**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **647 **

### Aluthge Transformation of (n,k) Quasi Class Q and

### (n,k) Quasi Class Q^* Operators

1_{S.Parvatham, }2_{D.Senthilkumar }
1_{Assistant Professor, }2_{Assistant Professor }

1_{Sri Ramakrishna institute of Technology, Coimbatore 10, Tamilnadu, India., }
2_{Govt. Arts College, Coimbatore-18. Tamilnadu, India. }

_____________________________________________________________________________________________________

**Abstract**** - In this paper, a new class of operators called (n,k) quasi class Q and (n,k) quasi class Q^* operators are **
**introduced and studied some properties. (n,k) Quasi class Q and (n,k) quasi class Q^* composition and weighted **
**composition operators on L^2 (λ) and H^2 (β) are characterized. Also we discuss (n,k) quasi class Q and (n,k) quasi class **
**Q^* composite multiplication operator on L^2 space and Aluthge transformation of these class of operators are obtained. **

**keywords**** - Class Q operators, class Q^* operators, composition operators, weighted composition operators, Aluthge **
**transformation. **

*____________________________________________________________________________________________________*

**I.INTRODUCTION **

Let ! be an infinite dimensional separable Complex Hilbert space. Let "(!) be the algebra of all bounded linear operators
acting on !. Let % be an operator on !. Every operator % can be decomposed into % = '|%| with a partial isometry ', where
% is the square root of (%∗_{%). If ' is determined uniquely by the kernel condition *(') = * %} _{, then this decomposition is }
called the polar decomposition, which is one of the most important results in operator theory.

Recall that an operator % is said to be paranormal if %+ ,_{≤ %},_{+ +} _{ for every + ∈ ! [7]. An operator % is said to be }
0-paranormal if %+ 123 _{≤ %}123_{+ +}1 _{ for every + ∈ ! [17] and normaloid if 4(%) = %} _{, where 4(%) denotes the spectral }
radius of %. An operator % is of class 5 [6], if %∗,_{%},_{− 2%}∗_{% + 9 ≥ 0. Equivalently % ∈ 5 if } _{%+} ,_{≤}3

, %

,_{+} ,_{+ +} , _{ for }
every + ∈ !. Class 5 operators are introduced and studied by B. P. Duggal et al and it is well known that every class 5 operator
is not necessarily normaloid and every paranormal operator is a normaloid of class 5. ie < ⊆ 5 ∩ *, where < and * denotes the
class of paranormal and normaloid operators respectively. Also he showed that the restriction of % to an invariant subspace is
again a class 5 operator.

Devika, Suresh [4], introduced a new class of operators which we call the quasi class 5 operators and it is defined as, for
% ∈ "(!) %,_{+} ,_{≤}3

, %

?_{+} ,_{+ %+} , _{ for every + ∈ ! }

In [8], A @-quasi class 5 operator is defined as follows, An operator % is of @-quasi class 5 if
%A23_{+} ,_{≤}3

,( %

A2,_{+} ,_{+ %}A_{+} ,_{) for every + ∈ ! }

and @ is a natural number. D. Senthil Kumar, Prasad. T in [15], has defined the new class of operators which we call B-class 5
operators. An operator % is of B class 5 if for a fixed real number B ≥ 1, % satisfies B,_{%}∗,_{%},_{− 2%}∗_{% + 9 ≥ 0 or equivalently }

%+ ,_{≤}3

,(B

, _{%},_{+} ,_{+ +} ,_{) for every + ∈ ! and a fixed real number B ≥ 1. }

In [18], Youngoh Yang and Cheoul Jun Kim introduced a class 5∗_{ operators. If %}∗,_{%},_{− 2%%}∗_{+ 9 ≥ 0, then % is called }
class 5∗_{ operators. He also proved that if % is class 5}∗_{ if and only if } _{%}∗_{+} ,_{≤}3

, %

,_{+} ,_{+ +} , _{ for every + ∈ !. In [12], D. }
Senthil Kumar et. al. introduced quasi class 5∗_{ operators. If %}∗?_{%}?_{− 2 %}∗_{%} ,_{+ %}∗_{% ≥ 0 $, then % is called quasi class 5}∗
operators. He also proved that if % is quasi class 5∗_{ if and only if } _{%}∗_{%+} ,_{≤}3

,( %

?_{+} ,_{+ %+} ,_{) for every + ∈ !. }

In this paper, we study some properties of (0, @) quasi class 5 and (0, @) quasi class 5∗_{ operators and we derive conditions }
for composition and weighted composition operators to be 0, @ quasi class 5 and (0, @) quasi class 5∗_{. Aluthge transformation }
of (0, @) quasi class 5 and (0, @) quasi class 5∗_{ operators are derived. Conditions for Composite multiplication operators to be }
(0, @) quasi class 5 and (0, @) quasi class 5∗_{ are also obtained. A characterization of (0, @) quasi class 5 and (0, @) quasi class }
5∗_{ composition and weighted composition operators on weighted Hardy space are obtained. }

**II.**(E, F) QUASI CLASS G OPERATORS

In this section, we define new class of operators called (0, @) quasi class 5, which is a super class of 0 class 5 and quasi 0 class 5 operators and studied some properties of this class of operators.

