#  spectrums of equivalent Schauder operators

Luo Yi Shi  Department of Mathematics, Tianjin Polytechnic University,
Tianjin, 300160,CHINA<EMAIL_ADDRESS>,  Yang Cao  Institution of
Mathematics, Jilin University, Changchun, 130012, China<EMAIL_ADDRESS>and  Geng Tian  Institution of Mathematics, Jilin University, Changchun,
130012, China<EMAIL_ADDRESS>
#  spectrums of equivalent Schauder operators

Luo Yi Shi  Department of Mathematics, Tianjin Polytechnic University,
Tianjin, 300160,CHINA<EMAIL_ADDRESS>,  Yang Cao  Institution of
Mathematics, Jilin University, Changchun, 130012, China<EMAIL_ADDRESS>and  Geng Tian  Institution of Mathematics, Jilin University, Changchun,
130012, China<EMAIL_ADDRESS>
###### Abstract.

Assume that $T_{1},T_{2}$ are equivalent Schauder operators. In this paper, we
show that even in this case their Schauder spectrum may be very different in
the view of operator theory. In fact, we get that if a self-adjoint Schauder
operator $A$ has more than one points in its essential spectrum
$\sigma_{e}(A)$, then there exists a unitary spread operator $U$ such that the
Schauder spectrum $\sigma_{S}(UA)$ contains a ring which is depended by the
essential spectrum; if there is only one point in $\sigma_{e}(A)$ and
satisfies some conditions then there exists a unitary spread operator $U$ such
that the Schauder spectrum $\sigma_{S}(UA)$ contains the circumference which
is depended by the essential spectrum.

###### 2000 Mathematics Subject Classification:

Primary 47B37, 47B99; Secondary 54H20, 37B99

##  1\. Introduction

In their paper [3], Cao give an operator theory description of bases on a
separable Hilbert space $\mathcal{H}$. To study operators on $\mathcal{H}$
from a basis theory viewpoint, it is naturel to consider the behavior of
operators related by equivalent bases. For examples, they show that there
always be some strongly irreducible operators in the orbit of equivalent
Schauder matrices([4]). However, in the usual way a spectral method
consideration of operators in the equivalent orbit is also important to the
joint research both on operator theory and Schauder bases. Cao introduces the
conception Schauder spectrum to do this work. The main purpose of this paper
is to show that the Schauder spectrum of Schauder operators in a given orbit
can be very different.

Recall that a sequence of vectors $\\{f_{n}\\}_{n=1}^{\infty}$ in
$\mathcal{H}$ is said to be a Schauder basis [13, 9] for $\mathcal{H}$ if
every element $f\in\mathcal{H}$ has a unique series expansion $f=\sum
c_{n}f_{n}$ which converges in the norm of $\mathcal{H}$. If $\\{f_{n}\\}$ is
Schauder basic for $\mathcal{H}$, the sequence space associated with
$\\{f_{n}\\}$ is defined to be the linear space of all sequences $\\{c_{n}\\}$
for which $f=\sum c_{n}f_{n}$ is convergent. Two Schauder bases
$\\{f_{n}\\}_{n=1}^{\infty}$ and $\\{g_{n}\\}_{n=1}^{\infty}$ are equivalent
to each other if they have the same sequence space(cf, [13], definition 12.1,
p131, [5], p163). Denote by $\omega$ the countable infinite cardinal. In paper
[2], Cao.e.t considered the $\omega\times\omega$ matrix whose column vectors
comprise a Schauder basis and call them the Schauder matrix. An operator has a
Schauder matrix representation under some ONB is called a Schauder operator.
Given an orthonormal basis(ONB in short) $\varphi=\\{e_{n}\\}_{n=1}^{\infty}$,
the vector $f_{n}$ in a Schauder basis sequence
$\psi=\\{f_{n}\\}_{n=1}^{\infty}$ corresponds an $l^{2}$ sequence
$\\{f_{mn}\\}_{m=1}^{\infty}$ defined uniquely by the series
$f_{n}=\sum_{m=1}^{\infty}f_{mn}e_{m}$. The matrix
$F_{\psi}=(f_{mn})_{\omega\times\omega}$ is called the Schauder matrix of
basis $\psi$ under the ONB $\varphi$.

Assume that $\psi_{1},\psi_{2}$ are equivalent Schauder bases and
$T_{\psi_{1}},T_{\psi_{2}}$ are the operators defined by Schauder matrices
$F_{\psi_{1}}$ and $F_{\psi_{2}}$ respectively under the same ONB. Then there
are no difference between $\psi_{1}$ and $\psi_{2}$ from the view of bases of
the Hilbert space. Are there some notable differences between the operators
$T_{\psi_{1}}$ and $T_{\psi_{2}}$ from the view of operator theory? From the
Arsove’s theorem([1], or theorem 2.12 in [2]), there is some invertible
operator $X\in L(\mathcal{H})$ such that $XT_{\psi_{1}}=T_{\psi_{2}}$ holds.
Hence for a Schauder basis $\psi=\\{f_{n}\\}_{n=1}^{\infty}$, the set defined
as

$\mathcal{O}_{gl}(\psi)=\\{X\psi;X\in gl(\mathcal{H})\\}$

in which $X\psi=\\{Xf_{n}\\}_{n=1}^{\infty}$ and $gl(\mathcal{H})$ consists of
all invertible operators in $L(\mathcal{H})$ contains exactly all equivalent
bases to $\psi$. Moreover, the set

$\mathcal{O}_{gl}(F_{\psi})=\\{M_{X}F_{\psi};M_{X}\hbox{ is the matrix of some
operator }X\in gl(\mathcal{H})\\}$

consists of all Schauder matrix equivalent to $F_{\psi}$. In the operator
level, we define

$\mathcal{O}_{gl}(T_{\psi})=\\{XT_{\psi};X\in gl(\mathcal{H})\\}.$

Then the set $\mathcal{O}_{gl}(T_{\psi})$ consists of operators related to
bases equivalent to $\psi$. Similarly, we consider following sets:

$\begin{array}[]{c}\mathcal{O}_{u}(\psi)=\\{U\psi;U\in U(\mathcal{H})\\},\\\
\mathcal{O}_{u}(F_{\psi})=\\{M_{U}F_{\psi};M_{U}\hbox{ is the matrix of some
unitary operator }U\\},\\\ \mathcal{O}_{u}(T_{\psi})=\\{UT_{\psi};U\in
U(\mathcal{H})\\},\end{array}$

where $U(\mathcal{H})$ consists of all unitary operators in $L(\mathcal{H})$.
Roughly speaking, by these set we bind operators related to equivalent bases
of the basis $\psi$ with the same basis const. It is easy to check that a
Schauder operator $T_{\psi}$ must be injective and having a dense range.
Denote by $T_{\psi}=UA_{\psi}$ the polar decomposition of $T_{\psi}$, then the
partial isometry $U$ must be a unitary operator. Then the orbit
$\mathcal{O}_{u}(T_{\psi})$ is just the orbit $\mathcal{O}_{u}(A_{\psi})$ in
which $A_{\psi}$ is the self-adjoint operator defined by the polar
decomposition of $T_{\psi}$. In this paper we focus on unitary operators with
a nice basis theory understanding, that is, a slight generalization of spread
form defined by W. T. Gowers and B. Maurey([6], [7]).

For a complex number $\lambda,\lambda$ will be called in the Schauder spectrum
of $T$ denoted by $\sigma_{S}(T)$ if and only if there is no ONB such that
$\lambda I-T$ has a matrix representation as a Schauder matrix. It is
obviously, $\sigma(T)\supset\sigma_{S}(T)=\sigma_{p}(T)\cup\sigma_{r}(T)$ in
which $\sigma_{r}(T)=\\{\lambda\in\mathbb{C},\overline{\textup{Ran}(\lambda
I-T)}\neq\mathcal{H}\\}$.

Now we state our main theorem:

######  Theorem 1.1.

Assume that $A$ is a self-adjoint Schauder operator.

(i) If $\sigma(A)\subseteq[\lambda_{1},\lambda_{2}],\lambda_{1}>0$ and
$\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$, then there exists a unitary spread
operator $U$ such that the Schauder spectrum $\sigma_{S}(UA)\supseteq R$ for
any rings $R$ in the ring $R_{\lambda_{1},\lambda_{2}}^{o}$;

(ii) If $\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$ and
$0<\lambda_{1}<\lambda_{2}$, then there exists a unitary spread operator $U$
such that the Schauder spectrum $\sigma_{S}(UA)\supseteq R$ for any rings $R$
in the ring $R_{\lambda_{1},\lambda_{2}}^{o}$;

(iii) If there exists only one point $\lambda_{1}\in\sigma_{e}(A)$,
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{1}$, and
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{t_{k}}{\lambda_{1}})^{2}<\infty$,
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{\lambda_{1}}{r_{k}})^{2}<\infty$.
Then there exists a unitary spread operator $U$ such that the Schauder
spectrum $\sigma_{S}(UA)\supseteq\\{\lambda,|\lambda|=\lambda_{1}\\}$.

That is, if $T$ is a Schauder operator, then there exist operator
$T_{1}\in\mathcal{O}_{u}(T)$ such that $\sigma_{S}(T_{1})$ has a certain
thickness. Related concept will be clear in later section.

We organize our paper as follows. In section 2, we introduce some notations
and lemmas which will be used in the main theorem; in section 3, we research
the case that the spectrum of self-adjoint Schauder operator has only two
points; In section 4 we research the case that the essential spectrum of self-
adjoint Schauder operator has only two points; In section 5, we research the
case that there is no point spectrum in the spectrum of self-adjoint Schauder
operator. At last, we get that if $A$ is a self-adjoint Schauder operator with
at least two essential spectrum, then exists $UA\in\mathcal{O}_{u}(A)$ such
that $\sigma_{S}(A)$ is thin and $\sigma_{S}(UA)$ has a certain thickness.

######  Remark 1.2.

In the seminar held at Jilin university, Cao shows that for a Schauder
operator $T$ there must be some unitary spread $U$ such that the Schauder
operator $UT$ has an empty Schauder spectrum. In this sense, our result in
this paper show that the Schauder spectrum of $UT$ may be very bad.

##  2\. Notation and auxiliary results

In this section we will introduce some notation for convenience, and some
lemmas which will be used in the main theorem.

Throughout this paper, let
$R_{\lambda_{1},\lambda_{2}}=\\{\lambda,\lambda_{1}\leq|\lambda|\leq\lambda_{2}\\}$,
$R_{\lambda_{1}}=\\{\lambda,|\lambda|=\lambda_{1}\\}$,
$R_{\lambda_{2}}=\\{\lambda,|\lambda|=\lambda_{2}\\}$ and
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,\lambda_{1}<|\lambda|<\lambda_{2}\\}$
for $0<\lambda_{1}<\lambda_{2}$. If $E$ is a subset of complex plane
$\mathbb{C}$ and $0\notin E$, let
$E^{-1}=\\{\lambda,\frac{1}{\lambda}\in\textup{E}\\}$, $\textup{Card}\\{E\\}$
denote the cardinal number of $E$.

Recall the definition of the spread from $A$ to $B$ given by W. T. Gowers and
B. Maurey.

######  Definition 2.1.

([7], p549) Given an ONB $\\{e_{n}\\}_{n=1}^{\infty}$ and two infinite subsets
$A,B$ of $\mathbb{N}$. Let $c_{00}$ be the vector space of all sequences of
finite support. Let the elements of $A$ and $B$ be written in increasing order
respectively as $\\{a_{1},a_{2},\cdots\\}$ and $\\{b_{1},b_{2},\cdots\\}$.
Then $e_{n}$ maps to $0$ if $n\notin A$, and $e_{a_{k}}$ maps to $e_{b_{k}}$
for every $k\in N$. Denote this map by $S_{A,B}$ and call it the spread from
$A$ to $B$.

Using spread forms, we can write some unitary operator into their linear
combination. See the Example 4.13 in [2].

######  Definition 2.2.

([2], Definition 4.14) A unitary operator $U$ on $\mathcal{H}$ is said to be a
unitary spread if there is a sequence $\\{S_{A_{n},B_{n}}\\}_{n=1}^{\infty}$
of spreads such that the series $\sum_{n=1}^{\infty}S_{A_{n},B_{n}}$ converges
to $U$ in strongly operator topology (SOT). Moreover, $U$ will be called a
finite unitary spread if $U$ can be written as a finite linear combination.

