I. INTRODUCTION
Let be a -finite measure space. Then a mapping T from X into X is said to be a measurable transformation if for every . A measurable transformation T is said to be non-singular if whenever . If T is non-singular then the measure defined as for every E in , is an absolutely continuous measure on with respect to . Since is a -finite measure, then by the Radon-Nikodym theorem, there exists a non-negative function in such that for every . The function is called the Radon-Nikodym derivative of with respect to .
Every non- singular measurable transformation T from X into itself induces a linear transformation on defined as for every f in . In case is continuous from into itself, then it is called a composition operator on induced by T. We restrict our study of the composition operators on which has Hilbert space structure. If u is an essentially bounded complex-valued measurable function on X, then the mapping on defined by , is a continuous operator with range in . The operator is known as the multiplication operator induced by u. A composite multiplication operator is linear transformation acting on a set of complex valued measurable functions f of the form
\[\mathrm{M} _ {\mathrm{u}, \mathrm{T}} (\mathrm{f}) = \mathrm{C} _ {\mathrm{T}} \mathrm{M} _ {\mathrm{u}} (\mathrm{f}) = \mathrm{u} \circ \mathrm{T} \, \mathrm{f} \circ \mathrm{T}\]
Where u is a complex valued, measurable function. In case u = 1 almost everywhere, becomes a composition operator, denoted by .
In the study considered is the using conditional expectation of composite multiplication operator on -spaces. For each , , there exists an unique -measurable function such that
\[\int_ {A} g f d \mu = \int_ {A} g E (f) d \mu\]
for every -measurable function g, for which the left integral exists. The function is called the conditional expectation of f with respect to the subalgebra . As an operator of , E is the projection onto the closure of range of T and E is the identity on , if and only if . Detailed discussion of E is found in [1-4].
1.1 Normal operator
Let H be a Complex Hilbert Space. An operator T on H is called normal operator if
1.2 Quasi-normal operator
Let H be a Complex Hilbert Space. An operator T on H is called Quasi-normal operator if T TT = TT T ie, T*T commute with T
1.3 Quasi p-normal operator [13]
Let H be a Complex Hilbert Space. An operator T on H is called Quasi-normal operator if
1.4 Power -normal operator
Let H be a Complex Hilbert Space. An operator T on H is called 2 power-normal operator if
1..5 Class Q-operator [14]
Let H be a Complex Hilbert Space. An operator T on H is called Quasi-normal operator if
The study of weighted composition operators on 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 , spaces, where the measure spaces are -finite, appeared already in [6]. Also boundedness of weighted operators on C(X,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 paranormal of composite multiplication operators on spaces [8-12]. In this paper we investigate composite multiplication operators on -space become Quasi-P-Normal operators and n-Power class Q operator have been obtained in terms of radon-nikodym derivative .
III. CHARACTERIZATION ON COMPOSITE MULTIPLICATION OF QUASI P NORMAL OPERATORS ON SPACE
3.1 Proposition
Let the composite multiplication operator . Then for
(i)
(ii)
\[(iii) M^{n}_{u,T}(\mathbf{f}) = (C_{T}M_{u})^{n}(\mathbf{f}) = u_{n}\left(\mathbf{f} \circ T^{n}\right) \quad u_{n} = u \circ T \cdot u \circ T^{2} \cdot u \circ T^{3} \dots \dots \dots u \circ T^{n}\]
(iv)
(v)
where
Theorem 3.1
Let the be a composite multiplication operator on . Then the following statements are equivalent (i) is Quasi p-normal operator
\[\mathrm{u} \circ \mathrm{T} \mathrm{u} ^ {2} \circ \mathrm{Th} \circ \mathrm{Tf} \circ \mathrm{T} + \mathrm{huE} (\mathrm{hu} ^ {2} \mathrm{f}) \circ \mathrm{T} ^ {- 1} = \mathrm{hu} ^ {2} \mathrm{u} \circ \mathrm{Tf} \circ \mathrm{T} + \mathrm{h} ^ {2} \mathrm{u} ^ {3} \mathrm{E} (\mathrm{f})\]
Proof:
For , is Quasi P-normal operator if
\[(\mathbf{M}_{\mathrm{u}, \mathrm{T}} + \mathbf{M}^*_{\mathrm{u}, \mathrm{T}}) (\mathbf{M}^*_{\mathrm{u}, \mathrm{T}} \mathbf{M}_{\mathrm{u}, \mathrm{T}}) \mathrm{f} = (\mathbf{M}^*_{\mathrm{u}, \mathrm{T}} \mathbf{M}_{\mathrm{u}, \mathrm{T}}) (\mathbf{M}_{\mathrm{u}, \mathrm{T}} + \mathbf{M}^*_{\mathrm{u}, \mathrm{T}}) \mathrm{f} \text{and we have,} \\(\mathbf{M}_{\mathrm{u}, \mathrm{T}} + \mathbf{M}^*_{\mathrm{u}, \mathrm{T}}) (\mathbf{M}^*_{\mathrm{u}, \mathrm{T}} \mathbf{M}_{\mathrm{u}, \mathrm{T}}) \mathrm{f} = \mathbf{M}_{\mathrm{u}, \mathrm{T}} (\mathbf{M}^*_{\mathrm{u}, \mathrm{T}} \mathbf{M}_{\mathrm{u}, \mathrm{T}}) \mathrm{f} + \mathbf{M}^*_{\mathrm{u}, \mathrm{T}} (\mathbf{M}^*_{\mathrm{u}, \mathrm{T}} \mathbf{M}_{\mathrm{u}, \mathrm{T}}) \mathrm{f} \\= \mathbf{M}_{\mathrm{u}, \mathrm{T}} \mathbf{M}^*_{\mathrm{u}, \mathrm{T}} (\mathbf{u} \circ \mathrm{T} \mathbf{f} \circ \mathrm{T}) + \mathbf{M}^*_{\mathrm{u}, \mathrm{T}} \mathbf{M}^*_{\mathrm{u}, \mathrm{T}} (\mathbf{u} \circ \mathrm{T} \mathbf{f} \circ \mathrm{T}) \\= \mathbf{M}_{\mathrm{u}, \mathrm{T}} [ h u E (u f \circ T) \circ T^{-1} ] + \mathbf{M}^*_{\mathrm{u}, \mathrm{T}} [ h u E (u f \circ T) \circ T^{-1} ] \\= \mathbf{M}_{\mathrm{u}, \mathrm{T}} [ h u^{2} f ] + \mathbf{M}^*_{\mathrm{u}, \mathrm{T}} [ h u^{2} f ] \\= u \circ T (h u^{2} f) \circ T + h u E (h u^{2} f) \circ T^{-1} \\= u \circ T u^{2} \circ T h \circ T f \circ T + h u E (h u^{2} f) \circ T^{-1}\]
Consider
\[(\mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathrm{M} _ {\mathrm{u}, \mathrm{T}}) (\mathrm{M} _ {\mathrm{u}, \mathrm{T}} + \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*}) \mathrm{f} = (\mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathrm{M} _ {\mathrm{u}, \mathrm{T}}) \mathrm{M} _ {\mathrm{u}, \mathrm{T}} \mathrm{f} + (\mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathrm{M} _ {\mathrm{u}, \mathrm{T}}) \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathrm{f} \\= (\mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathrm{M} _ {\mathrm{u}, \mathrm{T}}) (\mathrm{u} \circ \mathrm{T} \mathrm{f} \circ \mathrm{T}) + (\mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathrm{M} _ {\mathrm{u}, \mathrm{T}}) (\mathrm{h} \mathrm{u} \mathrm{E}(\mathrm{f}) \circ \mathrm{T} ^ {- 1}) \\= \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathrm{u} \circ \mathrm{T} (\mathrm{u} \circ \mathrm{T} \mathrm{f} \circ \mathrm{T}) \circ \mathrm{T} + \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathrm{u} \circ \mathrm{T} (\mathrm{h} \mathrm{u} \mathrm{E}(\mathrm{f}) \circ \mathrm{T} ^ {- 1}) \circ \mathrm{T} \\= \mathrm{h} \mathrm{u} \mathrm{E} [ \mathrm{u} \circ \mathrm{T} \mathrm{u} \circ \mathrm{T} ^ {2} \mathrm{f} \circ \mathrm{T} ^ {2} ] \circ \mathrm{T} ^ {- 1} + \mathrm{h} \mathrm{u} \mathrm{E} [ \mathrm{u} \circ \mathrm{T} \mathrm{h} \circ \mathrm{T} \mathrm{u} \circ \mathrm{T} \mathrm{E}(\mathrm{f}) ] \circ \mathrm{T} ^ {- 1} \\= \mathrm{h} \mathrm{u} ^ {2} \mathrm{u} \circ \mathrm{T} \mathrm{f} \circ \mathrm{T} + \mathrm{h} ^ {2} \mathrm{u} ^ {3} \mathrm{E}(\mathrm{f})\]
Suppose, is Quasi P-normal operator. Then
\[\left(\mathbf{M}_{\mathrm{u},\mathrm{T}} + \mathbf{M}_{\mathrm{u},\mathrm{T}}^{*}\right)\left(\mathbf{M}_{\mathrm{u},\mathrm{T}}^{*}\mathbf{M}_{\mathrm{u},\mathrm{T}}\right)\mathrm{f} = \left(\mathbf{M}_{\mathrm{u},\mathrm{T}}^{*}\mathbf{M}_{\mathrm{u},\mathrm{T}}\right)\left(\mathbf{M}_{\mathrm{u},\mathrm{T}} + \mathbf{M}_{\mathrm{u},\mathrm{T}}^{*}\right)\mathrm{f}\]
\[\begin{array}{l} \Leftrightarrow u \circ T u ^ {2} \circ T h \circ T f \circ T + h u E (h u ^ {2} f) \circ T ^ {- 1} = h u ^ {2} u \circ T f \circ T + h ^ {2} u ^ {3} E (f) \\\text { almost everywhere. } \end{array}\]
Corollary 3.2
The composition operator on is Quasi p-normal operator if and only if almost everywhere. Proof:
The proof is obtained from Theorem 3.1 by putting u = 1.
Theorem 3.3
Let the be a composite multiplication operator on . Then the following statements are equivalent
\[\mathbf {M} _ {\mathrm{u,T}} ^ {*}\tag{i}\]
(ii)
\[\begin{array}{l} \mathrm{h} ^ {2} \mathrm{u} ^ {3} \mathrm{E} (\mathrm{f}) \circ \mathrm{T} ^ {- 1} + \mathrm{h} \circ \mathrm{T} ^ {2} \mathrm{u} \circ \mathrm{T} \mathrm{u} ^ {2} \circ \mathrm{T} ^ {2} \mathrm{E} (\mathrm{f}) \circ \mathrm{T} \\= \mathrm{h} \circ \mathrm{T} \mathrm{u} ^ {2} \circ \mathrm{T} \mathrm{E} (\mathrm{h}) \mathrm{E} (\mathrm{u}) \mathrm{E} (\mathrm{f}) \circ \mathrm{T} ^ {- 1} + \mathrm{h} \circ \mathrm{T} \mathrm{u} \circ \mathrm{T} \mathrm{u} ^ {2} \circ \mathrm{T} \mathrm{f} \circ \mathrm{T} \\\text {almost everywhere.} \end{array}\]
Proof:
\[\mathrm{f} \in \mathrm{L} ^ {2} (\mu), \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*}\]
\[\left(\mathbf {M} _ {\mathrm{u}, \mathrm{T}} ^ {*} + \mathbf {M} _ {\mathrm{u}, \mathrm{T}}\right) \left(\mathbf {M} _ {\mathrm{u}, \mathrm{T}} \mathbf {M} _ {\mathrm{u}, \mathrm{T}} ^ {*}\right) \mathrm{f} = \left(\mathbf {M} _ {\mathrm{u}, \mathrm{T}} \mathbf {M} _ {\mathrm{u}, \mathrm{T}} ^ {*}\right) \left(\mathbf {M} _ {\mathrm{u}, \mathrm{T}} ^ {*} + \mathbf {M} _ {\mathrm{u}, \mathrm{T}}\right) \mathrm{f}\]