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **648 **
An operator % ∈ "(!) is said to be (0, @) quasi class 5 if for every positive integer 0 and for every + ∈ !

%A23_{+} ,_{≤} 1

1 + 0 ( %A2321+ ,

+ 0 %A_{+} ,_{) }
when 0 = 1 it is of @ quasi class 5 operators.

**Theorem 2.2. **

An operator % is of (0, @) quasi class 5 if and only if

%∗A _{%}∗321_{%}321 _{− 1 + 0 %}∗_{ % + 09 %}A _{≥ 0 for every positive integer 0. }
**Proof **

Since % is (0, @) quasi class 5 operator, we have

%A23_{+} ,_{≤} 3
321 ( %

A2321_{+} ,_{+ 0 %}A_{+} ,_{) }

⇔ % A2321_{+} ,_{− 1 + 0 %}A23_{+} ,_{+ 0 %}A_{+} ,_{) ≥ 0 }

⇔ %A2321_{+, %}A2321 I _{ − 1 + 0 %}A23_{+, %}A23_{+ + 0 %}A_{+, %}A_{+ ≥ 0 }

⇔ %∗A2321_{%}A2321 _{− 1 + 0 %}∗A23 _{%}A23 _{+ 0%}∗A_{%}A _{≥ 0 }
⇔ %∗A _{%}∗321_{%}321 _{− 1 + 0 %}∗_{ % + 09 %}A_{≥ 0. }

For example: let + = +3, +,, … ∈ K,, Define %: K,→ K, by % + = 0, +3, +,, … , %∗(+) = (+,, +?, . . . ). Then
%∗A _{%}∗321_{%}321 _{− 1 + 0 %}∗_{ % + 09 %}A_{≥ 0. ie % is @ quasi 0 class 5 operators or (0, @) quasi class 5 operators. }

From the definition of quasi 0 class 5 operator we can easily say that every quasi 0 class 5 operator is also an operator of @ quasi 0 class 5. Hence we have the following implication

class 5 ⊂ 0 class 5 ⊂ quasi 0 class 5 ⊂ @ quasi 0 class 5.
**Theorem 2.3 **

Every @ quasi class 5 operator is (0, @) quasi class 5 operator.
**Proof **

By using induction principle and simple calculation we get the result.
**Corollary 2.4 **

If % ∈ "(!) is of (0, @) quasi class 5 then % is of (0 + 1, @) quasi class 5 operator
**Corollary 2.5 **

If % ∈ "(!) is of (0, @) quasi class 5 then P% is (0, @) quasi class 5 operator for any complex number P.
**Theorem 2.6 **

Let % ∈ "(!). If QRST % is an operator of (0, @) quasi class 5, then % is @ quasi 0 paranormal operator for all Q > 0.

**Proof **

Since QVST% $ is an operator of (0, @) quasi class 5, then

QVST%

∗A

QVST%

∗ 321 QVST%

321

− 1 + 0 QVST%

∗

QVST% + 09 QVST%

A ≥ 0.

QVST%

∗ A2321 QVST%

A2321

− 1 + 0 QVST%

∗A23 QVST%

A23

+ 0 QVST%

∗A QVST%

A ≥ 0.

QVST

, A2321

%∗A2321_{%}A2321_{− 1 + 0 Q}VS_{T} , A23 _{ %}∗ A23_{%}A23_{+ 0 Q}VS_{T} ,A_{%}∗A_{%}A_{≥ 0 }
By multiplying QA2321_{ and let Q = W, then }

%∗A2321_{%}A2321_{− 1 + 0 W}1_{%}∗A23_{%}A23 _{+ 0W}321_{%}∗A_{%}A _{≥ 0. }
Hence % is @ quasi 0 paranormal operator for all Q > 0.

**Theorem 2.7 **

If (0, @) quasi class 5 operator T doubly commutes with an isometric operator S, then TS is an operator of (0, @) quasi class 5.