In the paper [3], Cao.e.t proved that for each bijection $\sigma$ on the set
$\mathbb{N}$, the unitary operator $U_{\sigma}$ is a unitary spread.

######  Lemma 2.3.

Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)\subseteq[\lambda_{1},\lambda_{2}]$, $\lambda_{1}>0$, and there
exists $x\neq 0$ such that $||Ax||=\lambda_{i}||x||$. Then
$\lambda_{i}\in\sigma_{p}(A)$, and $x\in\textup{Ker}(\lambda_{i}I-A)$,
$i=1,2$.

###### Proof.

Indeed we only need to prove the case of $i=1$. The proof of the case of
$i=2$, is minor modifications of the proof of the analogous statements in the
case of $i=1$ by consider $A^{-1}$ and will be omitted.

Since $\sigma(A)\subseteq[\lambda_{1},\lambda_{2}]$, we know that
$(Ax,x)\geq\lambda_{1}||x||$ for any $x\neq 0$. Hence,
$||(\lambda_{1}I-A)x||^{2}=\lambda_{1}^{2}||x||+||Ax||^{2}-2\lambda_{1}(Ax,x)\leq
0$, it follows that $||(\lambda_{1}I-A)x||=0$. That is to say
$x\in\textup{Ker}(\lambda_{1}I-A)$ and $\lambda_{1}\in\sigma_{p}(A)$. ∎

######  Lemma 2.4.

Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)\subseteq[\lambda_{1},\lambda_{2}]$, $\lambda_{1}>0$. Then, for any
unitary operator $U$,

(i) $\sigma(UA)\subseteq R_{\lambda_{1},\lambda_{2}}$;

(ii) If $\lambda_{1},\lambda_{2}\notin\sigma_{p}(A)$, then $\sigma_{p}(UA)\cap
R_{\lambda_{i}}=\emptyset$;if $\lambda_{1},\lambda_{2}\in\sigma_{p}(A)$, then
$\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{i}}\\}\leq\textup{dim
Ker}(\lambda_{i}I-A),i=1,2$.

###### Proof.

(i) It is well known that if $T$ is an invertible operator, then
$\sigma(T^{-1})=\\{\lambda,\lambda^{-1}\in\sigma(T)\\}$, and $r(T)\leq||T||$
for any $T\in B(\mathcal{H})$. Thus $\sigma(UA)=\frac{1}{\sigma((UA)^{-1})}$
and $||UA||=||A||=\lambda_{2},||(UA)^{-1}||=||A^{-1}||=\frac{1}{\lambda_{1}}$.
It follows that $\lambda\notin\sigma(UA)$ when $|\lambda|>\lambda_{2}$ and
$\lambda\notin\sigma((UA)^{-1})$ when $|\lambda|>\frac{1}{\lambda_{1}}$.
Hence, $\sigma(UA)\subseteq R_{\lambda_{1},\lambda_{2}}$, for any unitary
operator $U$.

(ii)Indeed we only need to prove the case of $i=1$. The proof of the case of
$i=2$, is minor modifications of the proof of the analogous statements in the
case of $i=1$ by consider $A^{-1}$ and will be omitted.

Assume $U$ is a unitary operator and $\lambda\in\sigma_{p}(UA)\cap
R_{\lambda_{1}}$. Then there exists $x\neq 0$ such that $UAx=\lambda x$ and
$||Ax||=||UAx||=\lambda_{1}||x||$. By Lemma 2.3, $\lambda_{1}\in\sigma_{p}(A)$
and $x\in\textup{Ker}(\lambda_{1}I-A)$. Hence, $A$ and $U$ have the matrix
forms

$A=\left[\begin{array}[]{cc}\lambda_{1}I&\\\
&A_{1}\end{array}\right]\begin{matrix}\mbox{$\textup{Ker}(\lambda I-UA)$}\\\
\mbox{$\textup{Ker}(\lambda I-UA)^{\perp}$}\\\
\end{matrix},U=\left[\begin{array}[]{cc}U_{11}&U_{12}\\\
U_{21}&U_{22}\end{array}\right]\begin{matrix}\mbox{$\textup{Ker}(\lambda
I-UA)$}\\\ \mbox{$\textup{Ker}(\lambda I-UA)^{\perp}$}\\\ \end{matrix},$

and $\textup{Ker}(\lambda I-UA)\subseteq\textup{Ker}(\lambda_{1}I-A)$.

For any $x\in\textup{Ker}(\lambda I-UA)$, $UAx=\lambda x$. Since $U$ is a
unitary operator, it is easy to check that $U_{12}=U_{21}=0$. Hence, $UA$ has
the matrix form

$UA=\left[\begin{array}[]{cc}\lambda_{1}U_{11}&\\\
&U_{22}A_{1}\end{array}\right]\begin{matrix}\mbox{$\textup{Ker}(\lambda
I-UA)$}\\\ \mbox{$\textup{Ker}(\lambda I-UA)^{\perp}$}\\\ \end{matrix},$

in which $U_{11}$ and $U_{22}$ are unitary operators and

$\sigma_{P}(UA)=\sigma_{P}(\lambda_{1}U_{11})\cup\sigma_{P}(U_{22}A_{1}),$
$\textup{Card}\\{\sigma_{P}(\lambda_{1}U_{11})\\}\leq\textup{dim Ker}(\lambda
I-UA)\leq\textup{dim Ker}(\lambda_{1}I-A).$

If there exists another $\delta\in\sigma_{p}(U_{22}A_{1})\cap
R_{\lambda_{1}}$, repeating the above process, we can get that
$\textup{Ker}(\delta I-U_{22}A_{1})\subseteq\textup{Ker}(\lambda_{1}I-A)$ and
$\textup{Ker}(\delta I-U_{22}A_{1})\bot\textup{Ker}(\lambda I-UA)$.

Repeating the above process, we can obtain that
$\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq\textup{dim
Ker}(\lambda_{1}I-A).$

∎

######  Remark 2.5.

By the above lemma, we know that if the spectrum $\sigma(A)$ of a self-adjoint
Schauder operator is contained in an interval, then the Schauder spectrum
$\sigma_{S}(UA)$ must be contained in the ring which is depended by the
interval.

##  3\. Only two points in $\sigma(A)$

In this section, we will research the case that the spectrum of self-adjoint
Schauder operator $A$ has only two points $\lambda_{1},\lambda_{2}$ and
$0<\lambda_{1}<\lambda_{2}$.

According to Lemma 2.4, we know that for any unitary operator $U$, there
exists at most denumerable subsets $\sigma_{1}$ in $R_{\lambda_{1}}$ and
$\sigma_{2}$ in $R_{\lambda_{2}}$ such that
$\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup
R_{\lambda_{1},\lambda_{2}}^{o}$. In this section, we will show that if
ker$(\lambda_{i}-A)=\infty,i=1,2$, then for any at most denumerable subsets
$\sigma_{1}$ in $R_{\lambda_{1}}$, $\sigma_{2}$ in $R_{\lambda_{2}}$ and a
ring $R$ in $R_{\lambda_{1},\lambda_{2}}^{o}$, there exists a unity operator
$U$ such that $\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup R$. Hence,
there exists $UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is thin and
$\sigma_{S}(UA)$ has a certain thickness.

######  Lemma 3.1.

Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\\{\lambda_{1},\lambda_{2}\\}$, $0<\lambda_{1}<\lambda_{2}$ and dim
ker$(\lambda_{i}-A)=\infty$. Then, there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=R_{\lambda_{1},\lambda_{2}}^{o}$.

###### Proof.

By the classical spectral theory of normal operator, we have following
orthogonal decomposition of $A$,

$A=\oplus_{n\in\mathbb{Z}}A_{n},$

in which $A_{0}=\lambda_{2}I,A_{-1}=\lambda_{1}I,$ $A_{n}=\lambda_{1}I$ for
all $n\geq 1$ and $A_{n}=\lambda_{2}I$ for all $n\leq-2$.

Now we choose an ONB $\\{e_{k}^{(n)}\\}_{k=1}^{\infty}$, for each
$n\in\mathbb{Z}$. And let $U$ be the unitary spread operator defined as

$Ue_{k}^{(n)}=e_{k}^{(n+1)},n\in\mathbb{Z},k\in\mathbb{N}.$

For a vector $x\in\mathcal{H}$ now under the ONB constructed it has a
$l_{2}$-sequence coordinate in the form

$x=\sum_{n\in\mathbb{Z}}x^{(n)}=\sum_{n\in\mathbb{Z}}\sum_{k=1}^{\infty}x_{k}^{(n)}e_{k}^{(n)}.$

Now simply we have

$UAx=UA(\sum_{n\in\mathbb{Z}}x^{(n)})=\sum_{n\in\mathbb{Z}}UAx^{(n)}=\sum_{n\in\mathbb{Z}}A_{n-1}x^{(n-1)}.$

Now suppose for some $\lambda\neq 0$ we do have some vector $x$ such that
$(\lambda I-UA)x=0$, then we have

$\lambda x^{(n)}=A_{n-1}x^{(n-1)}.$

Therefore, following equations hold:

$x^{(n)}=\lambda^{-n}A_{n-1}A_{n-2}\cdots A_{0}x^{(0)},n\geq 1,$

$x^{(n)}=\lambda^{-n}A_{n}^{-1}A_{n+1}^{-1}\cdots A_{-1}^{-1}x^{(0)},n\leq-1.$

That is to say $x^{(n)}=\lambda^{-n}\lambda_{1}^{n-1}\lambda_{2}x^{(0)},n\geq
1$ and $x^{(n)}=\lambda^{-n}\lambda_{1}^{-1}\lambda_{2}^{n+1}x^{(0)},n\leq-1.$

Since $0<\lambda_{1}<\lambda_{2}$ it is easy to see that if
$\lambda_{1}<|\lambda|<\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}<\infty$; if $\lambda_{1}\leq|\lambda|$
then $||x^{(n)}||>1$ for $n\geq 1$, if $|\lambda|\geq\lambda_{2}$ then
$||x^{(-n)}||>1$ for $n\geq 1$, i.e. if $\lambda_{1}\leq|\lambda|$ or
$|\lambda|\geq\lambda_{2}$ then $\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}=\infty$.
Hence, $\sigma_{p}(UA)=\\{\lambda,\lambda_{1}<|\lambda|<\lambda_{2}\\}$.

∎

######  Proposition 3.2.

Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\\{\lambda_{1},\lambda_{2}\\}$, $0<\lambda_{1}<\lambda_{2}$ and dim
ker$(\lambda_{i}-A)=\infty$. Then there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$.

###### Proof.

Firstly, we prove that if
$R=\\{\lambda,(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}<|\lambda|<(\lambda_{1}^{m_{1}}\lambda_{2}^{m_{2}})^{\frac{1}{m_{1}+m_{2}}}\\}$
is a ring in $R_{\lambda_{1},\lambda_{2}}^{o}$ for some integers
$n_{1},n_{2},m_{1},m_{2}$, then there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=R$.

We assign the same notations used in the proof of Lemma 3.1.

$x^{(n)}=\lambda^{-n}A_{n-1}A_{n-2}\cdots A_{0}x^{(0)},n\geq 1,$

$x^{(n)}=\lambda^{-n}A_{n}^{-1}A_{n+1}^{-1}\cdots A_{-1}^{-1}x^{(0)},n\leq-1.$

Let

$A_{0}=A_{1}=\cdots A_{n_{1}}=\lambda_{1}I,$ $A_{n_{1}+1}=A_{n_{1}+2}=\cdots
A_{n_{1}+n_{2}}=\lambda_{2}I,$ $\vdots$
$A_{k_{1}n_{1}+k_{2}n_{2}+k_{3}}=\lambda_{1}I,$

for any $k_{1},k_{2}$ and $1\leq k_{3}\leq n_{1}$,

$A_{k_{1}n_{1}+k_{2}n_{2}+k_{3}}=\lambda_{2}I,$

for any $k_{1},k_{2}$ and $n_{1}+1\leq k_{3}\leq(n_{1}+n_{2})$.

