
if\,0\leq k\leq n}\\\ {0\qquad\qquad\qquad\qquad\qquad\qquad\qquad
if\,k>n}\end{array}\right.$ (1.4)

The Pascal mean $P$ [[7]] which is described as Pascal matrix
$\left[p_{nk}\right]$ is defined by

${P}=\left[p_{nk}\right]=\left\\{\begin{array}[]{l}{\left(\begin{array}[]{l}{n}\\\
{n-k}\end{array}\right)\qquad\qquad if\,0\leq k\leq n}\\\
{0\qquad\qquad\qquad\qquad\qquad if\,k>n}\end{array}\right.$

The inverse of Pascal matrix $P^{-1}=\left[p_{nk}\right]^{-1}$ is defined by

$P^{-1}=\left\\{\begin{array}[]{l}{\left(-1\right)^{n-k}{\left(\begin{array}[]{l}{n}\\\
{n-k}\end{array}\right)}\qquad if\,0\leq k\leq n}\\\
{0\qquad\qquad\qquad\qquad\qquad\qquad if\,k>n}\end{array}\right.\ $ (1.5)

Pascal sequence spaces ${p_{\infty}}\ ,{p_{0}}\ ,and\,{p_{c}}$ are introduced
by Polat[[7]] are as follows:

$\begin{array}[]{l}{p_{0}=\left\\{\zeta=\left(\zeta_{k}\right)\in\omega:{\mathop{\lim}\limits_{n\to\infty}}\sum_{k=0}^{n}{\left(\begin{array}[]{l}{n}\\\
{n-k}\end{array}\right)}\zeta_{k}=0\right\\}},\\\
{p_{c}=\left\\{\zeta=\left(\zeta_{k}\right)\in\omega:{\mathop{\lim}\limits_{n\to\infty}}\sum_{k=0}^{n}{\left(\begin{array}[]{l}{n}\\\
{n-k}\end{array}\right)}\zeta_{k}\ exists\right\\}},\\\
{p_{\infty}=\left\\{\zeta=\left(\zeta_{k}\right)\in\omega:{\mathop{\sup}\limits_{n}}{\left|\sum_{k=0}^{n}{\left(\begin{array}[]{l}{n}\\\
{n-k}\end{array}\right)}\zeta_{k}\right|}<\infty\right\\}}\end{array}$

Aydin and Polat [[18]] introduced Pascal difference sequence spaces of integer
order $m$ as,

$\begin{array}[]{l}{X\left(\Delta^{\left(m\right)}\right)=\left\\{\zeta=\left(\zeta_{k}\right)\in\omega:\Delta^{\left(m\right)}\left(\zeta\right)\in
X\right\\}},\\\ \end{array}\ $ (1.6)

where $X=\left\\{{p_{\infty}}\ ,{p_{0}}\ ,{p_{c}}\right\\}$.
Now our interest is to introduce Pascal difference sequence spaces of
fractional order. Motivated by equation 1.6 we introduce the spaces
${\ell_{p}}{\left(\hat{P}\right)}$, for $1\leq p\leq\infty$ . We prove certain
topological properties of these spaces along with $\alpha-,\beta-,\gamma-$
duals.

##  2 Main Results

Here we introduce the matrix
$\hat{P}={{P}\left(\Delta^{\left(\tau\right)}\right)}$ by taking the product
of Pascal matrix ${P}$ [[7]] and fractional ordered difference operator
$\Delta^{\left(\tau\right)}$ [9], as

$\hat{P}=\left({{P}\left(\Delta^{\left(\tau\right)}\right)}\right)_{nk}=\left\\{\begin{array}[]{l}{\sum_{i=k}^{n}(-1)^{i-k}}\left(\begin{array}[]{l}\
{n}\\\
{n-i}\end{array}\right)\frac{\Gamma\left({\tau}+1\right)}{\left(i-k\right)!\Gamma\left({\tau}-i+k+1\right)}\
,{\qquad if\,0\leq k\leq n}\\\
{0,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
if\,k>n}\end{array}\right.$ (2.1)

Equivalently, one may write,

$\left({\bar{P}\left(\Delta^{\left(\tau\right)}\right)}\right)=\
\left(\begin{array}[]{ccccccc}{1}&0&0&0&\dots\\\ ({2-\tau})&({1})&0&0&\dots\\\
({3-3\tau+\frac{\tau{\left(\tau-1\right)}}{2!}})&({3-\tau})&({1})&0&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right)$ (2.2)

######  Theorem 2.1.

The inverse of the product matrix
$\hat{P}=\left({{P}\left(\Delta^{\left(\tau\right)}\right)}\right)$ is given
by $\hat{P}^{-1}=\\\
\left\\{\begin{array}[]{l}{\sum_{j=k}^{n}\left(-1\right)^{n-k}\left(\begin{array}[]{l}{j}\\\
{j-k}\end{array}\right)\frac{\Gamma\left({-\tau}+1\right)}{\left(n-j\right)!\Gamma\left({-\tau}-n+j+1\right)},\qquad\qquad
if\,0\leq k\leq n}\\\
{0,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
if\,k>n}\end{array}\right.$

###### Proof.

The proof is a routine verification using results 1.4 and 1.5 and hence
omitted. ∎