and then we have
\[= \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \left[ \mathrm{u} \circ \mathrm{Th} \circ \mathrm{Tu} \circ \mathrm{TE(f)} \right] + \mathrm{M} _ {\mathrm{u}, \mathrm{T}} \left[ \mathrm{u} \circ \mathrm{Th} \circ \mathrm{Tu} \circ \mathrm{TE(f)} \right]\]
Consider
\[\begin{array}{r l} & {\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) \mathbf{f} = \left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} *\right) \mathbf{M} _ {\mathrm{u}, \mathrm{T}} * \mathbf{f} + \left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} *\right) \mathbf{M} _ {\mathrm{u}, \mathrm{T}} \mathbf{f}} \\& {= \left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} *\right) \mathbf{h u E (f)} \circ \mathbf{T} ^ {- 1} + \left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} *\right) (\mathbf{u} \circ \mathbf{T} \mathbf{f} \circ \mathbf{T})} \\& {= \mathbf{M} _ {\mathrm{u}, \mathrm{T}} \mathbf{h u E} (\mathbf{h u E (f)} \circ \mathbf{T} ^ {- 1}) \circ \mathbf{T} ^ {- 1} + \mathbf{M} _ {\mathrm{u}, \mathrm{T}} \mathbf{h u E} (\mathbf{u} \circ \mathbf{T} \mathbf{f} \circ \mathbf{T}) \circ \mathbf{T} ^ {- 1}} \\& {= \mathbf{M} _ {\mathrm{u}, \mathrm{T}} \mathbf{h u E (h)} \circ \mathbf{T} ^ {- 1} \mathbf{E (u)} \circ \mathbf{T} ^ {- 1} \mathbf{E (f)} \circ \mathbf{T} ^ {- 2} + \mathbf{M} _ {\mathrm{u}, \mathrm{T}} \mathbf{h u ^ {2}} \mathbf{f}} \\& {= \mathbf{u} \circ \mathbf{T} (\mathbf{h u E (h)} \circ \mathbf{T} ^ {- 1} \mathbf{E (u)} \circ \mathbf{T} ^ {- 1} \mathbf{E (f)} \circ \mathbf{T} ^ {- 2}) \circ \mathbf{T} + \mathbf{u} \circ \mathbf{T} (\mathbf{h u ^ {2}} \mathbf{f}) \circ \mathbf{T}} \\& {= \mathbf{h} \circ \mathbf{T u ^ {2}} \circ \texttt{T E (h)} \texttt{E (u)} \texttt{E (f)} \circ \texttt{T} ^ {- 1} + \mathbf{h} \circ \texttt{T u} \circ \texttt{T u ^ {2}} \circ \texttt{T f} \circ \texttt{T}} \\& {\textsf{S u p p o s e M} _ {\mathrm{u}, \mathrm{T}} * i s Q u a s i p - n o r m a l o p e r a t o r. then} \\& {\left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} * + \mathbf{M} _ {\mathrm{u}, \mathrm{T}}\right) (\mathbf{M} _ {\mathrm{u}, \mathrm{T}} * \mathbf{M} _ {\mathrm{u}, \mathrm{T}} *) \mathbf{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}}) \mathbf{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} \\& {\quad = 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} \\& {\textsf{a l m o s t e v e r y w h e r e}.} \end{array}\]
Corollary 3.4
The composition operator on is Quasi P-normal operator if and only if
\[\mathrm{h} ^ {2} \mathrm{E} (\mathrm{f}) \circ \mathrm{T} ^ {- 1} + \mathrm{h} \circ \mathrm{T} ^ {2} \mathrm{E} (\mathrm{f}) \circ \mathrm{T} = \mathrm{h} \circ \mathrm{T} \mathrm{E} (\mathrm{h}) \mathrm{E} (\mathrm{f}) \circ \mathrm{T} ^ {- 1} + \mathrm{h} \circ \mathrm{T} \mathrm{f} \circ \mathrm{T}\]
almost everywhere.
Proof:
The proof is obtained from Theorem 3.3 by putting u = 1.
