## I. INTRODUCTION Let $(\mathrm{X},\Sigma,\mu)$ 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 $\mathrm{T}^{-1}(\mathrm{E})\in \Sigma$ for every $E\in \Sigma$. 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 $\Sigma$, is an absolutely continuous measure on $\Sigma$ 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 $\mathrm{E}\in \Sigma$. 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 $\Sigma$ measurable functions $f$ of the form $$ \mathrm {M} _ {\mathrm {u}, \mathrm {T}} (\mathrm {f}) = \mathrm {C} _ {\mathrm {T}} \mathrm {M} _ {\mathrm {u}} (\mathrm {f}) = \mathrm {u} \circ \mathrm {T} \quad \mathrm {f} \circ \mathrm {T} $$ Where $\mathbf{u}$ is a complex valued, $\Sigma$ measurable function. In case $\mathbf{u} = 1$ almost everywhere, $\mathrm{M}_{\mathrm{u,T}}$ becomes a composition operator, denoted by $\mathrm{C_T}$. In the study considered is the using conditional expectation of composite multiplication operator on $\mathbf{L}^2$ -spaces. For each $f \in \mathrm{L}^p(\mathrm{X}, \Sigma, \mu)$, $1 \leq p \leq \infty$, there exists an unique $\mathrm{T}^{-1}(\Sigma)$ -measurable function $\mathrm{E}(f)$ such that $$ \int_ {A} g f d \mu = \int_ {A} g E (f) d \mu $$ for every $\mathrm{T}^{-1}(\Sigma)$ -measurable function $\mathbf{g}$, for which the left integral exists. The function $\operatorname{E}(\mathbf{f})$ is called the conditional expectation of $\mathbf{f}$ with respect to the subalgebra $\mathrm{T}^{-1}(\Sigma)$. 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 $\mathrm{L}^{\mathrm{p}}(\mu)$, $\mathfrak{p} \geq 1$ if and only if $\mathrm{T}^{-1}(\Sigma) = \Sigma$. 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 $\mathrm{T}^*\mathrm{T} = \mathrm{TT}^*$ ### 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 $\mathrm{T} \mathrm{T}^* \mathrm{T} = \mathrm{T}^* \mathrm{T} \mathrm{T}$, ie, $\mathrm{T}^* \mathrm{T}$ 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 $T^{*}T\left(T + T^{*}\right) = \left(T + T^{*}\right)T^{*}T$ ### 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 $T^2 T^* = T^* T^2$ ### 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 $T^{*2}T^2 = (T^* T)^2$. ## 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 $L^p(\sum)$, $(1 \leq p < \infty)$ 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 $u \geq 0$ - (i) $\mathbf{M}^{*}_{\mathrm{u,T}}\mathbf{M}_{\mathrm{u,T}}\mathbf{f} = \mathbf{u}^2\mathbf{f}_0\mathbf{f}$ - (ii) $\mathbf{M}_{\mathrm{u,T}}\mathbf{M}^{*}_{\mathrm{u,T}}\mathbf{f} = \mathbf{u}^{2}\circ \mathbf{T}\cdot \mathbf{f}_{0}\circ \mathbf{T}\cdot \mathbf{E}(\mathbf{f})$ - (iii) $\mathbf{M}^{\mathrm{n}}_{\mathrm{u},\mathrm{T}}(\mathbf{f}) = (\mathbf{C}_{\mathrm{T}}\mathbf{M}_{\mathrm{u}})^{\mathrm{n}}(\mathbf{f}) = \mathbf{u}_{\mathrm{n}}(\mathbf{f}\circ \mathbf{T}^{\mathrm{n}}),\quad \mathbf{u}_{\mathrm{n}} = \mathbf{u}\circ \mathbf{T}.