Now we introduce the new Pascal difference sequence spaces of fractional order
${\ell_{p}}{\left(\hat{P}\right)}$, for $1\leq p<\infty$ and
${\ell_{\infty}}{\left(\hat{P}\right)}$ as the set of all sequences such that
${P}{\left(\Delta^{\left(\tau\right)}\right)}-$ transforms are in the spaces
${\ell_{p}}$ and $\ell_{\infty}$ respectively, that is

$\begin{array}[]{l}{{\ell_{p}}\left({\hat{P}}\right)=\left\\{\zeta=\left(\zeta_{k}\right)\in\omega:\hat{P}\left(\zeta\right)\in\ell_{p}\right\\}},1\leq
p<\infty\ and\\\
{{\ell_{\infty}}{\left(\hat{P}\right)}=\left\\{\zeta=\left(\zeta_{k}\right)\in\omega:\hat{P}\left(\zeta\right)\in\ell_{\infty}\right\\}}.\end{array}$

Alternatively above spaces can be written as :

${\ell_{p}}{\left(\hat{P}\right)}=\left({\ell_{p}}\right)_{\left(\hat{P}\right)}\
and\
{\ell_{\infty}}{\left(\hat{P}\right)}=\left({\ell_{\infty}}\right)_{\left(\hat{P}\right)}.$
(2.3)

Now our introduced sequence spaces generalize the following sequence spaces as
follows:
${\left(i\right)}$. For ${\tau={0}}$ and $p=\infty$ , it reduces to
${\ell_{\infty}}{\left(P\right)}$, studied by Polat [[7]].
${\left(ii\right)}$. For ${\tau={1}}$ and $p=\infty$ , it reduces to
${\ell_{\infty}}{\left(P{\left(\Delta\right)}\right)}$, where $\
{\Delta}={x_{k}}-{x_{k-1}}$, studied by Aydin and Polat [[17]].
${\left(iii\right)}$. For ${\tau={m}}$ and $p=\infty$, it reduces to
${\ell_{\infty}}{\left(P{\left(\Delta^{\left(m\right)}\right)}\right)}$, and
$\
\Delta^{\left(m\right)}=\sum_{i=0}^{m}{-1}^{j}\left(\begin{array}[]{l}{m}\\\
{j}\end{array}\right){x_{m-j}}$ studied by Aydin and Polat [[18]].
Now with $\hat{P}$ \- transform of $x={\left(x_{k}\right)}$ we define the
sequence $y=\left(y_{n}\right)$ , $n\in N$ as follows :

$\begin{array}[]{l}{y_{n}=\left(\hat{P}x\right)_{n}}=\sum_{k=0}^{n}\sum_{i=k}^{n}\left(-1\right)^{i-k}\left(\begin{array}[]{l}{n}\\\
{n-i}\end{array}\right)\frac{\Gamma\left({\tau}+1\right)}{\left(i-k\right)!\Gamma\left({\tau}-i+k+1\right)}{x_{k}}\end{array}.$
(2.4)

##  3 Topological structure

This section deals with certain topological results of the spaces
${\ell_{p}}{\left(\hat{P}\right)}$ and
${\ell_{\infty}}{\left(\hat{P}\right)}$.

######  Theorem 3.1.

The sequence spaces ${\ell_{p}}{\left(\hat{P}\right)}$ and
${\ell_{\infty}}{\left(\hat{P}\right)}$ are $BK-$ spaces with the norm defined
by

$\parallel
x\parallel_{{\ell_{p}}{\left(\hat{P}\right)}}=\parallel{\left(\hat{P}x\right)}\parallel_{\ell_{p}}={\left(\sum_{k}{\left|{\left(\hat{P}\right)}_{k}{x}\right|^{p}}\right)}^{1/p}\,1\leq
p<\infty\ $ (3.1)

and

$\parallel
x\parallel_{{\ell_{\infty}}{\left(\hat{P}\right)}}=\parallel{\left(\hat{P}x\right)}\parallel_{\ell_{\infty}}=\sup_{k}{\left|{\left(\hat{P}\right)}_{k}{x}\right|}\
$ (3.2)

###### Proof.

The proof of linearity is a routine verification and hence omitted. The spaces
$\ell_{p}$ and $\ell_{\infty}$ are BK spaces with their natural norms. Since
equation (2.3) holds and $\hat{P}$ is a triangular matrix, Theorem 4.3.2 of
Wilansky [[1]] implies the fact that ${\ell_{p}}{\left(\hat{P}\right)}$ and
${\ell_{\infty}}{\left(\hat{P}\right)}$ are BK spaces. ∎

######  Theorem 3.2.

The sequence spaces ${\ell_{p}}{\left(\hat{P}\right)}$ and
${\ell_{\infty}}{\left(\hat{P}\right)}$ are linearly isomorphic to $\ell_{p}$
and $\ell_{\infty}$ respectively, for $1\leq p\leq\infty$.