And let

$A_{-1}=\cdots A_{-m_{2}}=\lambda_{2}I,$ $A_{-m_{2}-1}=A_{-m_{2}-2}=\cdots
A_{-m_{2}-m_{1}}=\lambda_{1}I,$ $\vdots$
$A_{-k_{1}m_{2}-k_{2}m_{1}-k_{3}}=\lambda_{2}I,$

for any $k_{1},k_{2}$ and $1\leq k_{3}\leq m_{1}$,

$A_{-k_{1}m_{2}-k_{2}m_{1}-k_{3}}=\lambda_{1}I,$

for any $k_{1},k_{2}$ and $m_{2}+1\leq k_{3}\leq(m_{1}+m_{2})$.

Then we have

$x^{(k(n_{1}+n_{2}))}=\lambda^{-k(n_{1}+n_{2})}\lambda_{1}^{kn_{1}}\lambda_{2}^{kn_{2}}x^{(0)},k\geq
1,$

and

$x^{-(k(m_{1}+m_{2}))}=\lambda^{-k(m_{1}+m_{2})}\lambda_{1}^{-km_{1}}\lambda_{2}^{-km_{2}}x^{(0)},k\geq
1.$

It is easy to see that if
$(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}<|\lambda|<(\lambda_{1}^{m_{1}}\lambda_{2}^{m_{2}})^{\frac{1}{m_{1}+m_{2}}}$
then $\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}<\infty$; if
$(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}\geq|\lambda|$
there exists $N$ such that $||x^{(n)}||>1$ for $n>N$, if
$|\lambda|\geq(\lambda_{1}^{m_{1}}\lambda_{2}^{m_{2}})^{\frac{1}{m_{1}+m_{2}}}$
there exists $N$ such that $||x^{(-n)}||>1$ for $n>N$, i.e. if
$\lambda_{1}\leq|\lambda|$ or $|\lambda|\geq\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}=\infty$. Hence, $\sigma_{p}(UA)=R$.

Now we turn to the more general situation.

Since
$\lim_{n_{1}\rightarrow\infty}(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}=\lambda_{1},\lim_{n_{2}\rightarrow\infty}(\lambda_{1}^{n_{1}}\lambda_{2}^{n_{2}})^{\frac{1}{n_{1}+n_{2}}}=\lambda_{2}$.
We can get that there exists a unitary spread operator $U$ such that
$\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$.

∎

######  Lemma 3.3.

Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\\{\lambda_{1},\lambda_{2}\\}$, $\lambda_{1}<\lambda_{2}$ and dim
ker$(\lambda_{i}-A)=\infty$. Then, there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=\sigma_{1}\cup\sigma_{2}$ for any at most
denumerable subsets $\sigma_{1}$ in $\\{\lambda,|\lambda|=\lambda_{1}\\}$ and
$\sigma_{2}$ in $\\{\lambda,|\lambda|=\lambda_{2}\\}$.

###### Proof.

Since $A$ is a self-adjoint operator, by the classical spectral theory of
normal operator, we have following orthogonal decomposition of $A$

$\begin{matrix}\begin{bmatrix}\lambda_{1}I&\\\
&\lambda_{2}I\end{bmatrix}&\end{matrix}.$

Let $U=U_{1}\oplus U_{2}$, in which $U_{1},U_{2}$ are unity operators such
that
$\sigma_{p}(\lambda_{1}U_{1})=\sigma_{1},\sigma_{p}(\lambda_{2}U_{2})=\sigma_{2}$.
Then $U$ is a unity operator and $\sigma_{p}(UA)=\sigma_{1}\cup\sigma_{2}$. ∎

According to the Lemmas 2.4, 3.1, 3.3 and the Proposition 3.2, we can get the
following theorem:

######  Theorem 3.4.

Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\\{\lambda_{1},\lambda_{2}\\}$, $0<\lambda_{1}<\lambda_{2}$ and dim
ker$(\lambda_{i}-A)=\infty$. Then, there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=\sigma_{1}\cup\sigma_{2}\cup R$ for any at most
denumerable subsets $\sigma_{1}$ in $\\{\lambda,|\lambda|=\lambda_{1}\\}$,
$\sigma_{2}$ in $\\{\lambda,|\lambda|=\lambda_{2}\\}$ and $R$ is a ring in the
ring $R_{\lambda_{1},\lambda_{2}}^{o}$. Moreover, for any unitary operator
$U$, there exists at most denumerable subsets $\sigma_{1}$ in
$R_{\lambda_{1}}$ and $\sigma_{2}$ in $R_{\lambda_{2}}$ such that
$\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup
R_{\lambda_{1},\lambda_{2}}^{o}$.

###### Proof.

Since $A$ is a self-adjoint operator, by the classical spectral theory of
normal operator, we have following orthogonal decomposition of $A=A_{1}\oplus
A_{1}$ where $A_{1}$ is a self-adjoint operator satisfying that
$\sigma(A_{1})=\\{\lambda_{1},\lambda_{2}\\}$ and dim
ker$(\lambda_{i}-A)=\infty$. By Lemmas 3.1, 3.3 and the Proposition 3.2, we
get that there exists a unitary operator $U$ such that
$\sigma_{p}(UA)=\sigma_{1}\cup\sigma_{2}\cup R$ for any at most denumerable
subsets $\sigma_{1}$ in $\\{\lambda,|\lambda|=\lambda_{1}\\}$, $\sigma_{2}$ in
$\\{\lambda,|\lambda|=\lambda_{2}\\}$ and $R$ is a ring in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$. The last part of this theorem is obvious by
the Lemma 2.4. ∎

######  Remark 3.5.

By the above theorem, we know that if the spectrum $\sigma(A)$ of a self-
adjoint Schauder operator has only two points $\lambda_{1},\lambda_{2}$ and
ker$(\lambda_{i}-A)=\infty,i=1,2$, then for any ring $R$ in
$R_{\lambda_{1},\lambda_{2}}^{o}$ and at most denumerable subsets $\sigma_{1}$
in $\\{\lambda,|\lambda|=\lambda_{1}\\}$, $\sigma_{2}$ in
$\\{\lambda,|\lambda|=\lambda_{2}\\}$, there exists $UA\in\mathcal{O}_{u}(A)$
such that $\sigma_{S}(UA)$ contains $\sigma_{1}\cup\sigma_{2}\cup R$. i.e.
$\sigma_{S}(UA)$ has a certain thickness, $\sigma_{S}(A)$ is thin. In other
words, there is no ONB such that $\lambda I-UA$ has a matrix representation as
a Schauder matrix for $\lambda\in\sigma_{1}\cup\sigma_{2}\cup R$.

##  4\. Only two points in $\sigma_{e}(A)$

In this section, we will research the case that the essential spectrum of
self-adjoint operator $A$ has only two points $\lambda_{1},\lambda_{2}$ and
$0<\lambda_{1}<\lambda_{2}$. We will show that for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$,
there exists a unitary spread operator $U$ such that
$R_{\lambda_{1}\lambda_{2}}\supseteq\sigma_{p}(UA)\supseteq R$. i.e. there
exists $UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is thin and
$\sigma_{S}(UA)$ has a certain thickness.

######  Theorem 4.1.

Assume that $A$ is a self-adjoint operator satisfying the following
properties:

(i) $\sigma(A)=\sigma_{p}(A)\cup\\{\lambda_{1},\lambda_{2}\\}$,
$0<\lambda_{1}<\lambda_{2}$ and $\lambda_{1},\lambda_{2}$ are the unique
accumulation points of $\sigma(A)$;

(ii) For each $t\in\sigma_{p}(A)$, dim $\ker(A-tI)=1$.

Then there exists a unitary spread operator $U$ such that
$R_{\lambda_{1}\lambda_{2}}\supseteq\sigma_{p}(UA)\supseteq R$ for any rings
$R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$.

Moreover, if $t_{k}>t_{k+1},r_{k}<r_{k+1}$ for all $k$, then there exists a
unitary spread operator $U$ such that $\sigma_{p}(UA)=R$ for any rings $R$ in
the ring
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$;
for any unitary operator $U$, $\sigma_{p}(UA)\subset
R_{\lambda_{1},\lambda_{2}}=\\{\lambda,|\lambda_{1}|\leq|\lambda|\leq\lambda_{2}\\}$
and $\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq
1,\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq 1$.

###### Proof.

We only prove the case that $R=R_{\lambda_{1}\lambda_{2}}$, the proof of the
more general situation is similar to the Proposition 3.2 and we omit it.

The self-adjoint operator satisfying the conditions appearing in the
proposition has a spectrum in the following form:

$\sigma(A)=\\{t_{1},t_{2},\cdots,t_{k},\cdots\\}\cup\\{r_{1},r_{2},\cdots,r_{k},\cdots\\}\cup\\{\lambda_{1},\lambda_{2}\\},$

in which $\lambda_{1}$ is the accumulation point of the sequence
$\\{t_{k}\\}$, $\lambda_{2}$ is the accumulation point of the sequence
$\\{r_{k}\\}$.

Choose the subsequences $\\{t_{nk}\\}_{k=1}^{\infty}$, $n\geq 0$ of
$\\{t_{k}\\}$ and $\\{r_{nk}\\}_{k=1}^{\infty}$, $n\geq 1$ of $\\{r_{k}\\}$
satisfying the following properties:

(i)
$\lim_{k\rightarrow\infty}t_{nk}=\lim_{n\rightarrow\infty}t_{nk}=\lambda_{1}$,
$\lim_{k\rightarrow\infty}r_{nk}=\lim_{n\rightarrow\infty}r_{nk}=\lambda_{2}$;

(ii) There exist $t_{nk}$ and $r_{nk}$ such that $t_{nk}=t_{k_{0}}$,
$r_{nk}=r_{k_{1}}$ for any $t_{k_{0}}\in\\{t_{k}\\},r_{k_{1}}\in\\{r_{k}\\}$;

(iii) $t_{n_{1}k_{1}}\neq t_{n_{2}k_{2}},r_{n_{1}k_{1}}\neq r_{n_{2}k_{2}}$
when $n_{1}\neq n_{2}$ or $k_{1}\neq k_{2}$.

Let $J_{n}=\\{t_{nk}\\},n\geq 0,J_{n}=\\{r_{-nk}\\},n\leq-1.$ We rearrange
these intervals as follows: $I_{0}=J_{0},I_{n}=J_{n+1}$ for $n\geq 1$,
$I_{-1}=J_{1},I_{n}=J_{n+1}$ for $n\leq-2$.

Denote $E_{n}=E_{I_{n}}$ the spectral projection on the interval $I_{n}$ and
by $H_{n}=$Ran$(E_{n})$ for $n\in\mathbb{Z}$. Now we choose an ONB
$\\{e_{k}^{(n)}\\}_{k=1}^{\infty}$, for each $n\in\mathbb{Z}$. Since each
$H_{n}$ is a reducing subspace of $A$, we can write $A$ into the direct sum:

$A=\oplus_{n=-\infty}^{+\infty}A_{n}.$

Let
$\sup_{k}\\{t_{nk}\\}=\alpha_{n}^{(1)},\inf_{k}\\{t_{nk}\\}=\alpha_{n}^{(2)}$
for $n\geq 0$,
$\sup_{k}\\{r_{nk}\\}=\beta_{n}^{(1)},\inf_{k}\\{r_{nk}\\}=\beta_{n}^{(2)}$
for $n\geq 1$. Then
$\lim_{n\rightarrow\infty}\alpha_{n}^{(1)}=\lim_{n\rightarrow\infty}\alpha_{n}^{(2)}=\lambda_{1},\lim_{n\rightarrow\infty}\beta_{n}^{(1)}=\lim_{n\rightarrow\infty}\beta_{n}^{(2)}=\lambda_{2}$.