\mathbf{u}\circ \mathbf{T}^{2}.\mathbf{u}\circ \mathbf{T}^{3}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots$ - (iv) $\mathbf{M}^{*}_{\mathrm{u,T}}\mathbf{f} = \mathbf{u}\mathbf{f}_{0}\cdot \mathbf{E}(\mathbf{f})\circ \mathbf{T}^{-1}$ - (v) $\mathbf{M}^{*^{\mathrm{n}}}{}_{\mathrm{u,T}}\mathbf{f} = \mathbf{u}\mathbf{f}_{0}\cdot \mathbf{E}(\mathbf{u}\mathbf{f}_{0})\circ \mathbf{T}^{-(\mathrm{n - 1})}\cdot \mathbf{E}(\mathbf{f})\circ \mathbf{T}^{-\mathrm{n}}$ 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 $\mathbf{M}_{\mathrm{u,T}}$ be a composite multiplication operator on $\mathrm{L}^2 (\mu)$. Then the following statements are equivalent (i) $\mathbf{M}_{\mathrm{u,T}}$ is Quasi p-normal operator $$ u \circ T u ^ {2} \circ T h \circ T f \circ T + h u E (h u ^ {2} f) \circ T ^ {- 1} = h u ^ {2} u \circ T f \circ T + h ^ {2} u ^ {3} E (f) $$ Proof: For $f \in L^2(\mu)$, $M_{u,T}$ is Quasi P-normal operator if $$ (M_{u,T} + M^*_{u,T})(M^*_{u,T}M_{u,T})f = (M^*_{u,T}M_{u,T})(M_{u,T} + M^*_{u,T})f \\= M_{u,T}M^*_{u,T}(u \circ T f \circ T) + M^*_{u,T}M^*_{u,T}(u \circ T f \circ T) \\= M_{u,T}[huE(uf \circ T) \circ T^{-1}] + M^*_{u,T}[huE(uf \circ T) \circ T^{-1}] \\= M_{u,T}[hu^2f] + M^*_{u,T}[hu^2f] \\= u \circ T(hu^2f) \circ T + huE(hu^2f) \circ T^{-1} \\= u \circ T u^2 \circ T h \circ T f \circ T + huE(hu^2f) \circ T^{-1} $$ $$ (\mathbf{M}_{\mathrm{u,T}}^{*} \mathbf{M}_{\mathrm{u,T}}) (\mathbf{M}_{\mathrm{u,T}} + \mathbf{M}_{\mathrm{u,T}}^{*}) \mathbf{f} = (\mathbf{M}_{\mathrm{u,T}}^{*} \mathbf{M}_{\mathrm{u,T}}) \mathbf{M}_{\mathrm{u,T}} \mathbf{f} + (\mathbf{M}_{\mathrm{u,T}}^{*} \mathbf{M}_{\mathrm{u,T}}) \mathbf{M}_{\mathrm{u,T}}^{*} \mathbf{f} $$ Suppose, $\mathbf{M}_{\mathrm{u,T}}$ is Quasi P-normal operator. Then $$ (\mathbf{M}_{\mathrm{u},\mathrm{T}} + \mathbf{M}^{*}_{\mathrm{u},\mathrm{T}})(\mathbf{M}^{*}_{\mathrm{u},\mathrm{T}}\mathbf{M}_{\mathrm{u},\mathrm{T}})\mathrm{f} = (\mathbf{M}^{*}_{\mathrm{u},\mathrm{T}}\mathbf{M}_{\mathrm{u},\mathrm{T}})(\mathbf{M}_{\mathrm{u},\mathrm{T}} + \mathbf{M}^{*}_{\mathrm{u},\mathrm{T}})\mathrm{f} $$ #### Theorem 3.2 Let the $\mathrm{M}_{\mathrm{u,T}}$ be a composite multiplication