###### Proof.

Now define a mapping $T:{\ell_{p}}{\left(\hat{P}\right)}\to\ell_{p}$ by $x\to
y=Tx$. Clearly , $T$ is a linear transformation. If $Tx=\theta\ then\
x=\theta$, where $\theta=\left(0\,,0\,,0\,,......\right)$ , so $T$ is one-one.
Let $y\in\ell_{p}$ , define a sequence $x=\left(x_{k}\right)$ as

$x_{k}=\sum_{j=0}^{k}\sum_{i=j}^{k}\left(-1\right)^{k-j}\left(\begin{array}[]{l}{i}\\\
{i-j}\end{array}\right)\frac{\Gamma\left(-{\tau}+1\right)}{\left(k-i\right)!\Gamma\left(-{\tau}-k+i+1\right)}y_{j},(k\in
N)$ (3.3)

Then

$\parallel
x\parallel_{{\ell_{p}}{\left(\hat{P}\right)}}=\left({\sum_{k}}\left|\sum_{j=0}^{k}\sum_{i=j}^{k}\left(-1\right)^{i-j}\left(\begin{array}[]{l}{k}\\\
{k-i}\end{array}\right)\frac{\Gamma\left({\tau}+1\right)}{\left(i-j\right)!\Gamma\left({\tau}-i+j+1\right)}{x_{j}}\right|^{p}\right)^{\frac{1}{p}}$
$=\left(\mathop{\sum}\limits_{k}{\left|{\delta_{kj}\
y_{j}}\right|}^{p}\right)^{\frac{1}{p}}=\left({\mathop{\sum}\limits_{k}}\left|y_{k}\right|^{p}\right)^{\frac{1}{p}}\
<\infty$ $where\ {\delta_{kj}}=\left\\{\begin{array}[]{l}{1,\qquad if\,k=j}\\\
{0,\qquad if\,k\neq j}\end{array}\right.$

Thus, $x\in{\ell_{p}}{\left(\hat{P}\right)}$ and $T$ is a linear bijection and
norm preserving. Hence the spaces ${\ell_{p}}{\left(\hat{P}\right)}$ and
$\ell_{p}$ are linearly isomorphic. i.e.
${\ell_{p}}{\left(\hat{P}\right)}\cong\ell_{p}$. The proof for other spaces
can be obtained in a similar manner. ∎

######  Theorem 3.3.

For $1\leq p<\infty$ the space ${\ell_{p}}{\left(\hat{P}\right)}$ is not a
Hilbert space for $p\neq 2$ .

###### Proof.

Here we prove that ${\ell_{2}}{\left(\hat{P}\right)}$ is the only Hilbert
space in ${\ell_{p}}{\left(\hat{P}\right)}$ for $1\leq p<\infty$. Since
${\ell_{p}}{\left(\hat{P}\right)}$ is a BK-space for $1\leq p<\infty$, so
${\ell_{2}}{\left(\hat{P}\right)}$ is also a BK-space with the norm

$\parallel
x\parallel_{{\ell_{2}}{\left(\hat{P}\right)}}\,=\parallel{\left(\hat{P}\right)}\parallel_{\ell_{2}}={\left(\sum_{k}{\left|{\left(\hat{P}\right)}_{k}{x}\right|^{2}}\right)}^{1/2}$
$={\left\langle{{\left(\hat{P}\right)}x},{{\left(\hat{P}\right)}x}\right\rangle}^{1/2}$

for every $x\in{\ell_{2}}{\left(\hat{P}\right)}$, we conclude that
${\ell_{2}}{\left(\hat{P}\right)}$ is a Hilbert space.
Let us consider two sequences
$u=(u_{k})=\left(1\,,\tau-1\,,\frac{\tau(9-\tau)}{2}-6\,,.......\right)$ and
$v=(v_{k})=\left(1\,,\tau-4\,,\frac{\tau(5-\tau)}{2}\,,.......\right)$ then we
have ${{\left(\hat{P}\right)}u}=\left(1\,,1\,,0\,,0......\right)$ and
${{\left(\hat{P}\right)}v}=\left(1\,,-1\,,0\,,0......\right)$. Thus, it can be
verify by the parallelogram law that is ,
L.H.S. = ${\parallel
u+v\parallel}^{2}_{{\ell_{p}}{\left(\hat{P}\right)}}+{\parallel
u-v\parallel}^{2}_{{\ell_{p}}{\left(\hat{P}\right)}}=8$
R.H.S. =$2\left({\parallel
u\parallel}^{2}_{{\ell_{p}}{\left(\hat{P}\right)}}+{\parallel
v\parallel}^{2}_{{\ell_{p}}{\left(\hat{P}\right)}}\right)=4\left(2^{2/p}\right).\\\
$ For $\,p\neq 2$, L.H.S. $\neq$ R.H.S. but for $\,p=2$, L.H.S. $=$ R.H.S. .
Hence ${\ell_{p}}{\left(\hat{P}\right)}$ is not a Hilbert space for $p\neq 2$
. ∎

######  Theorem 3.4.

The space ${\ell_{p}}{\left(\hat{P}\right)}$ for $1\leq p<\infty$ is non-
absolute type.