Now let $U$ be the unitary spread operator defined as

$Ue_{k}^{(n)}=e_{k}^{(n+1)},n\in\mathbb{Z},k\in\mathbb{N}.$

For a vector $x\in\mathcal{H}$ now under the ONB constructed it has a
$l_{2}$-sequence coordinate in the form

$x=\sum_{n\in\mathbb{Z}}x^{(n)}=\sum_{n\in\mathbb{Z}}\sum_{k=1}^{\infty}x_{k}^{(n)}e_{k}^{(n)}.$

Now simply we have

$UAx=UA(\sum_{n\in\mathbb{Z}}x^{(n)})=\sum_{n\in\mathbb{Z}}UAx^{(n)}=\sum_{n\in\mathbb{Z}}A_{n-1}x^{(n-1)}.$

Now suppose for some $\lambda\neq 0$ we do have some vector $x$ such that
$(\lambda I-UA)x=0$, then we have

$\lambda x^{(n)}=A_{n-1}x^{(n-1)}.$

Therefore, following equations hold:

$x^{(n)}=\lambda^{-n}A_{n-1}A_{n-2}\cdots A_{0}x^{(0)},n\geq 1,$

$x^{(n)}=\lambda^{-n}A_{n}^{-1}A_{n+1}^{-1}\cdots A_{-1}^{-1}x^{(0)},n\leq-1.$

Hence,

$\lambda^{-n}\beta_{0}^{(2)}\alpha_{2}^{(2)}\alpha_{3}^{(2)}\cdots\alpha_{n}^{(2)}\leq||x^{(n)}||\leq\lambda^{-n}\beta_{0}^{(1)}\alpha_{2}^{(1)}\alpha_{3}^{(1)}\cdots\alpha_{n}^{(1)},n\geq
1;$
$\frac{\lambda^{-n}}{\alpha_{1}^{(1)}\beta_{1}^{(1)}\beta_{2}^{(1)}\cdots\beta_{-n-1}^{(1)}}\leq||x^{(n)}||\leq\frac{\lambda^{-n}}{\alpha_{1}^{(2)}\beta_{1}^{(2)}\beta_{2}^{(2)}\cdots\beta_{-n-1}^{(2)}},n\leq-1.$

Since
$\lim_{n\rightarrow\infty}\alpha_{n}^{(1)}=\lim_{n\rightarrow\infty}\alpha_{n}^{(2)}=\lambda_{1},\lim_{n\rightarrow\infty}\beta_{n}^{(1)}=\lim_{n\rightarrow\infty}\beta_{n}^{(2)}=\lambda_{2}$,
it is easy to see that if $\lambda_{1}<|\lambda|<\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}<\infty$; if $\lambda_{1}<|\lambda|$
there exists $N$ such that $||x^{(n)}||>1$ for $n>N$, if
$|\lambda|>\lambda_{2}$ there exists $N$ such that $||x^{(-n)}||>1$ for $n>N$,
i.e. if $\lambda_{1}<|\lambda|$ or $|\lambda|>\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}=\infty$. That is to say
$\\{\lambda,\lambda_{1}\leq|\lambda|\leq\lambda_{2}\\}\supseteq\sigma_{p}(UA)\supseteq\\{\lambda,\lambda_{1}<|\lambda|<\lambda_{2}\\}$.

Moreover, if $t_{k}>t_{k+1},r_{k}<r_{k+1}$ for all $k$ then
$t_{k}>\lambda_{1},r_{k}<\lambda_{2}$. So
$\alpha_{n}^{(i)}\geq\lambda_{1},\beta_{n}^{(i)}\leq\lambda_{2}$ for all $n$
and $i=1,2$. It is easy to see that
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$.
Furthermore, by Lemma 2.4, we get that for any unitary operator $U$,
$\sigma_{p}(UA)\subset\\{\lambda,|\lambda_{1}|\leq|\lambda|\leq\lambda_{2}\\}$
and $\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq
1,\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{1}}\\}\leq 1$. ∎

######  Remark 4.2.

(i) Trivial modifications adapt the proof of Theorem 4.1, we can weaken the
condition dim $\ker(A-tI)=1$ to dim $\ker(A-tI)<\infty$.

(ii) In the Theorem 4.1, we obtained that there exists a unitary operator $U$
such that $\sigma_{p}(UA)\supseteq R$ for any rings $R$ in the ring
$R_{\lambda_{1}\lambda_{2}}$. Moreover, we got
$\sigma_{p}(UA)=R_{\lambda_{1}\lambda_{2}}$ if adding the condition that
$t_{k}>t_{k+1},r_{k}<r_{k+1}$ for all $k$. The following examples illustrate
that this condition is necessary.

######  Example 4.3.

We assign the same notations used in the Theorem 4.1.

(1) Let $\lambda_{1}=1$, $\lambda_{2}>1$, and $t_{n1}=1-\frac{1}{n}$,
$t_{nk}=\frac{k+n-1}{k+n}+\frac{\frac{k+n}{k+n+1}-\frac{k+n-1}{k+n}}{k+n-1}\cdot
n$ for $n\geq 1,k\geq 2$ and $r_{k}<r_{k+1}$ for all $k\geq 1$. Then according
to the proof of Theorem 4.1 and let $A_{n}=\oplus_{k=1}^{\infty}t_{nk}$,
$x^{(0)}=e_{0}^{(0)}$ in Theorem 4.1, we obtain that
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|\leq|\lambda|<\lambda_{2}\\}$.

(2) Let $\lambda_{2}=1$, $\lambda_{1}<1$, and $r_{n1}=1+\frac{1}{n}$,
$r_{nk}=\frac{k+n+1}{k+n}-\frac{\frac{k+n+1}{k+n}-\frac{k+n+2}{k+n+1}}{k+n-1}\cdot
n$ for $n\geq 1,k\geq 2$ and $t_{k}>t_{k+1}$ for all $k\geq 1$. Then according
to the proof of Theorem 4.1 and let $A_{n}=\oplus_{k=1}^{\infty}t_{nk}$,
$x^{(0)}=e_{0}^{(0)}$ in Theorem 4.1, we obtain that
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|<|\lambda|\leq\lambda_{2}\\}$.

(3) Let $\lambda_{1}=1$, $\lambda_{2}=2$, and $t_{n1}=1-\frac{1}{n}$,
$r_{n1}=2+\frac{2}{n}$,
$t_{nk}=\frac{k+n-1}{k+n}+\frac{\frac{k+n}{k+n+1}-\frac{k+n-1}{k+n}}{k+n-1}\cdot
n$,
$r_{nk}=(2+\frac{2}{k+n-1})-\frac{(2+\frac{2}{k+n-1})-(2+\frac{2}{k+n})}{k+n-1}\cdot
n$ for $n\geq 1,k\geq 2$. Then according to the proof of Theorem 4.1 and let
$A_{n}=\oplus_{k=1}^{\infty}t_{nk}$, $x^{(0)}=e_{0}^{(0)}$ in Theorem 4.1, we
obtain that
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|\leq|\lambda|\leq\lambda_{2}\\}$.

Trivial modifications adapt the proof of the Theorem 4.1, we can get the
following Proposition.

######  Corollary 4.4.

Assume that $A$ is a self-adjoint operator satisfying the following
properties:

(i) $\sigma(A)=\sigma_{p}(A)\cup\\{\lambda_{1}\\}$, $0<\lambda_{1}$ and
$\lambda_{1}$ is the unique accumulation point of $\sigma(A)$;

(ii) For each $t\in\sigma_{p}(A)$, dim $\ker(A-tI)<\infty$;

(iii)
$\sigma_{p}(A)=\\{t_{1},t_{2},\cdots,t_{k},\cdots\\}\cup\\{r_{1},r_{2},\cdots,r_{k},\cdots\\}$,
$t_{k}<t_{k+1},r_{k}>r_{k+1}$, and
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{t_{k}}{\lambda_{1}})^{2}<\infty$,
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{\lambda_{1}}{r_{k}})^{2}<\infty$.

Then, there exists a unitary spread operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=\lambda_{1}\\}$.

######  Example 4.5.

Let $A$ is a self-adjoint operator satisfying that
$\sigma(A)=\sigma_{p}(A)\cup\\{1\\}$,
$\sigma_{p}(A)=\\{t_{nk},r_{nk}\\}_{k,n=1}^{\infty}$, in which
$t_{n1}=1-\frac{1}{n}$,
$t_{nk}=\frac{k+n-1}{k+n}+\frac{\frac{k+n}{k+n+1}-\frac{k+n-1}{k+n}}{k+n-1}\cdot
n$, $r_{n1}=1+\frac{1}{n}$,
$r_{nk}=\frac{k+n+1}{k+n}-\frac{\frac{k+n+1}{k+n}-\frac{k+n+2}{k+n+1}}{k+n-1}\cdot
n$ for $n\geq 1,k\geq 2$, for each $t\in\sigma_{p}(A)$, dim $\ker(A-tI)=1$. By
Corollary 4.4, and (1), (2) of Example 4.3, we can get that there exists a
unitary spread operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=1\\}$.

######  Remark 4.6.

By the Theorem 4.1, we know that if the essential spectrum of self-adjoint
operator $A$ has only two points $\lambda_{1},\lambda_{2}$ and
$0<\lambda_{1}<\lambda_{2}$ and for each $t\in\sigma_{p}(A)$, dim
$\ker(A-tI)<\infty$, then for any ring $R$ in
$R_{\lambda_{1},\lambda_{2}}^{o}$, there exists $UA\in\mathcal{O}_{u}(A)$ such
that $\sigma_{S}(UA)$ contains $R$. i.e. $\sigma_{S}(UA)$ has a certain
thickness, $\sigma_{S}(A)$ is thin. In other words, there is no ONB such that
$\lambda I-UA$ has a matrix representation as a Schauder matrix for
$\lambda\in R$.

##  5\. No points spectrum in $\sigma(A)$

In this section, we will research the case that there is no point spectrum in
$\sigma(A)$. i.e. $\sigma(A)=[\lambda_{1},\lambda_{2}],0<\lambda_{1}$.

According to Lemma 2.4, we know that for any unitary operator $U$, there
exists at most denumerable subsets $\sigma_{1}$ in $R_{\lambda_{1}}$ and
$\sigma_{2}$ in $R_{\lambda_{2}}$ such that
$\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup
R_{\lambda_{1},\lambda_{2}}^{o}$. In this section, we will show that if
ker$(\lambda_{i}-A)=\infty,i=1,2$, then for any at most denumerable subsets
$\sigma_{1}$ in $R_{\lambda_{1}}$, $\sigma_{2}$ in $R_{\lambda_{2}}$ and a
ring $R$ in $R_{\lambda_{1},\lambda_{2}}^{o}$, there exists a unity operator
$U$ such that $\sigma_{p}(UA)\subseteq\sigma_{1}\cup\sigma_{2}\cup R$. i.e.
there exists $UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is thin and
$\sigma_{S}(UA)$ has a certain thickness.

######  Theorem 5.1.

Assume that $A$ is a self-adjoint operator satisfying that
$\sigma(A)=[\lambda_{1},\lambda_{2}]$, $\lambda_{1}>0$ and
$\sigma_{p}(A)=\varnothing$. Then, there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$.

###### Proof.

There is a sequence $\alpha_{n}\longrightarrow\lambda_{2}$ such that
$\alpha_{n+1}>\alpha_{n}$ for each $n\geq 1$. Moreover, the range of spectral
projection $E_{[\alpha_{n},\alpha_{n+1}]}$ is an infinite subspace; and a
sequence $\beta_{n}\longrightarrow\lambda_{1}$ such that
$\beta_{n}>\beta_{n+1}$ for each $n\geq 1$. Moreover, the range of spectral
projection $E_{[\beta_{n+1},\beta_{n}]}$ is an infinite subspace.

Now we rearrange these intervals as follows.

$J_{n}=[\alpha_{n},\alpha_{n+1}),n\geq 0,$
$J_{n}=[\beta_{-n+1},\beta_{-n}),n\leq-1.$

Let $I_{0}=J_{0},I_{n}=J_{-n+1}$ for $n\geq 1$, $I_{-1}=J_{-1},I_{n}=J_{-n}$
for $n\leq-1$.

Denote $E_{n}=E_{I_{n}}$ the spectral projection on the interval $I_{n}$ and
by $H_{n}=$Ran$(E_{n})$ for $n\in\mathbb{Z}$. Now we choose an ONB
$\\{e_{k}^{(n)}\\}_{k=1}^{\infty}$, for each $n\in\mathbb{Z}$. Since each
$H_{n}$ is a reducing subspace of $A$, we can write $A$ into the direct sum:

$A=\oplus_{n=-\infty}^{+\infty}A_{n}.$

And $\alpha_{0}||x||\leq||A_{0}x||\leq\alpha_{1}$ for $x\in H_{0}$,
$\beta_{0}|x||\leq||A_{-1}x||\leq\beta_{-1}||x||$ for $x\in H_{-1}$,
$\beta_{-n}||x||\leq||A_{n}x||\leq\beta_{-n-1}$ for $x\in H_{n}$, $n\geq 1$,
$\alpha_{n}||x||\leq||A_{0}x||\leq\alpha_{n+1}$ for $x\in H_{n}$ $n\leq-1$.

Now let $U$ be the unitary spread operator defined as

$Ue_{k}^{(n)}=e_{k}^{(n+1)},n\in\mathbb{Z},k\in\mathbb{N}.$

For a vector $x\in\mathcal{H}$ now under the ONB constructed it has a
$l_{2}$-sequence coordinate in the form

$x=\sum_{n\in\mathbb{Z}}x^{(n)}=\sum_{n\in\mathbb{Z}}\sum_{k=1}^{\infty}x_{k}^{(n)}e_{k}^{(n)}.$

Now simply we have

$UAx=UA(\sum_{n\in\mathbb{Z}}x^{(n)})=\sum_{n\in\mathbb{Z}}UAx^{(n)}=\sum_{n\in\mathbb{Z}}A_{n-1}x^{(n-1)}.$

Now suppose for some $\lambda\neq 0$ we do have some vector $x$ such that
$(\lambda I-UA)x=0$, then we have

$\lambda x^{(n)}=A_{n-1}x^{(n-1)}.$

Therefore, following equations hold:

$x^{(n)}=\lambda^{-n}A_{n-1}A_{n-2}\cdots A_{0}x^{(0)},n\geq 1,$

$x^{(n)}=\lambda^{-n}A_{n}^{-1}A_{n+1}^{-1}\cdots A_{-1}^{-1}x^{(0)},n\leq-1.$

Since $\alpha_{0}||x||\leq||A_{0}x||\leq\alpha_{1}$ for $x\in H_{0}$,
$\beta_{0}|x||\leq||A_{-1}x||\leq\beta_{-1}||x||$ for $x\in H_{-1}$,
$\beta_{-n}||x||\leq||A_{n}x||\leq\beta_{-n-1}$ for $x\in H_{n}$, $n\geq 1$,
$\alpha_{n}||x||\leq||A_{0}x||\leq\alpha_{n+1}$ for $x\in H_{n}$ $n\leq-1$ and
$\beta_{n}\longrightarrow\lambda_{1},\alpha_{n}\longrightarrow\lambda_{2}$, it
is easy to see that if $\lambda_{1}<|\lambda|<\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}<\infty$; if $\lambda_{1}\leq|\lambda|$
there exists $N$ such that $||x^{(n)}||>1$ for $n>N$, if
$|\lambda|\geq\lambda_{2}$ there exists $N$ such that $||x^{(-n)}||>1$ for
$n>N$, i.e. if $\lambda_{1}<|\lambda|$ or $|\lambda|>\lambda_{2}$ then
$\sum_{n\in\mathbb{Z}}||x^{(n)}||^{2}=\infty$. Hence,
$\sigma_{p}(UA)=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$.

The proof of the more general situation is similar to the Proposition 3.2. ∎

######  Remark 5.2.

By the Theorem 5.1, we know that if $A$ is a self-adjoint operator satisfying
that $\sigma(A)=[\lambda_{1},\lambda_{2}]$, $\lambda_{1}>0$ and
$\sigma_{p}(A)=\varnothing$, then for any ring $R$ in
$R_{\lambda_{1},\lambda_{2}}^{o}$, there exists $UA\in\mathcal{O}_{u}(A)$ such
that $\sigma_{S}(UA)$ contains $R$. i.e. $\sigma_{S}(UA)$ has a certain
thickness, $\sigma_{S}(A)$ is thin. In other words, there is no ONB such that
$\lambda I-UA$ has a matrix representation as a Schauder matrix for
$\lambda\in R$.

Trivial modifications adapt the proof of the Theorems of 3.4, 4.1 and 5.1, we
can get the following proposition:

######  Proposition 5.3.

Assume that $A$ is a self-adjoint operator.

(i) If $\sigma(A)\subseteq[\lambda_{1},\lambda_{2}],\lambda_{1}>0$ and
$\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$, then there exists a unitary spread
operator $U$ such that $\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$;

(ii) If $\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$ and
$0<\lambda_{1}<\lambda_{2}$, then there exists a unitary spread operator $U$
such that $\sigma_{p}(UA)\supseteq R$ for any rings $R$ in the ring
$R_{\lambda_{1},\lambda_{2}}^{o}$. Moreover, if there exist sequence
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}>t_{k+1},r_{k}<r_{k+1}$ for all $k$, then there exists a unitary spread
operator $U$ such that $\sigma_{p}(UA)=R$ for any rings $R$ in the ring
$R_{\lambda_{1}\lambda_{2}}=\\{\lambda,|\lambda_{1}|<|\lambda|<\lambda_{2}\\}$;
for any unitary operator $U$,
$\sigma_{p}(UA)\subset\\{\lambda,|\lambda_{1}|\leq|\lambda|\leq\lambda_{2}\\}$
and $\textup{Card}\\{\sigma_{p}(UA)\cap R_{\lambda_{i}}\\}\leq\textup{dim
Ker}(\lambda_{i}I-A),i=1,2$;

(iii) If there exists only one point $\lambda_{1}\in\sigma_{e}(A)$,
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{2}$, and
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{t_{k}}{\lambda_{1}})^{2}<\infty$,
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{\lambda_{1}}{r_{k}})^{2}<\infty$.
Then there exists a unitary spread operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=\lambda_{1}\\}$.

As we know,
$\sigma(T)\supset\sigma_{S}(T)=\sigma_{p}(T)\cup\\{\lambda\in\mathbb{C},\overline{\textup{Ran}(\lambda
I-T)}\neq\mathcal{H}\\}$ for every $T\in B(\mathcal{H})$. Hence, by the
Proposition 5.3, we obtain the main theorem:

######  Theorem 5.4.

Assume that $A$ is a self-adjoint Schauder operator.

(i) If $\sigma(A)\subseteq[\lambda_{1},\lambda_{2}],\lambda_{1}>0$ and
$\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$, then there exists a unitary spread
operator $U$ such that the Schauder spectrum $\sigma_{S}(UA)\supseteq R$ for
any rings $R$ in the ring $R_{\lambda_{1},\lambda_{2}}^{o}$;

(ii) If $\lambda_{1},\lambda_{2}\in\sigma_{e}(A)$ and
$0<\lambda_{1}<\lambda_{2}$, then there exists a unitary spread operator $U$
such that the Schauder spectrum $\sigma_{S}(UA)\supseteq R$ for any rings $R$
in the ring $R_{\lambda_{1},\lambda_{2}}^{o}$;

(iii) If there exists only one point $\lambda_{1}\in\sigma_{e}(A)$,
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{1}$, and
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{t_{k}}{\lambda_{1}})^{2}<\infty$,
$\sum_{n=1}^{\infty}\prod_{k=1}^{n}(\frac{\lambda_{1}}{r_{k}})^{2}<\infty$.
Then there exists a unitary spread operator $U$ such that the Schauder
spectrum $\sigma_{S}(UA)\supseteq\\{\lambda,|\lambda|=\lambda_{1}\\}$.

According to the Proposition 5.3 and Theorem 5.4, we know that if a self-
adjoint operator $A$ has more than one points in its essential spectrum, then
there exists a unitary spread operator $U$ such that $\sigma_{p}(UA)$ contains
a ring which is depended by the essential spectrum, i.e. there exists
$UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is thin and
$\sigma_{S}(UA)$ has a certain thickness; if there is only one point in the
essential spectrum and satisfies some conditions, then there exists a unitary
spread operator $U$ such that $\sigma_{p}(UA)$ contains the circumference
which is depended by the essential spectrum, i.e. there exists
$UA\in\mathcal{O}_{u}(A)$ such that $\sigma_{S}(A)$ is at most denumerable and
$\sigma_{S}(UA)$ is uncountable. Furthermore, by Lemma 2.4, we know that if
$\sigma_{e}(A)$ has only one point $\lambda_{1}$ and $\\{t_{k}\\}$ (or
$\\{r_{k}\\}$) contained in $\sigma(A)$ and satisfy that $t_{k}<t_{k+1}$ (or
$r_{k}>r_{k+1}$), $t_{k}\rightarrow\lambda_{1}$(or
$r_{k}\rightarrow\lambda_{1}$), then for any unity operator $U$,
$\sigma_{p}(UA)\neq R_{\lambda_{1}}$. However, we don’t know if there exist
$\\{t_{k}\\}$ and $\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{2}$, does there exist a
unitary operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=\lambda_{1}\\}$. It is easy to know that
if $A=\lambda I$, then the point spectrum of $UA$ is at most denumerable for
any unitary operator. We call a normal operator $A$ is non-trivial, if
$A\neq\lambda I$ for any $\lambda\in\mathbb{C}$. Hence, we have the following
question:

######  Question 5.5.

Assume that $A$ is a non-trivial invertible self-adjoint operator, and there
exists only one point $\lambda_{1}\in\sigma_{e}(A)$, $\\{t_{k}\\}$ and
$\\{r_{k}\\}$ contained in $\sigma(A)$ and satisfy that
$t_{k}<t_{k+1},r_{k}>r_{k+1}$,
$t_{k}\rightarrow\lambda_{1},r_{k}\rightarrow\lambda_{2}$. Whether there must
be a unity operator $U$ such that
$\sigma_{p}(UA)=\\{\lambda,|\lambda|=\lambda_{1}\\}$?

## References

  * [1] Arsove, Maynard G. Similar bases and isomorphisms in Fr chet spaces. Math. Ann. 135, 1958, 283-293.
  * [2] Y. Cao G. Tian and B. Z. Hou, Schauder Bases and Operator Theory, preprint. Avaliable at http://arxiv.org/abs/1203.3603.
  * [3] Y. Cao B. Z. Hou and G. Tian, On unitary operators in spread form(in Chinese), accepted.
  * [4] Y. Q. Ji G. Tian and Y. Cao, Strongly Irreducible Schauder Operators, preprint.
  * [5] Garling, D. J. H., Symmetric bases of locally convex spaces, Studia Math. 30, 1968, 163-181.
  * [6] W. T. Gowers and B. Maurey, The unconditional basic sequence problem. J. Amer. Math. Soc. 6 (1993), no. 4, 851-874.
  * [7] W. T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997) no. 4, 543–568.
  * [8] S. Jaffard and R. M. Young, A representation theorem for Schauder bases in hilbert space, Proc. Ame. Math. soc. 126 (1998) 553–560.
  * [9] C. W. McArthur, developments in schauder basis theory, Bulletin of American Mathematical Society, 78 (1972) no. 6, 877–901.
  * [10] Robert E. Megginson, An introuduction to Banach Space Theory, GTM183, Springe-Verlag, 1998.
  * [11] A. M. Olevskii, On operators generating conditional bases in a Hilbert space, Translated from Matematicheskie Zametki, Vol(12), No.1, pp. 73-84, July, 1972.
  * [12] Allen L. Shields, Weighted shift operators and analytic function theory , in: Topics in Operator Theory, Math. Surveys No. 13, 49-128, Amer. Math. Soc., Providence (1974).
  * [13] I. Singer, Bases in Banach Space I, Springer-verlag, 1970.


setting—assume a fixed set of $k$ respondents for each instance.

If the variability in the estimates is predominantly due to apartments’
features the ICC value would be high. Conversely, if there is a high variance
in the magnitude of the estimates due to differences in scales
($\sigma_{\alpha}$) or noise ($\sigma_{\epsilon}$) ICC would be lower.
Essentially, the ICC captures how responses cluster for each apartment. A
value of zero implies that there are no clusters, and each response is likely
to be independent. A value of one implies that all the responses are the same.

Descriptively, we observe that the ICC of people’s pre-review estimates is
similar across all five treatments, as shown in Figure 12(a). In treatments T1
and T3 the ICC of people’s post-review estimates remained similar to the ICC
of their pre-review estimates. However, in treatments T2, T4 and T5 people’s
post-review estimates exhibited a higher ICC than their pre-review estimates.
Namely, we observe an increase of 0.21, 0.14 and 0.13 in T2, T4 and T5
respectively.

