I. INTRODUCTION

Let $(\mathrm{X},\mathrm{\Sigma },\mu )$ be a $\sigma$ -finite measure space. Then a mapping $\mathrm{T}$ from $\mathrm{X}$ into $\mathrm{X}$ is said to be a measurable transformation if ${\mathrm{T}}^{-1}(\mathrm{E})\in \mathrm{\Sigma }$ for every $E\in \mathrm{\Sigma }$. A measurable transformation $\mathrm{T}$ is said to be non-singular if $\mu ({\mathrm{T}}^{-1}(\mathrm{E}))=0$ whenever $\mu (\mathrm{E})=0$. If $\mathrm{T}$ is non-singular then the measure $\mu {\mathrm{T}}^{-1}$ defined as $\mu {\mathrm{T}}^{-1}(\mathrm{E})=\mu ({\mathrm{T}}^{-1}(\mathrm{E}))$ for every $\mathrm{E}$ in $\mathrm{\Sigma }$, is an absolutely continuous measure on $\mathrm{\Sigma }$ with respect to $\mu$. Since $\mu$ is a $\sigma$ -finite measure, then by the Radon-Nikodym theorem, there exists a non-negative function ${\mathbf{f}}_0$ in ${\mathrm{L}}^1(\mu )$ such that $\mu {\mathrm{T}}^{-1}(\mathrm{E})={\int }_{\mathrm{E}}{\mathbf{f}}_0\mathrm{d}\mu$ for every $\mathrm{E}\in \mathrm{\Sigma }$. The function ${\mathbf{f}}_0$ is called the Radon-Nikodym derivative of $\mu {\mathrm{T}}^{-1}$ with respect to $\mu$.

Every non-singular measurable transformation $\mathrm{T}$ from $\mathrm{X}$ into itself induces a linear transformation ${\mathbf{C}}_{\mathrm{T}}$ on ${\mathrm{L}}^{\mathrm{p}}(\mu )$ defined as ${\mathbf{C}}_{\mathrm{T}}\mathbf{f}=\mathbf{f}\circ \mathbf{T}$ for every $\mathbf{f}$ in ${\mathrm{L}}^{\mathrm{p}}(\mu )$. In case ${\mathbf{C}}_{\mathrm{T}}$ is continuous from ${\mathrm{L}}^{\mathrm{p}}(\mu )$ into itself, then it is called a composition operator on ${\mathrm{L}}^{\mathrm{p}}(\mu )$ induced by $\mathrm{T}$. We restrict our study of the composition operators on ${\mathrm{L}}^2(\mu )$ which has Hilbert space structure. If $\mathbf{u}$ is an essentially bounded complex-valued measurable function on $\mathrm{X}$, then the mapping ${\mathbf{M}}_{\mathbf{u}}$ on ${\mathrm{L}}^2(\mu )$ defined by ${\mathbf{M}}_{\mathbf{u}}\mathbf{f}=\mathbf{u}\cdot \mathbf{f}$, is a continuous operator with range in ${\mathrm{L}}^2(\mu )$. The operator ${\mathbf{M}}_{\mathbf{u}}$ is known as the multiplication operator induced by $\mathbf{u}$.

A composite multiplication operator is linear transformation acting on a set of complex valued $\mathrm{\Sigma }$ measurable functions $f$ of the form

Where $\mathbf{u}$ is a complex valued, $\mathrm{\Sigma }$ measurable function. In case $\mathbf{u}=1$ almost everywhere, ${\mathrm{M}}_{\mathrm{u},\mathrm{T}}$ becomes a composition operator, denoted by ${\mathrm{C}}_{\mathrm{T}}$.