###### Proof.

Let consider a sequence $w\in{\ell_{p}}$ and
$w=\left(1\,,1\,,0\,,0......\right)$. Then,

${\left(\hat{P}\right)}(w)=\left(1\,,\,1-\tau\,,\,\frac{\tau(\tau-1)}{2}-{2\tau}\,,.......\right)$

and

${\left(\hat{P}\right)}(\left|w\right|)=\left(1\,,\,3-\tau\,,\,\frac{\tau(\tau-2)}{2}-{4\tau}+6\,,.......\right)$

which implies that ,

$\parallel
w\parallel_{{\ell_{p}}{\left(\hat{P}\right)}}\neq\parallel{\left|w\right|}\parallel_{{\ell_{p}}{\left(\hat{P}\right)}}.$

∎

##  4 The inclusion relation

In this section we prove some inclusion relations concerning with the spaces
${\ell_{p}}{\left(\hat{P}\right)}$ and
${\ell_{\infty}}{\left(\hat{P}\right)}$.

######  Theorem 4.1.

The inclusion ${\ell_{p}}\subset{\ell_{p}}{\left(\hat{P}\right)}$ strictly
holds for $1\leq p<\infty$.

###### Proof.

To prove the validity of the theorem, it is sufficient to show that there is a
number $M>0$ such that $\parallel
x\parallel_{{\ell_{p}}{\left(\hat{P}\right)}}=M.\parallel
x\parallel_{\ell_{p}}$ . By applying H$\ddot{o}$lder’s inequality on $y_{k}$
we have

$\left|y_{k}\right|^{p}=\left|\left({\left(\hat{P}\right)}x\right)_{k}\right|^{p}$
$=\left|\sum_{j=0}^{k}\sum_{i=j}^{k}\left(-1\right)^{i-j}\left(\begin{array}[]{l}{k}\\\
{k-i}\end{array}\right)\frac{\Gamma\left({\tau}+1\right)}{\left(i-j\right)!\Gamma\left({\tau}-i+j+1\right)}{x_{j}}\right|^{p}$
$\leq\left(\sum_{j=0}^{k}\sum_{i=j}^{k}\left(\begin{array}[]{l}{k}\\\
{k-i}\end{array}\right)\frac{\Gamma\left({\tau}+1\right)}{\left(i-j\right)!\Gamma\left({\tau}-i+j+1\right)}{\left|x_{j}\right|}^{p}\right)$
$\,\,\times\left(\sum_{j=0}^{k}\sum_{i=j}^{k}\left(\begin{array}[]{l}{k}\\\
{k-i}\end{array}\right)\frac{\Gamma\left({\tau}+1\right)}{\left(i-j\right)!\Gamma\left({\tau}-i+j+1\right)}\right)^{p-1}$
$=\left(\sum_{j=0}^{k}\sum_{i=j}^{k}\left(\begin{array}[]{l}{k}\\\
{k-i}\end{array}\right)\frac{\Gamma\left({\tau}+1\right)}{\left(i-j\right)!\Gamma\left({\tau}-i+j+1\right)}{\left|x_{j}\right|}^{p}\right),$

Then,

${\parallel
x\parallel}^{p}_{{\ell_{p}}{\left(\hat{P}\right)}}=\left({\sum_{k}}\left|\sum_{j=0}^{k}\sum_{i=j}^{k}\left(-1\right)^{i-j}\left(\begin{array}[]{l}{k}\\\
{k-i}\end{array}\right)\frac{\Gamma\left({\tau}+1\right)}{\left(i-j\right)!\Gamma\left({\tau}-i+j+1\right)}{x_{j}}\right|^{p}\right)$
$\leq{\sum_{k}}\left(\sum_{j=0}^{k}\sum_{i=j}^{k}\left(\begin{array}[]{l}{k}\\\
{k-i}\end{array}\right)\frac{\Gamma\left({\tau}+1\right)}{\left(i-j\right)!\Gamma\left({\tau}-i+j+1\right)}{\left|x_{j}\right|}^{p}\right)$
$\leq{\sum_{k}}\left(\sum_{j=0}^{k}\left(\begin{array}[]{l}{k}\\\
{k-j}\end{array}\right){\left|x_{j}\right|}^{p}\right)=M.{\sum_{k}}\left({\left|x_{k}\right|}^{p}\right)=M.{\parallel
x\parallel}^{p}_{\ell_{p}},$

where $M=\sum_{j=0}^{k}\left(\begin{array}[]{l}{k}\\\
{k-j}\end{array}\right).$ This yields us that, ${\parallel
x\parallel}^{p}_{{\ell_{p}}{\left(\hat{P}\right)}}\leq M.{\parallel
x\parallel}^{p}_{\ell_{p}}$ . Thus,
${\ell_{p}}\subset{\ell_{p}}{\left(\hat{P}\right)}$ strictly holds for $1\leq
p<\infty$. ∎

##  5 Basis for the spaces

In this section the Schauder basis [29] for ${\ell_{p}}{\left(\hat{P}\right)}$
is constructed. In theorem $\left(2.2\right)$ the spaces
${\ell_{p}}{\left(\hat{P}\right)}$ and $\ell_{p}$ are linearly isomorphic,
therefore the inverse of the basis
${\left\\{e^{\left(k\right)}\right\\}}_{k\in N}$ of the space $\ell_{p}$ forms
the basis of ${\ell_{p}}{\left(\hat{P}\right)}$.