III. CHARACTERIZATIONS ON N POWER CLASS Q COMPOSITE MULTIPLICATION OPERATOS ON L2-SPACE
Theorem 4.1
Let the be a composite multiplication operator on . Then is n power class Q composite multiplication operator if and only if
\[\begin{array}{r l} \mathrm {h u E(h)\circ T^ {- 1} E(u)\circ T^ {- 1} E(u_ {2n})\circ T^ {- 2} f\circ T^ {2n - 2}} \\& = \mathrm {h u E(u_ {n})\circ T^ {- 1} h\circ T ^ {n - 1} u\circ T ^ {n - 1} E(u_ {n})\circ T ^ {n - 2} f\circ T^ {2n - 2}} \end{array}\]
Proof:
Now Consider,
\[\begin{array}{l} \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {* 2} \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {2 \mathrm{n}} \mathrm{f} = \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {* 2} \left[ \mathrm{u} _ {2 \mathrm{n}} \mathrm{f} \circ \mathrm{T} ^ {2 \mathrm{n}} \right] \\\text{where} \mathrm{u} _ {2 \mathrm{n}} = \mathrm{u} \circ \mathrm{T} ^ {2} \mathrm{u} \circ \mathrm{T} ^ {4} \dots \dots \dots \dots \dots \mathrm{u} \circ \mathrm{T} ^ {2 \mathrm{n}} \\= \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \left( \text{huE} \left( \mathrm{u} _ {2 \mathrm{n}} \mathrm{f} \circ \mathrm{T} ^ {2 \mathrm{n}} \right) \circ \mathrm{T} ^ {- 1} \right) \\= \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \text{huE} (\mathrm{u} _ {2 \mathrm{n}}) \circ \mathrm{T} ^ {- 1} \mathrm{f} \circ \mathrm{T} ^ {2 \mathrm{n} - 1} \\= \text{huE} \left( \text{huE} (\mathrm{u} _ {2 \mathrm{n}}) \circ \mathrm{T} ^ {- 1} \mathrm{f} \circ \mathrm{T} ^ {2 \mathrm{n} - 1} \right) \circ \mathrm{T} ^ {- 1} \\= \text{huE(h)} \circ \mathrm{T} ^ {- 1} \text{E(u)} \circ \mathrm{T} ^ {- 1} \text{E(u_{2n})} \circ \mathrm{T} ^ {- 2} \mathrm{f} \circ \mathrm{T} ^ {2 \mathrm{n} - 2} \end{array}\]
Next we consider,
\[\left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {\mathrm{n}}\right) ^ {2} f = \left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {\mathrm{n}}\right) \left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {\mathrm{n}}\right) f \\= \left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {\mathrm{n}}\right) \mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \left(u _ {\mathrm{n}} f \circ T ^ {\mathrm{n}}\right) \\\text{where} u _ {\mathrm{n}} = u \circ T u \circ T ^ {2} \dots \dots \dots \dots \dots u \circ T ^ {\mathrm{n}} \\= \left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {\mathrm{n}}\right) h u E \left(u _ {\mathrm{n}} f \circ T ^ {\mathrm{n}}\right) \circ T ^ {- 1} \\= \left(\mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {\mathrm{n}}\right) h u E (u _ {\mathrm{n}}) \circ T ^ {- 1} f \circ T ^ {\mathrm{n-1}} \\= \mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} u _ {\mathrm{n}} \left(h u E (u _ {\mathrm{n}}) \circ T ^ {- 1} f \circ T ^ {\mathrm{n-1}}\right) \circ T ^ {\mathrm{n}} \\= \mathbf{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} u _ {\mathrm{n}} h \circ T ^ {\mathrm{n}} u \circ T ^ {\mathrm{n}} E (u _ {\mathrm{n}}) \circ T ^ {\mathrm{n-1}} f \circ...\]
Given is n power class Q composite multiplication operator
\[\Leftrightarrow \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {* 2} \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {2 \mathrm{n}} \mathrm{f} = \left(\mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {*} \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {\mathrm{n}}\right) ^ {2} \mathrm{f}\]
\[\begin{array}{l} \Leftrightarrow \mathrm {h u E(h)\circ T^ {- 1} E(u)\circ T^ {- 1} E(u_ {2n})\circ T^ {- 2} f\circ T^ {2n - 2}} \\= \mathrm {h u E(u_ {n})\circ T^ {- 1} h\circ T^ {n - 1} u\circ T^ {n - 1} E(u_ {n})\circ T^ {n - 2} f\circ T^ {2n - 2}} \end{array} \text { almost everywhere. }\]
Corollary 4.2
The composition operator on is n power class Q if and only if
\[\mathrm{h} \mathrm{E}(\mathrm{h}) \circ \mathrm{T}^{-1} \mathrm{f} \circ \mathrm{T}^{2n-2} = \mathrm{h h} \circ \mathrm{T}^{n-1} \mathrm{f} \circ \mathrm{T}^{2n-2}\]
almost everywhere.