In the study considered is the using conditional expectation of composite multiplication operator on ${\mathbf{L}}^2$ -spaces. For each $f\in {\mathrm{L}}^p(\mathrm{X},\mathrm{\Sigma },\mu )$, $1\le p\le \mathrm{\infty }$, there exists an unique ${\mathrm{T}}^{-1}(\mathrm{\Sigma })$ -measurable function $\mathrm{E}(f)$ such that

for every ${\mathrm{T}}^{-1}(\mathrm{\Sigma })$ -measurable function $\mathbf{g}$, for which the left integral exists. The function $\mathrm{E}(\mathbf{f})$ is called the conditional expectation of $\mathbf{f}$ with respect to the subalgebra ${\mathrm{T}}^{-1}(\mathrm{\Sigma })$. As an operator of ${\mathrm{L}}^{\mathrm{p}}(\mu )$, $\mathrm{E}$ is the projection onto the closure of range of $\mathrm{T}$ and $\mathrm{E}$ is the identity on ${\mathrm{L}}^{\mathrm{p}}(\mu )$, $\mathfrak{p}\ge 1$ if and only if ${\mathrm{T}}^{-1}(\mathrm{\Sigma })=\mathrm{\Sigma }$. Detailed discussion of $\mathrm{E}$ is found in [1-4].

a) Normal operator

Let H be a Complex Hilbert Space. An operator T on H is called normal operator if ${\mathrm{T}}^{\ast }\mathrm{T}={\mathrm{TT}}^{\ast }$

b) Quasi-normal operator

Let $\mathrm{H}$ be a Complex Hilbert Space. An operator $\mathrm{T}$ on $\mathrm{H}$ is called Quasi-normal operator if $\mathrm{T}{\mathrm{T}}^{\ast }\mathrm{T}={\mathrm{T}}^{\ast }\mathrm{T}\mathrm{T}$, ie, ${\mathrm{T}}^{\ast }\mathrm{T}$ commute with $\mathrm{T}$

c) Quasi $p$ -normal operator [13]

Let $\mathrm{H}$ be a Complex Hilbert Space. An operator $\mathrm{T}$ on $\mathrm{H}$ is called Quasi-normal operator if $T^{\ast }T(T+T^{\ast })=(T+T^{\ast })T^{\ast }T$

d) 2-Power -normal operator

Let $\mathbf{H}$ be a Complex Hilbert Space. An operator $\mathrm{T}$ on $\mathbf{H}$ is called 2 power-normal operator if $T^2{T}^{\ast }=T^{\ast }{T}^2$

e) Class $Q$ -operator [14]

Let $\mathrm{H}$ be a Complex Hilbert Space. An operator $\mathrm{T}$ on $\mathrm{H}$ is called Quasi-normal operator if $T^{\ast 2}{T}^2=(T^{\ast }T{)}^2$.

II. RELATED WORK IN THE FIELD

The study of weighted composition operators on ${\mathbf{L}}^2$ spaces was initiated by R. K. Singh and D. C. Kumar [5]. During the last thirty years, several authors have studied the properties of various classes of weighted composition operator. Boundedness of the composition operators in $L^p(\sum )$, spaces, where the measure spaces are $\sigma$ -finite, appeared already in [6]. Also boundedness of weighted operators on $\mathrm{C}(\mathrm{X},\mathrm{E})$ has been studied in [7]. Recently S. Senthil, P. Thangaraju, Nithya M, Surya devi B and D. C. Kumar, have proved several theorems on n-normal, n-quasi-normal, k-paranormal, and (n,k) paranormal of composite multiplication operators on ${\mathbf{L}}^2$ spaces [8-12]. In this paper we investigate composite multiplication operators on ${\mathrm{L}}^2(\mu )$ -space become Quasi-P-Normal operators and n-Power class Q operator have been obtained in terms of radon-nikodym derivative ${\mathbf{f}}_0$.

III. CHARACTERIZATION ON COMPOSITE MULTIPLICATION OF QUASI P NORMAL OPERATORS ON $L^2$ -SPACE

a) Proposition

Let the composite multiplication operator $M_{u,T}\in B(L^2(\mu ))$. Then for $u\ge 0$