######  Theorem 5.1.

let $\mu_{k}{={\left({\left(\hat{P}\right)}x\right)}_{k}}$. For $k\in N_{0}$
define
${b^{\left(k\right)}}={\left\\{{{b_{n}}^{\left(k\right)}}\right\\}}_{n\in
N_{0}}$ by

${\left\\{{{b_{n}}^{\left(k\right)}}\right\\}}=\left\\{\begin{array}[]{l}{\sum_{j=k}^{n}\left(-1\right)^{n-k}\left(\begin{array}[]{l}{j}\\\
{j-k}\end{array}\right)\frac{\Gamma\left({-\tau}+1\right)}{\left(n-j\right)!\Gamma\left({-\tau}-n+j+1\right)},\qquad\qquad
if\,0\leq k\leq n}\\\
{0,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
if\,k>n}\end{array}\right.$ (5.1)

Then $\left\\{{{b_{n}}^{\left(k\right)}}\right\\}$ is a Schauder basis for
${\ell_{p}}{\left(\hat{P}\right)}$ so each
$x\in{\ell_{p}}{\left(\hat{P}\right)}$ has a unique representation

$x=\sum_{k}\mu_{k}{{b_{n}}^{\left(k\right)}},\ for\ each\ k\in N.$

where $\mu_{k}={\left(\hat{P}\right)}_{k}.$

###### Proof.

$(i)$ By the definition of ${\left(\hat{P}\right)}$ and
${{b_{n}}^{\left(k\right)}}$ ,

${\left(\hat{P}\right)}{{b_{n}}^{\left(k\right)}}={e^{\left(k\right)}}\in\ell_{p},$

Let $x\in{\left(\hat{P}\right)}$, then
$x^{\left[r\right]}=\sum_{k=0}^{r}\mu_{k}{{b}^{\left(k\right)}}$ for an
integer $r\geq 0$.

By applying ${\left(\hat{P}\right)}$ we get
${\left(\hat{P}\right)}x^{\left[r\right]}=\sum_{k=0}^{r}\mu_{k}{{b}^{\left(k\right)}}$

$=\sum_{k=0}^{s}{\mu_{k}}{e^{\left(k\right)}}={\left({\left(\hat{P}\right)}x\right)}_{k}e^{\left(k\right)}$

and

${{\left(\hat{P}\right)}\left({x-x^{\left[r\right]}}\right)}_{s}=\left\\{\begin{array}[]{l}{0,\qquad\qquad\qquad\qquad
if\,0\leq s\leq r}\\\ {{\left({\left(\hat{P}\right)}x\right)}_{s},\qquad\qquad
if\,s>r}\end{array}\right.;for\ r,s\in{N_{0}}$ (5.2)

For $\epsilon>0$ there exist an integer $m_{o}$ s.t.

$\left({\mathop{\sum}\limits_{r}}\left|{\left({\left(\hat{P}\right)}x\right)}_{r}\right|^{p}\right)^{\frac{1}{p}}<{\frac{\epsilon}{2}}\
for\ all\ r\geq{m_{0}}.$

Hence

${\parallel\left({x-x^{\left[r\right]}}\right)\parallel}_{\left({\ell_{p}}{\left(\hat{P}\right)}\right)}={\mathop{\sum}\limits_{r}}{\left(\left|{\left({\left(\hat{P}\right)}x\right)}_{r}\right|^{p}\right)}^{\frac{1}{p}}<{\frac{\epsilon}{2}}<{\epsilon}\
,for\ all\ r\geq{m_{0}}.$

Assume that $x=\sum_{k}\eta_{k}{b}^{\left(k\right)}$. Since the linear mapping
$T$ from $\ell_{p}{\left(\hat{P}\right)}$ to $\ell_{p}$ is continuous we have,

${\left({\left(\hat{P}\right)}x\right)}_{k}=\sum_{k}\eta_{k}\left({\left(\hat{P}\right)}{b}^{\left(k\right)}\right)_{n}$
$=\sum_{k}\eta_{k}e^{\left(k\right)}=\eta_{n}\ .$

This contradicts to our assumption that
${\left({\left(\hat{P}\right)}x\right)}_{k}=\mu_{k}$ for each $k\in{N_{0}}$.
Thus, the representation is unique. ∎

##  6 ${\alpha-},\beta-$ and $\gamma-$ duals

Here we determine $\alpha-,\beta-$ and $\gamma-$ duals of
${\ell_{p}}{\left(\hat{P}\right)}$ and
${\ell_{\infty}}{\left(\hat{P}\right)}$. We define

$S(X,Y)=\left\\{u=\left(u\right)_{k}\in\omega:ux=\left(u_{k}x_{k}\right)\in
Y,whenever\,x=\left(x_{k}\right)\in X\right\\}$

as the multiplier sequence space for any two sequence spaces X and Y. Let
$\alpha-,\beta-$ and $\gamma-$duals be denoted by
$X^{\alpha}=S\left(X,l_{1}\right),\,X^{\beta}=S\left(X,cs\right),\,X^{\gamma}=S\left(X,bs\right)$
respectively. Throughout the collection of all finite subsets of $\mathbb{N}$
is denoted by $\kappa$ . We consider $K\in\kappa$. Some lemmas needed in
proving theorems.