Proof:
The proof is obtained from Theorem 4.1 by putting u = 1.
Theorem 4.3
Let the be a composite multiplication operator on . Then is n power class Q composite multiplication operator if and only if
\[\begin{array}{r l} \mathrm {u\circ Tu^ {2} \circ T^ {2} h\circ T^ {2} E(uh)\circ T ^ {- (2n - 3)} E(f)\circ T ^ {- (2n - 2)}} \\& = \mathrm {u^ {2} \circ Th\circ TE(uh)\circ T ^ {- (n - 2)} h\circ T ^ {- (n - 2)} E(uh)\circ T ^ {- (2n - 3)} E(f)\circ T ^ {- (2n - 2)}} \end{array}\]
Proof:
Now if we consider
\[\begin{array}{l} \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {2} \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {* 2 n} f = \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {2} \left(h u E (h u) \circ T ^ {- (2 n - 1)} E (f) \circ T ^ {- 2 n}\right) \\= \mathrm{M} _ {\mathrm{u}, \mathrm{T}} \left(u \circ T \left(h u E (h u) \circ T ^ {- (2 n - 1)} E (f) \circ T ^ {- 2 n}\right) \circ T\right) \\= \mathrm{M} _ {\mathrm{u}, \mathrm{T}} \left(h \circ T u ^ {2} \circ T E (h u) \circ T ^ {- (2 n - 2)} E (f) \circ T ^ {- (2 n - 1)}\right) \\= u \circ T \left(h \circ T u ^ {2} \circ T E (h u) \circ T ^ {- (2 n - 2)} E (f) \circ T ^ {- (2 n - 1)}\right) \circ T \\= u \circ T u ^ {2} \circ T ^ {2} h \circ T ^ {2} E (u h) \circ T ^ {- (2 n - 3)} E (f) \circ T ^ {- (2 n - 2)} \end{array}\]
and we consider
\[\begin{array}{l} \left(\mathrm{M} _ {\mathrm{u}, \mathrm{T}} \mathrm{M} ^ {* _ {\mathrm{n}}} _ {\mathrm{u}, \mathrm{T}}\right) ^ {2} \mathrm{f} = \left(\mathrm{M} _ {\mathrm{u}, \mathrm{T}} \mathrm{M} ^ {* _ {\mathrm{n}}} _ {\mathrm{u}, \mathrm{T}}\right) \left(\mathrm{M} _ {\mathrm{u}, \mathrm{T}} \mathrm{M} ^ {* _ {\mathrm{n}}} _ {\mathrm{u}, \mathrm{T}}\right) \mathrm{f} \\= \left(\mathrm{M} _ {\mathrm{u}, \mathrm{T}} \mathrm{M} ^ {* _ {\mathrm{n}}} _ {\mathrm{u}, \mathrm{T}}\right) \mathrm{M} _ {\mathrm{u}, \mathrm{T}} \text{uhE(uh)} \circ \mathrm{T} ^ {- (\mathrm{n} - 1)} \mathrm{E(f)} \circ \mathrm{T} ^ {- \mathrm{n}} \\= \left(\mathrm{M} _ {\mathrm{u}, \mathrm{T}} \mathrm{M} ^ {* _ {\mathrm{n}}} _ {\mathrm{u}, \mathrm{T}}\right) \mathrm{u} \circ \mathrm{T} \left(\text{uhE(uh)} \circ \mathrm{T} ^ {- (\mathrm{n} - 1)} \mathrm{E(f)} \circ \mathrm{T} ^ {- \mathrm{n}}\right) \circ \mathrm{T} \\= \mathrm{M} _ {\mathrm{u}, \mathrm{T}} \mathrm{M} ^ {* _ {\mathrm{n}}} _ {\mathrm{u}, \mathrm{T}} \left(\mathrm{u} ^ {2} \circ \mathrm{Th} \circ \mathrm{TE(uh)} \circ \mathrm{T} ^ {- (\mathrm{n} - 2)} \mathrm{E(f)} \circ \mathrm{T} ^ {- (\mathrm{n} - 1)}\right) \\= \mathrm{M} _ {\mathrm{u}, \mathrm{T}} \text{uhE(uh)} \circ \mathrm{T} ^ {- (\mathrm{n} - 1)} \mathrm{E} \left(\text{u} ^ {2} \circ \text{Th} \circ \text{TE(uh)} \circ \text{T} ^ {- (2 n - 3)} \text{E(f)} \circ \text{T} ^ {- (2 n - 2)}\right) \circ \text{T} ^ {- \mathrm{n}} \\= \mathrm{M} _ {\mathrm{u}, \mathrm{T}} \left(\text{uhE(uh)} \circ \mathrm{T} ^ {- (\mathrm{n} - 1)} u ^ {2} \circ \text{T} ^ {- (\mathrm{n} - 1)} h \circ \text{T} ^ {- (\mathrm{n} - 1)} \text{E(uh)} \circ \text{T} ^ {- (2 n - 2)} \text{E(f)} \circ \text{T} ^ {- (2 n - 1)}\right) \\= u \circ T \left(\text{uhE(uh)} \circ \text{T} ^ {- (\mathrm{n} - 1)} u ^ {2} \circ \text{T} ^ {- (\mathrm{n} - 1)} h \circ \text{T} ^ {- (\mathrm{n} - 1)} \text{E(uh)} \circ \text{T} ^ {- (2 n - 2)} \text{E(f)} \circ \text{T} ^ {- (2 n - 1)}\right) \circ T \\= u ^ {2} \circ T h o T E (u h) o T ^ {- (\mathrm{n} - 2)} h o T ^ {- (\mathrm{n} - 2)} E (u h) o T ^ {- (2 n - 3)} E (f) o T ^ {- (2 n - 2)} \\ \text{Since } M _ {\mathrm{u}, \mathrm{T}} \text{ is a composite multiplication operator, by definition} \\= U _ {0} \\= U _ {1} \\= U _ {2} \\= U _ {3} \\= U _ {4} \\= U _ {5} \\= U _ {6} \\= U _ {7} \\= U _ {8} \\= U _ {9}
\end{array}\]
\[\Leftrightarrow \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {2} \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {* 2 \mathrm{n}} \mathrm{f} = \left(\mathrm{M} _ {\mathrm{u}, \mathrm{T}} \mathrm{M} _ {\mathrm{u}, \mathrm{T}} ^ {* \mathrm{n}}\right) ^ {2} \mathrm{f}\]
\[\begin{array}{r l} \Leftrightarrow & u \circ T u ^ {2} \circ T ^ {2} h \circ T ^ {2} E (u h) \circ T ^ {- (2 n - 3)} E (f) \circ T ^ {- (2 n - 2)} \\& = u ^ {2} \circ T h \circ T E (u h) \circ T ^ {- (n - 2)} h \circ T ^ {- (n - 2)} E (u h) \circ T ^ {- (2 n - 3)} E (f) \circ T ^ {- (2 n - 2)} \end{array}\]
almost everywhere.
Corollary 4.4
The composition operator on is n power class Q if and only if
\[\begin{array}{r l} \mathrm {h\circ T^ {2} E(h)\circ T^ {- (2n - 3)} E(f)\circ T^ {- (2n - 2)}} \\& = \mathrm {h\circ T E(h)\circ T^ {- (n - 2)} h\circ T^ {- (n - 2)} E(h)\circ T^ {- (2n - 3)} E(f)\circ T^ {- (2n - 2)}} \end{array}\]
almost everywhere.
Proof:
The proof is obtained from Theorem 4.3 by putting u = 1.
ACKNOWLEDGEMENT
We would like to thank the reviewers for their constructive comments. We thank to Dr.R.David Chandrakumar, Professor, Department of Mathematics, Vickram College of Engineering for his encouragement and support given.