It is important to note that although ICC is a consistent statistic, it has a
positive bias, i.e., it overestimates the true value. Additionally, it relies
on several assumption such as $\alpha$, $\beta$ and $\epsilon$ having an
expected value of zero and $\beta$ being uncorrelated with $\alpha$ and
$\epsilon$. Below, we consider a metric that does not rely on such modeling
assumptions: the Gini coefficient.

Gini Coefficient. The Gini coefficient is a measure of dispersion commonly
used to quantify inequality within groups, such as wealth inequality within a
nation. Unlike the ICC, the Gini coefficient directly focuses on the
differences in respondents’ estimates for each apartment, without any modeling
assumptions. It is defined as follows:

(2) $G=\frac{\sum_{i}^{k}\sum_{j}^{k}|x_{i}-x_{j}|}{2\cdot
k\cdot\sum_{j}x_{j}},$

where $x_{*}$ denotes the estimate by respondent $*$ and $k$ is the number of
respondents. This value captures the dispersion of people’s responses for a
given apartment. A value of zero indicates that the responses are closely
clustered together, while a value of one means the estimates are completely
dispersed. To quantify the dispersion across all apartments, we calculate the
average Gini coefficient across all 30 apartments.

We report our findings in Figure 12(b). Descriptively, we found a similar
trend as for the ICC. For treatments T1 and T3, the Gini coefficients of
people’s pre- and post-review estimates remained similar. On the other hand,
in treatments T2, T4 and T5, people’s post-review estimates exhibit a lower
Gini coefficient, with a decrease of 0.08 in T2, and a decrease of 0.02 in T4
and T5.

(a) Inconsistency in people’s implicit relative judgments. The y-axis shows
the fraction of instances where people’s implicit relative ordering of
apartments (>,< or =) did not match the relative ordering of the majority of
respondents. A lower value indicates a higher degree of consistency. We report
mean values $\pm$ 1.96 standard errors of the mean (SEM).

(b) Kendall’s W. The y-axis shows the values of Kendall’s W statistic
calculated on the respondents’ implicit rankings. A higher value indicates a
higher degree of consistency.

Figure 13. Effect of the interventions on the consistency of people’s implicit
relative judgments. The experimental conditions T1–T5 are shown on the x-axis.

####  A.3.2. Consistency of People’s Implicit Relative Judgments

Pairwise Consistency. We consider a measure of consistency analogous to the
measure of accuracy described in Appendix A.2. We again derive people’s
implicit relative judgments from their absolute estimates. For each respondent
and for each pair of apartments (A,B), we check if Apartment A was estimated
to be more (>), less (<) or equally as expensive (=) as Apartment B. However,
instead of comparing a respondent’s implicit relative judgment with the ground
truth ordering, we compare it to the other respondents’ orderings—namely, to
the majority vote of others’ implicit relative judgments.

In Figure 13(a), we show the average degree of disagreement between individual
respondents’ relative judgments and the majority vote. We observe that
people’s pre-review estimates are similarly consistent across all experimental
conditions. In treatments T1 and T3, people’s post-review estimates exhibit a
similar degree of pairwise consistency as their pre-review estimates. However,
in treatments T2, T4 and T5 we find that the average disagreement with the
majority vote is decreased by 7.5, 6.9 and 7.1 percentage points respectively.
It is important to note that the pre-review inconsistency was already quite
low, leaving little room for improvement. The observed decrease in
inconsistency in T2, T4 and T5 respectively correspond to 38%, 32% and 35% of
the total possible decrease.

While this metric focused on pairwise consistency, below we consider a metric
that quantifies the consistency in the overall ordering of the apartments:
Kendall’s W.

Kendall’s W. In order to assess the consistency of respondents’ overall
ordering of apartments, we treat the provided price estimates as implicitly
ranking all of the apartments from the least expensive to the most expensive,
allowing for ties. We then quantify the consistency between the respondents’
implicit rankings utilizing Kendall’s W, a non-parametric statistic for rank
correlation.

Kendall’s W is commonly employed to evaluate agreement amongst respondents in
ranking tasks. At a high level, Kedall’s W corresponds to the normalized sum
of squared deviations from the mean in the rankings. We used the Pingouin
library (Vallat, 2018) to compute Kendall’s W with a correction for ties. A
value of one would indicate perfect agreement amongst respondents, while a
value of zero would indicate no agreement.

In Figure 13(b) we show the values of Kendall’s W statistic calculated on the
respondents’ implicit pre-review and post-review rankings. Descriptively, we
observe that in treatments T1 and T3, the respondents’ pre-review and post-
review rankings are consistent to a similar degree. However, treatments T2, T4
and T5 show an increase in Kendall’s W of 0.12, 0.16 and 0.15 in the post-
review rankings compared to the pre-review rankings.

Figure 14. Change in agreement with machine predictions. The y-axis shows the
fraction of pre-review estimates that were revised to match the decision aid’s
prediction exactly (Exact), that were revised in the direction of the
prediction, but did not match the prediction exactly (Closer), that were not
revised (No Change), and that were revised in the direction opposite of the
prediction (Farther), for the experimental conditions T2, T4 and T5, shown on
the x-axis.

###  A.4. Agreement with Machine Advice

In this section, we investigate the degree of agreement between the
respondents’ estimates and machine advice. That is, we explore how observing
machine advice impacted people’s estimates. Did respondents adjust their
initial estimates closer to machine predictions, or perhaps in the opposite
direction? When respondents followed machine advice, did they copy it exactly
or just move closer to it? We address these questions in Figure 14.

In T2, respondents observed the decision aid’s estimates of apartment prices.
The majority of respondents’ estimates were updated in the direction of
machine advice (65%). That is, the absolute difference between people’s
estimate and the decision aid’s estimate became smaller for 65% of responses.
However, many estimates were not updated (33%), and a few estimates were even
updated in the direction opposite of machine advice (2%). Amongst the 65% of
price estimates that were updated in the direction of the observed advice, 47%
were revised to exactly match the price predicted by the decision aid, while
the remaining 53% of estimates were just moved closer to the decision aid’s
predictions.

Unlike T2, where participants observed the decision aid’s estimates of
apartment prices, in T5 respondents observed only the decision aid’s
comparative valuation of pairs of apartments. Therefore, we cannot directly
compare people’s estimates and the decision aid’s predictions. However, we can
compare the decision aid’s relative ordering of pairs of apartments and the
respondents’ implicit relative ordering based on their price estimates. As a
running example, consider two apartments A and B, a decision aid that
estimated A to be more expensive than B, and respondents who estimated A to be
less or equally expensive as B. The price estimates of 85% of pairs of
apartments shown in the review phase were revised in the direction of machine
advice (i.e., the respondents from the running example either increased their
estimate for A, or decreased their estimate for B, or both). The implicit
relative ordering of 11% of pairs of apartments was not revised, and the
estimates of 4% of outlier pairs were revised in the direction opposite of
machine advice (i.e., the respondents from the running example either
decreased their estimate for A, or increased their estimate for B, or both).
Among the 85% of pairs that were revised in the direction of the decision
aid’s advice, 97% of the price estimates were updated so that the apartments’
implicit relative ordering matched the decision aid’s relative ordering (i.e.,
the respondents from the running example revised their responses such that A
is estimated to be more expensive than B), while for the remaining 3% of pairs
the price estimates were only moved closer to the decision aid’s prediction
(i.e., the respondents from the running example either increased their
estimate for A, or decreased their estimate for B, or both, but A remained
less or equally as expensive as B.)

In T4, participants did not observe any explicit machine advice. However, the
pairs of apartments respondents were asked to review were selected by the
decision aid that was also used in T5, thereby implicitly providing guidance.
Hence, we again compare this decision aid’s relative ordering with people’s
implicit relative ordering, for the pairs of apartments that were shown in the
review phase. Despite not having a chance to observe the decision aid’s
relative ordering of apartments, many respondents updated their estimates in
the direction of the decision aid’s predictions. For 69% of pairs estimates
were revised in the direction of machine predictions, 24% of implicit relative
orderings were not revised, and only 7% of estimates were revised in the
direction opposite of machine predictions. Amongst the 69% of pairs revised in
the direction of the machine predictions, 94% of the updates resulted in an
implicit relative ordering that matched the decision aid’s relative ordering
(even though the respondents did not observe the decision aid’s relative
ordering), while for the remaining 6% of pairs the price estimates were only
moved closer to the decision aid’s relative ordering.


# First and second-order dust-ion-acoustic rogue waves in non-thermal plasma

S. Banik∗1,2, R.K. Shikha∗∗,2, A.A. Noman∗∗∗,2, N.A. Chowdhury†,3, A.
Mannan‡,4, T.S. Roy§,5, and A.A. Mamun§§,2 1Health Physics Division, Atomic
Energy Centre, Dhaka-1000, Bangladesh
2Department of Physics, Jahangirnagar University, Savar, Dhaka-1342,
Bangladesh
3Plasma Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh
4Institut für Mathematik, Martin Luther Universität Halle-Wittenberg, D-06099
Halle, Germany
5Department of Physics, Bangladesh University of Textiles, Tejgaon Industrial
Area, Dhaka, Bangladesh
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A nonlinear Schrödinger equation (NLSE) has been derived by employing
reductive perturbation method for investigating the modulational instability
of dust-ion-acoustic waves (DIAWs) in a four-component plasma having
stationary negatively charged dust grains, inertial warm ions, and inertialess
non-thermal electrons and positrons. It is observed that under consideration,
the plasma system supports both modulationally stable and unstable domains,
which are determined by the sign of the dispersive and nonlinear coefficients
of NLSE, of the DIAWs. It is also found that the nonlinearity as well as the
height and width of the first and second-order rogue waves increases with the
non-thermality of electron and positron. The relevancy of our present
investigation to the observations in space plasmas is pinpointed.

###### keywords:

NLSE , Modulational instability , Dust-ion-acoustic waves , Rogue waves.

††journal: “The European Physical Journal D”

##  1 Introduction

The existence of massive dust grains in different electron-positron-ion (EPI)
plasmas (viz., Jupiter’s magnetosphere [1], Milky Way [2], auroral zone [3],
accretion disks near neutron stars [2], the hot spots on dust rings in the
galactic centre [3, 4, 5, 6], interstellar medium [2], and around pulsars [5],
etc.) does not only change the dynamics of the plasma medium but also
significantly modifies the basic properties of electrostatic dust-acoustic
(DA) waves (DAWs) [5, 6, 7] and dust-ion-acoustic (DIA) waves (DIAWs) [1, 2,
3]. Esfandyari-Kalejahi et al. [7] studied large amplitude DA solitary waves
in EPI plasma, and demonstrated that the amplitude of the DA solitary waves
increases with the charge state of dust grains. El-Tantawy et al. [8]
investigated DIAWs in EPI dusty plasma medium (EPIDPM), and observed that the
amplitude and width of the positive potential increases with the number
density and charge state of the dust grains.

The signature of non-thermal electrons in space plasmas has been observed by
the Viking [9] and Freja satellites [10]. Cairns et al. [11] first introduced
non-thermal distribution and associated parameter $\alpha$ demonstrating the
measurement of deviation from Maxwellian distribution for explaining the high-
energy tails in space plasmas. Banerjee and Maitra [3] investigated DIA
solitons and double layers in presence of non-thermal positrons and electrons.
Paul et al. [4] considered a four-component plasma model having warm ions,
stationary dust grains, non-thermal electrons and positrons, and studied
DIAWs, and found that the system supports only positive potential super-
solitons.

The modulational instability (MI) of electrostatic waves and formation of
associated rogue waves (RWs) have been governed by the nonlinear Schrödinger
equation (NLSE). A number of authors studied the MI of various kind of waves
in different plasma medium [12, 13, 14, 15, 16, 17]. Guo et al. [14]
investigated the MI of DIAWs in EPIDPM in presence of non-extensive electrons
and positrons. Bains et al. [15] considered iso-thermal electrons and
positrons to observe the MI condition of DIAWs, and found that the critical
wave number decreases with ion temperature. El-Labany et al. [16] studied the
MI of DAWs in a three-component plasma medium having non-thermal plasma
species, and found that the height of the DA RWs (DARWs) increases with the
non-thermality of plasma species. El-Tantawy et al. [17] examined ion-acoustic
(IA) super RWs in a two-component non-thermal plasma having inertial ions and
inerialess electrons, and reported that the nonlinearity as well as MI growth
rate of the IA waves increases with the non-thermality of electrons. To the
best of our knowledge, the effects of non-thermal electrons and positrons, and
stationary negatively charged massive dust grains on the MI of DIAWs and
associated DIA RWs (DIARWs) have not yet been investigated. Therefore, in our
present work, we will examine the MI of DIAWs and associated DIARWs in a four-
component EPIDPM.