######  Lemma 6.1.

[24] Let $A=\left(a_{n,k}\right)$ be an infinite matrix. Then,

  1. 1.

$A\in\left({\ell_{p},\ell_{1}}\right)$ iff

${\mathop{\sup}\limits_{K\in\kappa}}\sum_{k}{\left|\sum_{k\in
K}a_{nk}\right|}<\infty,\qquad 1<p\leq\infty;$ (6.1)

  2. 2.

$A\in\left(\ell_{p},c\right)$ iff

${\mathop{\lim}\limits_{n\to\infty}}a_{nk}\,exists\,for\,all\,k\in N;$ (6.2)
${\mathop{\sup}\limits_{n\in
N}}\sum_{k}\left|a_{nk}\right|^{q}<\infty,\,1<p<\infty.$ (6.3)

  3. 3.

$A\in\left(\ell_{\infty},c\right)$ iff $\left(5.2\right)$ holds and

${\mathop{\sup}\limits_{n\in N}}\sum_{k}\left|a_{nk}\right|<\infty,$ (6.4)
${\mathop{\lim}\limits_{n\to\infty}}\sum_{k}\left|a_{nk}-{\mathop{\lim}\limits_{n\to\infty}a_{nk}}\right|=0.$
(6.5)

  4. 4.

$A\in\left(\ell_{p},\ell_{\infty}\right)$ iff $\left(5.1\right)$ hplds with
$1<p\leq\infty$.

######  Theorem 6.1.

Define the set ${\tilde{D}_{1}}^{\left(\tau\right)}$ by

${\tilde{D}_{1}}^{\left(\tau\right)}=\left\\{a=\left({a_{k}}\right)\in\omega:{\mathop{\sup}\limits_{n\in
N}}\sum_{k}\left|\tilde{u}_{nk}\right|^{q}<\infty\right\\}$

Then,

$\left[{\ell_{p}}{\left(\hat{P}\right)}\right]^{\alpha}={\tilde{D}_{1}}^{\left(\tau\right)},\,1<p\leq\infty$

.

###### Proof.

Let $a=\left(a_{k}\right)\in\omega$ and the sequence $\left({x_{k}}\right)$
defined in equation $\left(3.3\right)$ , then we derive that

${a_{n}}{x_{n}}=\sum_{j=0}^{n}\sum_{i=j}^{n}\left(-1\right)^{n-j}\left(\begin{array}[]{l}{i}\\\
{i-j}\end{array}\right)\frac{\Gamma\left(-{\tau}+1\right)}{\left(n-i\right)!\Gamma\left(-{\tau}-n+i+1\right)}y_{j}a_{n}.$
(6.6) $={\tilde{U}}_{n}\left(y\right),\,for\,each\,n\in N.$

where the matrix ${\tilde{U}}_{n}=\left({\tilde{u}_{nk}}\right)$ is defined by

${\tilde{u}_{nk}}=\left\\{\begin{array}[]{l}{\sum_{j=k}^{n}\left(-1\right)^{n-k}\left(\begin{array}[]{l}{j}\\\
{j-k}\end{array}\right)\frac{\Gamma\left({-\tau}+1\right)}{\left(n-j\right)!\Gamma\left({-\tau}-n+j+1\right)}\qquad\qquad
if\,0\leq k\leq n},\\\
{0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,if\,k>n}\end{array}\right.$

for all $n,k\in N$. Therefore we conclude that
$ax=\left(a_{n}x_{n}\right)\in\ell_{1}$ when
$x=\left(x_{k}\right)\in{\ell_{p}}{\left(\hat{P}\right)}$ iff $Uy\in\ell_{1}$
as $y=\left({y_{k}}\right)\in{\ell_{p}}$ .
By lemma $\left(\ref{lm5}\right)$ we conclude that
$\left\\{{\ell_{p}}{\left(\hat{P}\right)}\right\\}^{\alpha}=\tilde{D}_{1}^{\left(\tau\right)}$.
∎

######  Theorem 6.2.

Define the sets ${\tilde{D}_{2}}^{\left(\tau\right)}$,
${\tilde{D}_{3}}^{\left(\tau\right)}$, ${\tilde{D}_{4}}^{\left(\tau\right)}$
by

${\tilde{D}_{2}}^{\left(\tau\right)}=\left\\{a=\left({a_{k}}\right)\in\omega:{\mathop{\sum}\limits_{k}}\left|\Delta^{\left(\tau\right)}{\left({a_{k}}{d_{k}}\right)}\right|^{q}<\infty\right\\};$
${\tilde{D}_{3}}^{\left(\tau\right)}=\left\\{a=\left({a_{k}}\right)\in\omega:{\mathop{\sup}\limits_{k}}\left|{a_{k}}{d_{k}}\right|<\infty\right\\};$

and

${\tilde{D}_{4}}^{\left(\tau\right)}=\left\\{a=\left({a_{k}}\right)\in\omega:{\mathop{\lim}\limits_{k}}\left({a_{k}}{d_{k}}\right)=0\right\\};$

where
$\Delta^{\left(\tau\right)}{\left({a_{n}}{d_{n}}\right)}={\sum_{i=0}^{n}\sum_{i=j}^{n}\left(-1\right)^{n-j}\left(\begin{array}[]{l}{i}\\\
{i-j}\end{array}\right)\frac{\Gamma\left({-\tau}+1\right)}{\left(n-i\right)!\Gamma\left({-\tau}-n+i+1\right)}{a_{j}}}$,
then