operator on $\mathrm{L}^2 (\mu)$. Then the following statements are equivalent (i) $\mathbf{M}_{\mathrm{u,T}}^{*}$ is Quasi p-normal operator $$ \mathrm {h} ^ {2} \mathrm {u} ^ {3} \mathrm {E} (\mathrm {f}) \circ \mathrm {T} ^ {- 1} + \mathrm {h} \circ \mathrm {T} ^ {2} \mathrm {u} \circ \mathrm {T} \mathrm {u} ^ {2} \circ \mathrm {T} ^ {2} \mathrm {E} (\mathrm {f}) \circ \mathrm {T} $$ (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 $f \in L^{2}(\mu)$, $\mathbf{M}_{\mathfrak{u},\mathrm{T}}^{*}$ is Quasi P-normal operator if $$ \left(\mathbf {M} _ {\mathrm {u}, \mathrm {T}} ^ {*} + \mathbf {M} _ {\mathrm {u}, \mathrm {T}}\right) \left(\mathbf {M} _ {\mathrm {u}, \mathrm {T}} \mathbf {M} _ {\mathrm {u}, \mathrm {T}} ^ {*}\right) \mathrm {f} = \left(\mathbf {M} _ {\mathrm {u}, \mathrm {T}} \mathbf {M} _ {\mathrm {u}, \mathrm {T}} ^ {*}\right) \left(\mathbf {M} _ {\mathrm {u}, \mathrm {T}} ^ {*} + \mathbf {M} _ {\mathrm {u}, \mathrm {T}}\right) \mathrm {f} $$ and then we have $$ (\mathbf{M}_{u,T}^{*} + \mathbf{M}_{u,T}) (\mathbf{M}_{u,T} \mathbf{M}_{u,T}^{*}) \mathrm{f} = \mathbf{M}_{u,T}^{*} (\mathbf{M}_{u,T} \mathbf{M}_{u,T}^{*}) \mathrm{f} + \mathbf{M}_{u,T} (\mathbf{M}_{u,T} \mathbf{M}_{u,T}^{*}) \mathrm{f} $$ $$ \begin{array}{l} = h u E \left(u \circ T h \circ T u \circ T E (f)\right) \circ T ^ {- 1} + u \circ T \left[ u \circ T h \circ T u \circ T E (f) \right] \circ T \\= h ^ {2} u ^ {3} E (f) \circ T ^ {- 1} + h \circ T ^ {2} u \circ T u ^ {2} \circ T ^ {2} E (f) \circ T \\\end{array} $$ Consider $$ \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*}_{\mathrm{u},\mathrm{T}}\right) \left(\mathbf{M}^{*}_{\mathrm{u},\mathrm{T}} + \mathbf{M}_{\mathrm{u},\mathrm{T}}\right) \mathrm{f} = \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*}_{\mathrm{u},\mathrm{T}}\right) \mathbf{M}^{*}_{\mathrm{u},\mathrm{T}} \mathrm{f} + \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*}_{\mathrm{u},\mathrm{T}}\right) \mathbf{M}_{\mathrm{u},\mathrm{T}} \mathrm{f} \\= \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*}_{\mathrm{u},\mathrm{T}}\right) \mathrm{h u E} (\mathrm{f}) \circ \mathrm{T}^{-1} + \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*}_{\mathrm{u},\mathrm{T}}\right) (\mathrm{u} \circ \mathrm{T f} \circ \mathrm{T}) \\= \mathrm{M}_{\mathrm{u},\mathrm{T}} \mathrm{h u E} (\mathrm{h u E} (\mathrm{f}) \circ \mathrm{T}^{-1}) \circ \mathrm{T}^{-1} + \mathrm{M}_{\mathrm{u},\mathrm{T}} \mathrm{h u E} (\mathrm{u} \circ \mathrm{T f} \circ \mathrm{T}) \circ \mathrm{T}^{-1} \\= \mathrm{M}_{\mathrm{u},\mathrm{T}} \mathrm{h u E} (\mathrm{h}) \circ \mathrm{T}^{-1} \mathrm{E} (\mathrm{u}) \circ \mathrm{T}^{-1} \mathrm{E} (\mathrm{f}) \circ \mathrm{T}^{-2} + \mathrm{M}_{\mathrm{u},\mathrm{T}} \mathrm{h u}^{2} \mathrm{f} \\= \mathrm{u} \circ \mathrm{T} \left(\mathrm{h u E} (\mathrm{h}) \circ \mathrm{T}^{-1} \mathrm{E} (\mathrm{u}) \circ \mathrm{T}^{-1} \mathrm{E} (\mathrm{f}) \circ \mathrm{T}^{-2}\right) \circ \mathrm{T} + \mathrm{u} \circ \mathrm{T} (\mathrm{h u}^{2} \mathrm{f}) \circ \mathrm{T} \\= \mathrm{h} \circ \mathrm{T} \mathrm{u}^{2} \circ \mathrm{T} \mathrm{E} (\mathrm{h}) \mathrm{E} (\mathrm{u}) \mathrm{E} (\mathrm{f}) \circ \mathrm{T}^{-1} + \mathrm{h} \circ \mathrm{T} \mathrm{u} \circ \mathrm{T} \mathrm{u}^{2} \circ \mathrm{T} \mathrm{f} \circ \mathrm{T} $$ Suppose $\mathbf{M}^{*}_{\mathrm{u,T}}$ is Quasi p-normal operator. Then $$ (M_{u,T}^* + M_{u,T})(M_{u,T} M_{u,T}^*)f = (M_{u,T} M_{u,T}^*)(M_{u,T}^* + M_{u,T})f \\Leftrightarrow h^2 u^3 E(f) \circ T^{-1} + h \circ T^2 u \circ T u^2 \circ T^2 E(f) \circ T \= h \circ T u^2 \circ T E(h) E(u) E(f) \circ T^{-1} + h \circ T u \circ T u^2 \circ T f \circ T $$ ## IV. CHARACTERIZATIONS ON N POWER CLASS Q COMPOSITE MULTIPLICATION OPERATOS ON $\mathbf{L}^2$ -SPACE Theorem 4.1 Let the $\mathbf{M}_{\mathrm{u,T}}$ be a composite multiplication operator on $\mathrm{L}^2 (\mu)$. Then $\mathbf{M}_{\mathrm{u,T}}$ is n power class Q composite multiplication operator if and only if $$ h u E(h) \circ T^{-1} E(u) \circ T^{-1} E(u_{2n}) \circ T^{-2} f \circ T^{2n-2} = h u E(u_{n}) \circ T^{-1} h \circ T^{n-1} u \circ T^{n-1} E(u_{n}) \circ T^{n-2} f \circ T^{2n-2} $$ Proof: Now Consider, $$ \mathbf {M} ^ {* 2} _ {\mathrm {u}, \mathrm {T}} \mathbf {M} ^ {2 \mathrm {n}} _ {\mathrm {u}, \mathrm {T}} f = \mathbf {M} ^ {* 2} _ {\mathrm {u}, \mathrm {T}} \left[ u _ {2 \mathrm {n}} f \circ \mathrm {T} ^ {2 \mathrm {n}} \right] $$ where $\mathbf{u}_{2\mathrm{n}} = \mathbf{u}\circ \mathrm{T}^2$ uoT4.....uoT2n $$ = \mathbf{M}^{*}_{\mathrm{u}, \mathrm{T}} \left( \mathrm{h u E} \left( \mathrm{u}_{2\mathrm{n}} \circ \mathrm{f} \circ \mathrm{T}^{2\mathrm{n}} \right) \circ \mathrm{T}^{-1} \right) $$ Next we consider, $$ \begin{array}{l} \left(\mathbf {M} ^ {*} _ {\mathrm {u}, \mathrm {T}} \mathbf {M} ^ {\mathrm {n}} _ {\mathrm {u}, \mathrm {T}}\right) ^ {2} \mathrm {f} = \left(\mathbf {M} ^ {*} _ {\mathrm {u}, \mathrm {T}} \mathbf {M} ^ {\mathrm {n}} _ {\mathrm {u}, \mathrm {T}}\right) \left(\mathbf {M} ^ {*} _ {\mathrm {u}, \mathrm {T}} \mathbf {M} ^ {\mathrm {n}} _ {\mathrm {u}, \mathrm {T}}\right) \mathrm {f} \\= \left(\mathbf {M} ^ {*} _ {\mathrm {u}, \mathrm {T}} \mathbf {M} ^ {\mathrm {n}} _ {\mathrm {u}, \mathrm {T}}\right) \mathbf {M} ^ {*} _ {\mathrm {u}, \mathrm {T}} \left(\mathbf {u} _ {\mathrm {n}} \mathbf {f} \circ \mathbf {T} ^ {\mathrm {n}}\right) \\\end{array} $$ where $\mathbf{u}_{\mathrm{n}} = \mathbf{u}\circ \mathrm{T}\mathbf{u}\circ \mathrm{T}^{2}$. $$ =\left(\mathbf{M}^{*}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{\mathrm{n}}_{\mathrm{u},\mathrm{T}}\right) \mathrm{h u E} \left(\mathrm{u}_{\mathrm{n}} f \circ \mathrm{T}^{\mathrm{n}}\right) \circ \mathrm{T}^{-1} $$ Given $\mathbf{M}_{\mathrm{u,T}}$ is n power class Q composite multiplication operator $$ \begin{array}{l} \Leftrightarrow \mathrm {M} ^ {* 2} _ {\mathrm {u}, \mathrm {T}} \mathrm {M} ^ {2 \mathrm {n}} _ {\mathrm {u}, \mathrm {T}} f = \left(\mathrm {M} ^ {*} _ {\mathrm {u}, \mathrm {T}} \mathrm {M} ^ {\mathrm {n}} _ {\mathrm {u}, \mathrm {T}}\right) ^ {2} f \\\Leftrightarrow \mathrm {h u E} (\mathrm {h}) \circ \mathrm {T} ^ {- 1} \mathrm {E} (\mathrm {u}) \circ \mathrm {T} ^ {- 1} \mathrm {E} \left(\mathrm {u} _ {2 n}\right) \circ \mathrm {T} ^ {- 2} \mathrm {f} \circ \mathrm {T} ^ {2 n - 2} \\= h u E \left(u _ {n}\right) \circ T ^ {- 1} h \circ T ^ {n - 1} u \circ T ^ {n - 1} E \left(u _ {n}\right) \circ T ^ {n - 2} f \circ T ^ {2 n - 2} \\\end{array} $$ Theorem 4.2 Let the $\mathbf{M}_{\mathbf{u},\mathrm{T}}$ be a composite multiplication operator on $\mathrm{L}^2 (\mu)$. Then $\mathbf{M}_{\mathbf{u},\mathrm{T}}^{*}$ is n power class Q composite multiplication operator if and only if $$ u\circ T\,u^{2}\circ T^{2}\,h\circ T^{2}\,E(uh)\circ T^{-(2n-3)}\,E(f)\circ T^{-(2n-2)} $$ Now if we consider $$ \begin{array}{l} \mathbf{M}^{2}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*2\mathrm{n}}_{\mathrm{u},\mathrm{T}} f = \mathbf{M}^{2}_{\mathrm{u},\mathrm{T}} \left(\mathrm{h} \mathrm{u} \mathrm{E}(\mathrm{h} \mathrm{u}) \circ \mathrm{T}^{-(2\mathrm{n}-1)} \mathrm{E}(\mathrm{f}) \circ \mathrm{T}^{-2\mathrm{n}}\right) \\= \mathrm{M}_{\mathrm{u},\mathrm{T}} \left(\mathrm{u} \circ \mathrm{T} \left(\mathrm{h} \mathrm{u} \mathrm{E}(\mathrm{h} \mathrm{u}) \circ \mathrm{T}^{-(2\mathrm{n}-1)} \mathrm{E}(\mathrm{f}) \circ \mathrm{T}^{-2\mathrm{n}}\right) \circ \mathrm{T}\right) \\= \mathrm{M}_{\mathrm{u},\mathrm{T}} \left(\mathrm{h} \circ \mathrm{T} \quad \mathrm{u}^{2} \circ \mathrm{T} \quad \mathrm{E}(\mathrm{h} \mathrm{u}) \circ \mathrm{T}^{-(2\mathrm{n}-2)} \quad \mathrm{E}(\mathrm{f}) \circ \mathrm{T}^{-(2\mathrm{n}-1)}\right) \\= \mathrm{u} \circ \mathrm{T} \left(\mathrm{h} \circ \mathrm{T} \mathrm{u}^{2} \circ \mathrm{T} \mathrm{E}(\mathrm{h} \mathrm{u}) \circ \mathrm{T}^{-(2\mathrm{n}-2)} \mathrm{E}(\mathrm{f}) \circ \mathrm{T}^{-(2\mathrm{n}-1)}\right) \circ \mathrm{T} \\= \mathrm{u} \circ \mathrm{T} \mathrm{u}^{2} \circ \mathrm{T}^{2} \mathrm{h} \circ \mathrm{T}^{2} \mathrm{E}(\mathrm{u} \mathrm{h}) \circ \mathrm{T}^{-(2\mathrm{n}-3)} \mathrm{E}(\mathrm{f}) \circ \mathrm{T}^{-(2\mathrm{n}-2)} \\end{array} $$ and we consider $$ \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*_{\mathrm{n}}}_{\mathrm{u},\mathrm{T}}\right)^{2} \mathrm{f} = \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*_{\mathrm{n}}}_{\mathrm{u},\mathrm{T}}\right) \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*_{\mathrm{n}}}_{\mathrm{u},\mathrm{T}}\right) \mathrm{f} \\= \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*_{\mathrm{n}}}_{\mathrm{u},\mathrm{T}}\right) \mathbf{M}_{\mathrm{u},\mathrm{T}} \mathrm{u} \mathrm{h} \mathrm{E}(\mathrm{u} \mathrm{h}) \circ \mathrm{T}^{- (n - 1)} \mathrm{E}(\mathrm{f}) \circ \mathrm{T}^{- n} \\= \left(\mathbf{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*_{\mathrm{n}}}_{\mathrm{u},\mathrm{T}}\right) \mathrm{u} \circ \mathrm{T} \left(\mathrm{u} \mathrm{h} \mathrm{E}(\mathrm{u} \mathrm{h}) \circ \mathrm{T}^{- (n - 1)} \mathrm{E}(\mathrm{f}) \circ \mathrm{T}^{- n}\right) \circ \mathrm{T} \\= \mathrm{M}_{\mathrm{u},\mathrm{T}} \mathbf{M}^{*_{\mathrm{n}}}_{\mathrm{u},\mathrm{T}} \left(\mathrm{u}^{2} \circ \mathrm{T} \mathrm{h} \circ \mathrm{T} \mathrm{E}(\mathrm{u} \mathrm{h}) \circ \mathrm{T}^{- (\mathrm{n} - 2)} \mathrm{E}(\mathrm{f}) \circ \mathrm{T}^{- (\mathrm{n} - 1)}\right) \\= \mathrm{M}_{\mathrm{u},\mathrm{T}} \mathrm{u} \mathrm{h} \mathrm{E}(\mathrm{u} \mathrm{h}) \circ \mathrm{T}^{- (\mathrm{n} - 1)} \mathrm{E} \left(\mathrm{u}^{2} \circ \mathrm{T} \mathrm{h} \circ \mathrm{T} \mathrm{E}(\mathrm{u} \mathrm{h}) \circ \mathrm{T}^{- (\mathrm{n} - 2)} \mathrm{E}(\mathrm{f}) \circ \mathrm{T}^{- (\mathrm{n} - 1)}\right) \circ \mathrm{T}^{- \mathrm{n}} $$ Since $\mathbf{M}_{\mathrm{u,T}}$ is a Composite multiplication operator, by definition $$ \Leftrightarrow \mathbf{M}^{2}_{\mathrm{u,T}} \mathbf{M}^{*2n}_{\mathrm{u,T}} f = \left(\mathbf{M}_{\mathrm{u,T}} \mathbf{M}^{*n}_{\mathrm{u,T}}\right)^{2} f $$

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