The manuscript is organized in the following order: The basic governing
equations are presented in Section 2. The derivation of the NLSE is
demonstrated in Section 3. The MI of DIAWs is provided in Section 4. The
DIARWs are exhibited in Section 5. Finally, the conclusion is presented in
Section 6.

##  2 Model Equations

We consider a four-component unmagnetized plasma model having inertial warm
ions, inertialess non-thermal electrons and positrons, and stationary
negatively charged massive dust grains. At equilibrium, the quasi-neutrality
condition can be expressed as $n_{e0}+Z_{d}n_{d0}=n_{p0}+Z_{i}n_{i0}$, where
$n_{e0}$, $n_{d0}$, $n_{p0}$, and $n_{i0}$ are, respectively, the equilibrium
number densities of electrons, dust grains, positrons, and ions. $Z_{i}$ is
the charge state of the positive ion and $Z_{d}$ is the number of electrons
residing on the dust grains surface. The normalized governing equations can be
written in the following form

$\displaystyle\frac{\partial n_{+}}{\partial t}+\frac{\partial}{\partial
x}(n_{+}u_{+})=0,$ (1) $\displaystyle\frac{\partial u_{+}}{\partial
t}+u_{+}\frac{\partial u_{+}}{\partial x}+\sigma n_{+}\frac{\partial
n_{+}}{\partial x}=-\frac{\partial\phi}{\partial x},$ (2)
$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}=\mu_{e}n_{e}+\mu_{d}-n_{+}-(\mu_{e}+\mu_{d}-1)n_{p},\ $ (3)

where $n_{+}$ is the warm ions number density normalized by it’s equilibrium
value $n_{+0}$; $u_{+}$ is the ion fluid speed normalized by the IA wave speed
$C_{+}=(Z_{+}k_{B}T_{e}/m_{+})^{1/2}$ (with $T_{e}$ being the electron
temperature, $m_{+}$ being the ion mass, and $k_{B}$ being the Boltzmann
constant); $\phi$ is the electrostatic wave potential normalized by
$k_{B}T_{e}/e$ (with $e$ being the magnitude of single electron charge); the
time and space variables are normalized by $\omega_{p+}^{-1}=(m_{+}/4\pi
Z_{+}^{2}e^{2}n_{+0})^{1/2}$ and $\lambda_{D+}=(k_{B}T_{+}/4\pi
Z_{+}n_{+0}e^{2})^{1/2}$, respectively, and $T_{+}$ being the ion temperature;
$p_{+}=p_{+0}(N_{+}/n_{+0})^{\gamma}$ [with $p_{+0}$ being the equilibrium
adiabatic pressure of the ion, and $\gamma=(N+2)/N$, where $N$ is the degree
of freedom. For one-dimensional case: $N=1$ then $\gamma=3$, and
$p_{+0}=n_{+0}k_{B}T_{+}$]. Other plasma parameters are considered as
$\sigma=3T_{+}/Z_{+}T_{e}$, $\mu_{e}=n_{e0}/Z_{+}n_{+0}$, and
$\mu_{d}=Z_{d}n_{d0}/Z_{+}n_{+0}$. The expression for the number density of
electron (following the Cairns’ non-thermal distribution [11, 18]) can be
written as

$\displaystyle
n_{e}=\Bigg{[}1-\frac{4\alpha_{e}}{1+3\alpha_{e}}\phi+\frac{4\alpha_{e}}{1+3\alpha_{e}}\phi^{2}\Bigg{]}\mbox{exp}(\phi),$
$\displaystyle=1+F_{1}\phi+F_{2}\phi^{2}+F_{3}\phi^{3}+\cdot\cdot\cdot,\ $ (4)

where $F_{1}=(1-\alpha_{e})/(1+3\alpha_{e})$, $F_{2}=1/2$,
$F_{3}=(15\alpha_{e}+1)/(18\alpha_{e}+6)$, and $\alpha_{e}$ being the non-
thermality of electrons. The number density of positron (following the Cairns’
non-thermal distribution [11, 18]) can be written as

$\displaystyle
n_{p}=\Bigg{[}1+\frac{4\alpha_{p}}{1+3\alpha_{p}}\delta\phi+\frac{4\alpha_{p}}{1+3\alpha_{p}}\delta^{2}\phi^{2}\Bigg{]}\mbox{exp}(-\delta\phi),$
$\displaystyle=1+F_{4}\phi+F_{5}\phi^{2}+F_{6}\phi^{3}+\cdot\cdot\cdot,\ $ (5)

where $F_{4}=[(\alpha_{p}-1)\delta]/(1+3\alpha_{p})$, $F_{5}=\delta^{2}/2$,
$F_{6}=[(15\alpha_{p}+1)\delta^{3}]/(18\alpha_{p}+6)$, and $\alpha_{p}$ being
the non-thermality of positrons. The ratio of $T_{e}$ to $T_{p}$ (positron
temperature) is defined by $\delta=T_{e}/T_{p}$. By substituting Eq. (4) and
(5) into Eq. (3) and expanding up to third order in $\phi$, we can write

$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}+n_{+}=1+\mu_{d}+F_{7}\phi+F_{8}\phi^{2}+F_{9}\phi^{3}+\cdot\cdot\cdot,\
$ (6)

where

$\displaystyle F_{8}=\frac{\mu_{e}-\eta\delta^{2}}{2},$ $\displaystyle
F_{7}=\frac{\mu_{e}(1-\alpha_{e})(1+3\alpha_{p})-\eta(\alpha_{p}-1)(1+3\alpha_{e})\delta}{(1+3\alpha_{e})(1+3\alpha_{p})},$
$\displaystyle
F_{9}=\frac{\mu_{e}(15\alpha_{e}+1)(18\alpha_{p}+6)+\eta(15\alpha_{p}+1)(18\alpha_{e}+6)\delta^{3}}{(18\alpha_{e}+6)(18\alpha_{p}+6)},\
$

where $\eta=\mu_{e}+\mu_{d}-1$, and the terms containing $F_{7}$, $F_{8}$, and
$F_{9}$ in Eq. (6) are due to the contribution of the non-thermal electrons
and positrons.

##  3 Derivation of the NLSE

To study the MI of DIAWs, we will derive a standard NLSE by employing the
reductive perturbation method. So, we first introduce the stretched co-
ordinates [18, 19]

$\displaystyle\xi=\epsilon(x-v_{g}t),$ (7) $\displaystyle\tau=\epsilon^{2}t,\
$ (8)

where $v_{g}$ is the group speed and $\epsilon$ is a small parameter. We can
write the dependent variables as [20, 21, 22, 23]

$\displaystyle
n_{+}=1+\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}n_{+l}^{(m)}(\xi,\tau)\mbox{exp}[il(kx-\omega
t)],$ (9) $\displaystyle
u_{+}=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}u_{+l}^{(m)}(\xi,\tau)\mbox{exp}[il(kx-\omega
t)],$ (10)
$\displaystyle\phi=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}\phi_{l}^{(m)}(\xi,\tau)\mbox{exp}[il(kx-\omega
t)],\ $ (11)

where $k$ ($\omega$) is real variable representing the carrier wave number
(frequency). The derivative operators in the above equations are treated as
follows [24, 25, 26, 27]:

$\displaystyle\frac{\partial}{\partial x}\rightarrow\frac{\partial}{\partial
x}+\epsilon\frac{\partial}{\partial\xi},$ (12)
$\displaystyle\frac{\partial}{\partial t}\rightarrow\frac{\partial}{\partial
t}-\epsilon
v_{g}\frac{\partial}{\partial\xi}+\epsilon^{2}\frac{\partial}{\partial\tau}.\
$ (13)

Now, by substituting Eqs. (7)$-$(13), into Eqs. (1), (2), and (6), and
selecting the terms containing $\epsilon$, the first order ($m=1$ and $l=1$)
reduced equations can provide the dispersion relation of DIAWs

$\displaystyle\omega^{2}=\frac{k^{2}}{k^{2}+F_{7}}+\sigma k^{2}.\ $ (14)

We again consider the second harmonic with ($m=2$ and $l=1$) and with the
compatibility condition, we have obtained the group velocity of DIAWs

$\displaystyle v_{g}=\frac{\omega^{2}-\beta^{2}}{k\omega},\ $ (15)

where $\beta=\omega^{2}-\sigma k^{2}$. The amplitude of the second-order
harmonics is found to be proportional to $|\phi_{1}^{(1)}|^{2}$

$\displaystyle
n_{+2}^{(2)}=F_{10}|\phi_{1}^{(1)}|^{2},~{}~{}~{}~{}~{}n_{+0}^{(2)}=F_{13}|\phi_{1}^{(1)}|^{2},$
(16) $\displaystyle
u_{+2}^{(2)}=F_{11}|\phi_{1}^{(1)}|^{2},~{}~{}~{}~{}~{}u_{+0}^{(2)}=F_{14}|\phi_{1}^{(1)}|^{2},$
(17)
$\displaystyle\phi_{2}^{(2)}=F_{12}|\phi_{1}^{(1)}|^{2},~{}~{}~{}~{}~{}\phi_{0}^{(2)}=F_{15}|\phi_{1}^{(1)}|^{2},\
$ (18)

where

$\displaystyle F_{10}=\frac{\sigma
k^{6}+3\omega^{2}k^{4}+2F_{12}\beta^{2}k^{2}}{2\beta^{3}},$ $\displaystyle
F_{11}=\frac{\omega F_{10}\beta^{2}-\omega k^{4}}{k\beta^{2}},$ $\displaystyle
F_{12}=\frac{k^{4}(3\omega^{2}+\sigma
k^{2})-2F_{8}\beta^{3}}{2\beta^{2}[(4k^{2}+F_{7})\beta-k^{2}]},$
$\displaystyle F_{13}=\frac{2v_{g}\omega k^{3}+k^{2}\omega^{2}+\sigma
k^{4}+F_{15}\beta^{2}}{(v_{g}^{2}-\sigma)\beta^{2}},$ $\displaystyle
F_{14}=\frac{v_{g}F_{13}\beta^{2}-2\omega k^{3}}{\beta^{2}},$ $\displaystyle
F_{15}=\frac{k^{2}(2\omega v_{g}k+\sigma
k^{2}+\omega^{2})-2F_{8}(v_{g}^{2}-\sigma)\beta^{2}}{\beta^{2}[F_{7}(v_{g}^{2}-\sigma)-1]}.\
$

Finally, the third harmonic modes ($m=3$) and ($l=1$), with the help of Eqs.
(14)$-$(18), give a set of equations, which can be reduced to the following
NLSE:

$\displaystyle
i\frac{\partial\Phi}{\partial\tau}+P\frac{\partial^{2}\Phi}{\partial\xi^{2}}+Q\Phi|\Phi|^{2}=0,\
$ (19)

we have considered $\Phi$=$\phi_{1}^{(1)}$ for simplicity, and in Eq. (19),
$P$ is the dispersion coefficient which can be written as

$\displaystyle P=\frac{\beta(4\sigma\omega k^{2}-3v_{g}k\omega^{2}-\sigma
v_{g}k^{3})}{2\omega^{2}k^{2}},\ $ (20)

and $Q$ is the nonlinear coefficient which can be written as

$\displaystyle
Q=\frac{\beta^{2}(3F_{9}+2F_{8}F_{12}+2F_{8}F_{15})-Q^{\prime}}{2\omega
k^{2}},\ $ (21)

where $Q^{\prime}=k^{2}[(\sigma k^{2}+\omega^{2})(F_{10}+F_{13})+2\omega
k(F_{11}+F_{14})]$. The space and time evolution of the DIAWs are directly
governed by the coefficients $P$ and $Q$.