$\left[{\ell_{p}}{\left(\hat{P}\right)}\right]^{\beta}={\tilde{D}_{2}}^{\left(\tau\right)}\bigcap{\tilde{D}_{3}}^{\left(\tau\right)},\,1<p\leq\infty$

and

$\left[{\ell_{p}}{\left(\hat{P}\right)}\right]^{\gamma}={\tilde{D}_{2}}^{\left(\tau\right)}\bigcap{\tilde{D}_{3}}^{\left(\tau\right)},\,1<p\leq\infty$

.

###### Proof.

Let $a=\left(a_{k}\right)\in\omega$ and the sequence $\left({x_{k}}\right)$
defined in equation $\left(3.3\right)$ , consider

$\sum_{k=0}^{n}{a_{k}}{x_{k}}=\sum_{k=0}^{n}{a_{k}}\left[\sum_{i=0}^{k}\sum_{j=i}^{k}\left(-1\right)^{k-j}\left(\begin{array}[]{l}{i}\\\
{i-j}\end{array}\right)\frac{\Gamma\left(-{\tau}+1\right)}{\left(k-i\right)!\Gamma\left(-{\tau}-k+i+1\right)}y_{j}\right]$
$\sum_{k=0}^{n}\left[\sum_{i=0}^{k}\sum_{j=i}^{k}\left(-1\right)^{k-j}\left(\begin{array}[]{l}{i}\\\
{i-j}\end{array}\right)\frac{\Gamma\left(-{\tau}+1\right)}{\left(k-i\right)!\Gamma\left(-{\tau}-k+i+1\right)}{a_{j}}\right]y_{k}$
$=\sum_{k=0}^{n}\Delta^{\left(\tau\right)}{\left({a_{k}}{d_{k}}\right)}{y_{k}}$
$=\tilde{V}_{n}{\left(y\right)},\,for\,each\,n\in N.$ (6.7)

where the matrix $\tilde{V}_{n}={\left({\tilde{v}_{nk}}\right)}$ defined by

${\tilde{u}_{nk}}=\left\\{\begin{array}[]{l}{\Delta^{\left(\tau\right)}{\left({a_{k}}{d_{k}}\right)}\qquad\qquad
if\,0\leq k<n},\\\ {1\qquad\qquad\qquad\qquad\qquad\,if\,k=n}\\\
{0\qquad\qquad\qquad\qquad\qquad\,if\,k>n}\end{array}\right.$

for all $n,k\in N$. Therefore we deduce that $ax=\left(a_{n}x_{n}\right)\in
cs$ whenever $x=\left(x_{k}\right)\in{\ell_{p}}{\left(\hat{P}\right)}$ iff
$\tilde{V}y\in c$ as $y=\left({y_{k}}\right)\in{\ell_{p}}$ . By using lemma
$\left(\ref{lm5}\right)$ we conclude that
$\left\\{{\ell_{p}}{\left(\hat{P}\right)}\right\\}^{\beta}=\tilde{D}_{2}^{\left(\tau\right)}\bigcap{\tilde{D}_{3}}^{\left(\tau\right)},\,1<p\leq\infty$.
Hence the theorem proved and the duals of other spaces can be obtained in a
similar manner using lemma $\left(\ref{lm5}\right)$ . ∎

##  7 Conclusion

In this paper, certain results on Pascal difference sequence spaces of order
m,$(m\in\mathbb{N})$ have been extended to Pascal difference sequence spaces
of fractional order $\tau$. Also we introduced certain topological properties
of new difference sequence spaces. For future work we will study certain
geometrical properties and matrix transformation of the spaces.