• (i) ${\mathbf{M}}_{\mathrm{u},\mathrm{T}}^{\ast }{\mathbf{M}}_{\mathrm{u},\mathrm{T}}\mathbf{f}={\mathbf{u}}^2{\mathbf{f}}_0\mathbf{f}$
• (ii) ${\mathbf{M}}_{\mathrm{u},\mathrm{T}}{\mathbf{M}}_{\mathrm{u},\mathrm{T}}^{\ast }\mathbf{f}={\mathbf{u}}^2\circ \mathbf{T}\cdot {\mathbf{f}}_0\circ \mathbf{T}\cdot \mathbf{E}(\mathbf{f})$
• (iii) ${\mathbf{M}}_{\mathrm{u},\mathrm{T}}^{\mathrm{n}}(\mathbf{f})=({\mathbf{C}}_{\mathrm{T}}{\mathbf{M}}_{\mathrm{u}}{)}^{\mathrm{n}}(\mathbf{f})={\mathbf{u}}_{\mathrm{n}}(\mathbf{f}\circ {\mathbf{T}}^{\mathrm{n}}),{\mathbf{u}}_{\mathrm{n}}=\mathbf{u}\circ \mathbf{T}.\mathbf{u}\circ {\mathbf{T}}^2.\mathbf{u}\circ {\mathbf{T}}^3\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots$
• (iv) ${\mathbf{M}}_{\mathrm{u},\mathrm{T}}^{\ast }\mathbf{f}=\mathbf{u}{\mathbf{f}}_0\cdot \mathbf{E}(\mathbf{f})\circ {\mathbf{T}}^{-1}$
• (v) ${\mathbf{M}}^{{\ast }^{\mathrm{n}}}{}_{\mathrm{u},\mathrm{T}}\mathbf{f}=\mathbf{u}{\mathbf{f}}_0\cdot \mathbf{E}(\mathbf{u}{\mathbf{f}}_0)\circ {\mathbf{T}}^{-(\mathrm{n}-1)}\cdot \mathbf{E}(\mathbf{f})\circ {\mathbf{T}}^{-\mathrm{n}}$

where $\mathrm{E}({\mathrm{uf}}_0)\circ {\mathrm{T}}^{-(\mathrm{n}-1)}=\mathrm{E}({\mathrm{uf}}_0)\circ {\mathrm{T}}^{-1}\cdot \mathrm{E}({\mathrm{uf}}_0)\circ {\mathrm{T}}^{-2}\dots \mathrm{E}({\mathrm{uf}}_0)\circ {\mathrm{T}}^{-(\mathrm{n}-1)}$

Theorem 3.1

Let the ${\mathbf{M}}_{\mathrm{u},\mathrm{T}}$ be a composite multiplication operator on ${\mathrm{L}}^2(\mu )$. Then the following statements are equivalent (i) ${\mathbf{M}}_{\mathrm{u},\mathrm{T}}$ is Quasi p-normal operator

Proof:

For $f\in L^2(\mu )$, $M_{u,T}$ is Quasi P-normal operator if

Suppose, ${\mathbf{M}}_{\mathrm{u},\mathrm{T}}$ is Quasi P-normal operator. Then

Theorem 3.2

Let the ${\mathrm{M}}_{\mathrm{u},\mathrm{T}}$ be a composite multiplication operator on ${\mathrm{L}}^2(\mu )$. Then the following statements are equivalent (i) ${\mathbf{M}}_{\mathrm{u},\mathrm{T}}^{\ast }$ is Quasi p-normal operator

(ii) $=\mathrm{h}\circ \mathrm{T}{\mathrm{u}}^2\circ \mathrm{T}\mathrm{E}(\mathrm{h})\mathrm{E}(\mathrm{u})\mathrm{E}(\mathrm{f})\circ {\mathrm{T}}^{-1}+\mathrm{h}\circ \mathrm{T}\mathrm{u}\circ \mathrm{T}{\mathrm{u}}^2\circ \mathrm{T}\mathrm{f}\circ \mathrm{T}$

Proof:

For $f\in L^2(\mu )$, ${\mathbf{M}}_{\mathfrak{u},\mathrm{T}}^{\ast }$ is Quasi P-normal operator if

and then we have

Consider

Suppose ${\mathbf{M}}_{\mathrm{u},\mathrm{T}}^{\ast }$ is Quasi p-normal operator. Then

IV. CHARACTERIZATIONS ON N POWER CLASS Q COMPOSITE MULTIPLICATION OPERATOS ON ${\mathbf{L}}^2$ -SPACE