Figure 1: The variation of $P/Q$ with $k$ for different values of $\sigma$
when $\alpha_{e}=0.5$, $\alpha_{p}=0.5$, $\delta=1$, $\mu_{d}=0.02$, and
$\mu_{e}=1.5$. Figure 2: The variation of $\Gamma$ with $\widetilde{k}$ for
different values of $\mu_{d}$ when $\alpha_{e}=0.5$, $\alpha_{p}=0.5$,
$\delta=1$, $\mu_{e}=1.5$, $\sigma=0.3$, $\phi_{0}=0.5$, and $k=0.7$. Figure
3: The variation of $|\phi_{1}|$ with $\xi$ for different values of
$\alpha_{e}$ when $\alpha_{p}=0.5$, $\delta=1$, $\mu_{d}=0.02$, $\mu_{e}=1.5$,
$\sigma=0.3$, $\tau=0$, and $k=0.7$. Figure 4: The variation of $|\phi_{2}|$
with $\xi$ for different values of $\alpha_{p}$ when $\alpha_{e}=0.5$,
$\delta=1$, $\mu_{d}=0.02$, $\mu_{e}=1.5$, $\sigma=0.3$, $\tau=0$, and
$k=0.7$. Figure 5: The variation of first-order (dashed green curve) and
second-order (solid red curve) rational solutions of NLSE at $k=0.7$ and
$\tau=0$.

##  4 Instability analysis

To study the MI of DIAWs, we consider the linear solution of the Eq. (19) in
the form $\Phi=\widetilde{\Phi}e^{iQ|\widetilde{\Phi}|^{2}\tau}$+c.c., where
$\widetilde{\Phi}=\widetilde{\Phi}_{0}+\epsilon\widetilde{\Phi}_{1}$ and
$\widetilde{\Phi}_{1}=\widetilde{\Phi}_{1,0}e^{i(\widetilde{k}\xi-\widetilde{\omega}{\tau})}+c.c$.
We note that the amplitude depends on the frequency, and that the perturbed
wave number $\widetilde{k}$ and frequency $\widetilde{\omega}$ which are
different from $k$ and $\omega$. Now, substituting these into Eq. (19), one
can easily obtain the following nonlinear dispersion relation [18, 19]

$\displaystyle\widetilde{\omega}^{2}=P^{2}\widetilde{k}^{2}\Big{(}\widetilde{k}^{2}-\frac{2|\widetilde{\Phi}_{0}|^{2}}{P/Q}\Big{)}.\
$ (22)

It is observed here that the ratio $P/Q$ is negative (i.e., $P/Q<0$), the
DIAWs will be modulationally stable. On the other hand, if the ratio $P/Q$ is
positive (i.e., $P/Q>0$), the DIAWs will be modulationally unstable. We have
graphically examined the effect of temperature of the ion and electron as well
as the charge state of the warm positive ion in recognizing the stable (i.e.,
$k<k_{c}$) and unstable (i.e., $k>k_{c}$) domains of DIAWs in Fig. 1, and it
is clear from this figure that (a) the plasma system under consideration
supports the DIAWs with either stable (i.e., $P/Q<0$) or unstable (i.e.,
$P/Q>0$); (b) the stable domain increases with the increase in the value of
the charge state of the warm positive ion when $T_{+}$ and $T_{e}$ are
invariant; (c) the $k_{c}$ decreases with the increase in the value of the ion
temperature while increases with $T_{e}$ for a fixed value of $Z_{+}$, and
this result agrees with the result of Bains et al. [15].

It is obvious from Eq. (22) that the DIAWs becomes modulationally unstable
when $\widetilde{k}_{c}>\widetilde{k}$ in the regime $P/Q>0$, where
$\widetilde{k}_{c}=\sqrt{2(Q/P)}{|\widetilde{\Phi}_{0}|}$. The growth rate
$\Gamma$ of the modulationally unstable DIAWs is [14, 18, 19] given by

$\displaystyle\Gamma=|P|\widetilde{k}^{2}\sqrt{\frac{\widetilde{k}_{c}^{2}}{\widetilde{k}^{2}}-1}.\
$ (23)

The variation of the $\Gamma$ with $\widetilde{k}$ for different values of
$\mu_{d}$ can be seen in Fig. 2. It is obvious from this figure that (a) the
maximum value of $\Gamma$ decreases (increases) with the increase in the
values of $Z_{d}$ ($Z_{+}$) for a constant value of $n_{d0}$ and $n_{+0}$; (b)
as we increase the value of $n_{d0}$ ($n_{+0}$), the maximum value of the
$\Gamma$ decreases (increases) when $Z_{d}$ and $Z_{+}$ remain constant. The
physics of this result is that the nonlinearity of the plasma system increases
with the charge state and number density of the warm ion, but decreases with
the charge state and number density of the stationary negatively charged
massive dust grains.

##  5 Rogue waves

The NLSE (19) has a variety of rational solutions, among them there is a
hierarchy of rational solution that are localized in both the $\xi$ and $\tau$
variables. The first-order rational solution of Eq. (19) can be written as
[28]

$\displaystyle\Phi_{1}(\xi,\tau)=\sqrt{\frac{2P}{Q}}\Big{[}\frac{4(1+4iP\tau)}{1+16P^{2}\tau^{2}+4\xi^{2}}-1\Big{]}\mbox{exp}(2iP\tau).\
$ (24)

Equation (24) reveals that a significant amount of DIAWs energy is
concentrated into a comparatively small region in EPIDPM. We have numerically
analysed Eq. (24) in Fig. 3 to illustrate the influence of non-thermal
electrons on the formation of DIARWs, and it can be seen from the figure that
(i) the height and width of the DIARWs increase as we increase in the value of
non-thermality of the electrons (via $\alpha_{e}$); (ii) The physics of this
result is that with increasing the value of $\alpha_{e}$, the nonlinearity of
the plasma system is increasing, which leads to increase the height and the
width of the DIARWs. This result agrees with the result of El-Labany et al.
[16].

The interaction of the two or more first-order RWs can generate higher-order
RWs which has a more complicated nonlinear structure. The second-order
rational solution of Eq. (19) can be written as [28]

$\displaystyle\Phi_{2}(\xi,\tau)=\sqrt{\frac{P}{Q}}\Big{[}1+\frac{G_{2}(\xi,\tau)+iM_{2}(\xi,\tau)}{D_{2}(\xi,\tau)}\Big{]}\mbox{exp}(i\tau
P),\ $ (25)

where

$\displaystyle
G_{2}(\xi,\tau)=\frac{-\xi^{4}}{2}-6(P\xi\tau)^{2}-10(P\tau)^{4}$
$\displaystyle\hskip 5.69046pt-\frac{3\xi^{2}}{2}-9(P\tau)^{2}+\frac{3}{8},$
$\displaystyle
M_{2}(\xi,\tau)=-P\tau\Big{[}\xi^{4}+4(P\xi\tau)^{2}+4(P\tau)^{4}$
$\displaystyle\hskip 5.69046pt-3\xi^{2}+2(P\tau)^{2}-\frac{15}{4}\Big{]},$
$\displaystyle
D_{2}(\xi,\tau)=\frac{\xi^{6}}{12}+\frac{\xi^{4}(P\tau)^{2}}{2}+\xi^{2}(P\tau)^{4}$
$\displaystyle\hskip
5.69046pt+\frac{\xi^{4}}{8}+\frac{9(P\tau)^{4}}{2}-\frac{3(P\xi\tau)^{2}}{2}$
$\displaystyle\hskip
5.69046pt+\frac{9\xi^{2}}{16}+\frac{33(P\tau)^{2}}{8}+\frac{3}{32}.\ $

Figure 4 represents the second-order DIARWs associated with DIAWs in the
modulationally unstable domain (i.e., $P/Q>0$). The increase in the value of
$\alpha_{p}$ does not only cause to change the height of the DIARWs but also
causes to change the width of the DIARWs. Figure 5 indicates the first-order
and second-order solution of the NLSE at $\tau=0$, and it is clear from this
figure that (a) the second-order rational solution has double structures
compared with the first-order rational solution; (b) the height of the second-
order rational solution is always greater than the height of the first-order
rational solution; (c) the potential profile of the second-order rational
solution becomes more spiky (i.e., the taller height and narrower width) than
the first-order rational solution; (d) the second (first) order rational
solution has four (two) zeros symmetrically located on the $\xi$-axis; (e) the
second (first) order rational solution has three (one) local maxima.

##  6 Conclusion

In this study, we have performed a nonlinear analysis of DIAWs in an
unmagnetized EPIDPM having stationary massive nagetively charged dust grains,
inertial warm positive ions, and inertialess non-thermal Cairns’ distributed
electrons and positrons. The evolution of DIAWs is governed by the standard
NLSE, and the coefficients $P$ and $Q$ of NLSE can recognize the
modulationally stable and unstable domains of DIAWs. It is observed that the
critical wave number, for which the MI sets in, decreases with ion temperature
but increases with electron temperature. The nonlinearity as well as the
height and width of the DIARWs increases with the non-thermality of electrons
and positrons. The limitation of this work is that the gravitational and
magnetic fields are not consider. In future and for better understanding,
someone can investigate the nonlinear propagation in a four-component EPIDPM
by considering the gravitational and magnetic fields. However, these results
may be applicable in understanding the conditions of the MI of DIAWs and
associated DIARWs in Jupiter’s magnetosphere [1], Milky Way [2], auroral zone
[3], accretion disks near neutron stars [2], the hot spots on dust rings in
the galactic centre [3, 4, 5, 6], interstellar medium [2], and around pulsars
[5], etc.

## References

  * [1] A. Paul and A. Bandyopadhyay, Astrophys Space Sci. 361, 172 (2016).
  * [2] S. Sardar, et al., Phys. Plasmas 23, 073703 (2016).
  * [3] G. Banerjee and S. Maitra, Phys. Plasmas 23, 123701 (2016).
  * [4] A. Paul, et al., Phys. Plasmas 24, 013707 (2017).
  * [5] N. Jehan, et al., Phys. Scr. 80, 035506 (2009).
  * [6] E. Saberian, et al., Plasma Phys. Rep. 43, 83 (2017).
  * [7] A. Esfandyari-Kalejahi, et al., Phys. Plasmas 19, 082308 (2012).
  * [8] S.A. El-Tantawy, et al., Phys. Plasmas 18, 052113 (2011).
  * [9] R. Boström, IEEE Trans. Plasma Sci. 20, 756 (1992).
  * [10] P.O. Dovner, et al., Geophys. Res. Lett. 21, 1827 (1994).
  * [11] R. Cairns, et al., J. Geophys. Res. 22, 2709 (1995).
  * [12] I. Kourakis and P.K. Shukla, Phys. Plasmas 10, 3459 (2003).
  * [13] R. Fedele, Phys. Scr. 65, 502 (2002).
  * [14] S. Guo, et al., Ann. Phys. 332, 38 (2012).
  * [15] A.S. Bains, et al., Astrophys Space Sci. 343, 293 (2013).
  * [16] S.K. El-Labany, et al., Phys. Plasmas 22, 073702 (2015).
  * [17] S.A. El-Tantawy, et al., Phys. Plasmas 20, 072102 (2013).
  * [18] N.A. Chowdhury, et al., Chaos 27, 093105 (2017).
  * [19] N.A. Chowdhury, et al., Phys. plasmas 24, 113701 (2017).
  * [20] M.H. Rahman, et al., Phys. Plasmas 25, 102118 (2018).
  * [21] N. Ahmed, et al., Chaos 28, 123107 (2018).
  * [22] S. Jahan, et al., Commun. Theor. Phys. 71, 327 (2019).
  * [23] M. Hassan, et al., Commun. Theor. Phys. 71, 1017 (2019).
  * [24] R.K. Shikha, et al., Eur. Phys. J. D 73, 177 (2019).
  * [25] M.H. Rahman, et al., Chinese J. Phys. 56, 2061 (2018).
  * [26] N.A. Chowdhury, et al., Plasma Phys. Rep. 45, 459 (2019).
  * [27] S.K. Paul, et al., Pramana-J. Phys. 94, 58 (2020).
  * [28] A. Ankiewicz, et al., J. Phys. A 43, 12002 (2010).