## References

  * [1] A. Wilansky , "Summability Through Functional Analysis", North Holland Mathematics Studies, vol. 85, Elsevier , Amsterdam , 1984
  * [2] B. Altay and F. Başar, "On some Euler sequence spaces of nonabsolute type", Ukr. Math. J. 57,1-17, 2005.
  * [3] B. Altay, F. Başar and M. Mursaleen, "On the Euler sequence spaces which include the spaces lp and l$\mathrm{\infty}$, I" Inf. Sci. 176, 1450-1462, 2006.
  * [4] B. Altay and F. Başar, "On the paranormed Riesz sequence space of non-absolute type", Southeast Asian Bulletin of Mathematics.vol. 26, pp. 701–7152002.
  * [5] B. Altay and F. Baş̧ar, "Some paranormed Riesz sequence spaces of non-absolute type", Southeast Asian Bull. Math. 30, 591–608, 2006.
  * [6] B.Altay and H. Polat, "On some new Euler difference sequence spaces", Southeast Asian Bull. Math. 30, 209-220, 2006.
  * [7] H. Polat, Some new Pascal sequence spaces, Fundam. J. Math. Appl., 1(1) (2018), 61-68
  * [8] P. Baliarsingh, "Some new difference sequence spaces of fractional order and their dual spaces", Appl. Math. Comput. 219, 9737-9742, 2013.
  * [9] P. Baliarsingh and S. Dutta, "A unifying approach to the difference operators and their applications", Bol. Soc. Paran. 33, 49-57, 2015.
  * [10] P. Baliarsingh and S. Dutta, "On the classes of fractional order difference sequence spaces and their matrix transformations", Appl. Math. Comput. 250, 665-674, 2015.
  * [11] P. Baliarsingh and S. Dutta, "On an explicit formula for inverse of triangular matrices", J. Egypt. Math. Soc. 23, 297-302, 2015.
  * [12] F. Başar and N.L. Braha, "Euler- Cesaro Difference Spaces of Bounded, Convergent and Null Sequences" Tamkang J. Math. 47, 4, 405–420, 2016.
  * [13] M. Başarir, "On the generalized Riesz B-difference sequence spaces", Filomat 24, 4. 35–52, 2010.
  * [14] M.Başarir and M. Öztürk, "On the Riesz difference sequence space", Rend. Circ. Mat. Palermo, 2,57, 377–389, 2008.
  * [15] M. Başarir and M. Öztürk, "On some generalized Bm-difference Riesz sequence spaces and uniformopial property.,̈ J. Inequal. Appl., Article ID 485730, 17 pages, 2011.
  * [16] R. Brawer, Potenzen der Pascal matrix und eine Identitat der Kombinatorik, Elem. der Math., 45 (1990), 107-110.
  * [17] S. Aydın and H. Polat, “Difference sequence spaces derived by using Pascal transform”, Fundam. J. Math. Appl. 2:1 (2019), 56–62.
  * [18] S. Aydın and H. Polat, “Some Pascal spaces of difference sequence spaces of order m”, Conf. Proc. Sci. Technol. 2:1 (2019), 97–103.
  * [19] S. Dutta and P. Baliarsingh, "A note on paranormed difference sequence spaces of fractional order and their matrix transformations" J. Egypt. Math. Soc. 22, 249-253, 2014.
  * [20] S. Dutta and P. Baliarsingh, "On some Toeplitz matrices and their inversion", J. Egypt. Math. Soc. 22, 420-423, 2014.
  * [21] H.B. Ellidokuzoglu and S. Demiriz, "Euler-Riesz difference sequence spaces", Turkish J. Math. Comput. Sci., 7, 63–72, 2017.
  * [22] M. Et and R. Çolak, "On generalized difference sequence spaces", Soochow J. Math., 21, 377-386, 1995.
  * [23] M. Et and M. Basarir, "On some new generalized difference sequence spaces", Periodica Math. Hungar. 35: 169–175, 1997.
  * [24] K.G. Grosse-Erdmann,1993. "Matrix transformation between the sequence spaces of Maddox", J. Math. Anal. Appl., 180, 223–238, 1993.
  * [25] U. Kadak and P. Baliarsingh, "On certain Euler difference sequence spaces of fractional order and related dual properties", J. Nonlinear Sci. Appl. 8: 997-1004, 2015.
  * [26] H. Kizmaz, "On certain sequence spaces", Can. Math. Bull. 24: 169-176, 1981.
  * [27] I.J. Maddox, "Spaces of strongly summable sequences", Q. J. Math., 18, 345–355, 1967.
  * [28] I.J. Maddox, "Paranormed sequence spaces generated by infinite matrices" Mathematical Proceedings of the Cambridge Philosophical Society, 64, 335–340, 1968.
  * [29] I.J. Maddox, "Elements of Functional Analysis", 2nd ed., Cambridge University Press, Cambridge, 1988.
  * [30] M. Mursaleen and K. Noman, "On Some new sequence spaces of non-absolute type related to the spaces $l_{p}$ and $l_{\infty}$\- I", Filomat., 25, 33-51, 2, 2011.
  * [31] M. Mursaleen and K. Noman, "On Some new sequence spaces of non-absolute type related to the spaces $l_{p}$ and $l_{\infty}$\- II" Mathematical Communications, 16, 383–398, 2011.
  * [32] M. Mursaleen, "Generalized spaces of difference sequences" J. Math. Anal. Appl., 203, 738–745, 1996.
  * [33] H. Polat and F. Başar, "Some Euler spaces of difference sequences of order m" Acta Math. Sci., 27, 254-266, 2007.
  * [34] S. Singh and S. Dutta, "On certain generalized mth order geometric difference sequence spaces" FJMS., 116, 1. 83-100, 2019.
  * [35] S. Singh, S. Dutta, D. Dash, and R. Sharma, "Strongly summable Fibonacci Difference Geometric Sequences derfined by Orlicz functions" GANITA., 71, 2, 99-109, 2021.
  * [36] A. Sönmez and F. Başar "Generalized Difference Spaces of Non-Absolute Type of Convergent and Null Sequences", Abstr. Appl. Anal., Article ID 435076, 20 pages, 2012.
  * [37] T. Yaying, "Paranormed Riesz difference sequence spaces of fractional order", Kragujevac J. Math., 46, 175-191, 2022.
  * [38] Aydın, C. and Ba¸sar, F., Some new sequence spaces which include the spaces $\ell_{p}$ and $\ell_{\infty}$. Demonstratio Math. 38 (2005), no. 3, 641-656