**Proof **

Since % is (0, @) quasi class 5 operator, then

%∗A_{ %}∗ 321 _{%}321_{− 1 + 0 %}∗_{% + 09 %}A _{≥ 0. }

Suppose % doubly commutes with an isometric operator X, then %X = X%, X∗_{% = %X}∗_{ and X}∗_{X = 9. Now let Y = %X. So we }
get Y∗A _{Y}∗ 321_{Y}321_{− 1 + 0 Y}∗_{Y + 09 Y}A _{≥ 0. Therefore %X is a (0, @) quasi class 5 operator. }

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **649 **
If a ( , @) quasi class 5 operator % ∈ "(!) is unitarily equivalent to operator X, then X is an operator of (0, @) quasi class 5.
**Proof **

Assume T is unitarily equivalent to operator X Then there exists an unitary operator ' such that X = '∗_{%' and % is (0, @) }
quasi class 5 operator, then

X∗A _{X}∗ 321 _{X}321_{− 1 + 0 X}∗_{X + 09 X}A

= '∗_{%'} ∗A _{'}∗_{%'} ∗ 321 _{'}∗_{%'} 321_{− 1 + 0 '}∗_{%'} ∗ _{'}∗_{%' + 09 '}∗_{%'} A _{≥ 0. Therefore X is (0, @) quasi class 5 }

operator.
**Theorem 2.9 **

Let % ∈ "(!) be an invertible operator and * be an operator such that * commutes with %∗_{%. Then operator * is (0, @) quasi }
class 5 if and only if operator %*%V3_{ is of (0, @) quasi class 5. }

**Proof **

Let * be (0, @) quasi class 5 operator, then

*∗A _{ *}∗ 32 _{ *}321_{− 1 + 0 *}∗_{* + 09 *}A_{≥ 0. }

Since operator * commutes with operator %∗_{%, we have } _{%*%}V3 ∗A _{%*%}V3 ∗321 _{%*%}V3 321 _{− 1 +}
0 %*%V3 ∗ _{%*%}V3 _{+ 09 %*%}V3 A

= % *∗A _{ *}∗321_{ *}321 _{− 1 + 0 *}∗_{* + 09 *}A _{%}V3_{. }
Since * is (0, @) quasi class 5 operator, then

% *∗A _{ *}∗321_{*}321 _{− 1 + 0 *}∗_{* + 09 *}A _{%}∗_{≥ 0. }
Which implies (%%∗_{) commutes with % *}∗A _{ *}∗ 321_{ *}321_{− 1 + 0 *}∗_{* + 09 *}A _{%}∗_{. }
Also %%∗ V3_{ is also commutes with % *}∗A _{ *}∗ 321_{ *}321_{ − 1 + 0 *}∗_{* + 09 *}A _{%}∗_{. }
Then % *∗A _{ *}∗ 321_{ *}32 _{− 1 + 0 *}∗_{* + 09 *}A _{%}V3_{≥ 0. }

Hence %*%V3_{ is (0, @) quasi class 5 operator. }

Conversely suppose that (%*% V3_{) is (0, @) quasi class 5 operator, then }

*∗A _{ *} ∗321_{ *}321 _{− 1 + 0 *}∗_{* + 09 *}A _{≥ 0. }
**Corollary 2.10 **

Let X be (0, @) quasi class 5 operator and Y any positive operator such that YV3_{= Y}∗_{. Then % = Y}V3_{XY is (0, @) quasi class }
5 operator.

**Theorem 2.11 **

Let % be (0, @) quasi class 5 operator. Then the tensor product % ⊗ 9 and 9 ⊗ % are both (0, @) quasi class 5 operators.
**Proof **

By the definition of (0, @) quasi class 5 and tensor product and by the simple calculation we get the result.
**Theorem 2.12 **

If % ∈ "(!) is of (0, @) quasi class 5 operator for some positive integers @ and 0, the range of % does not have dense range then
% has the following 2×2 matrix representation % = %_{0}3 %_{%},

? on ! = 4\0 %

A _{⊕ @^4 %}∗A_{, if and only if %}

3 is 0 class 5
operator on 4\0 %A _{ and %}

?A = 0. Further more _ % = _ %3 ∪ { 0} where _(%) denotes the spectrum of %.

**Proof **

Let < be an orthogonal projection of ! onto 4\0 %A _{. Then %}

3= %< = <%<. By Theorem 2.2 we have that
%∗A_{ %}∗ 321_{%}321_{− 1 + 0 %}∗_{% + 09 %}A_{≥ 0 }

Which implies

< %∗321_{%}321_{− 1 + 0 %}∗_{% + 09 < ≥ 0 }
Then %3∗321%3321− 1 + 0 %3∗%3+ 09 ≥ 0

So %3 is 0-class 5 operator on 4\0 %A . Also for any + = +3, +, ∈ !,

%_{?}A_{+}

,, +, = %A(9 − <)+, (9 − <)+
= (9 − <)+, %∗A_{(9 − <)+ = 0 }

This implies %_{?}A_{= 0. }

Since _ % ∪ c = _ %3 ∪ _(%? ) where c is the union of certain holes in _(%), which happens to be a subset of _ %3 ∩
_(%? ) [by corollary 7, [9] ] and _(%? ) = 0. _ %3 ∩ _(%?) has no interior points. So we have _ % = _ %3 ∪ { 0}.
Suppose that % = %_{0}3 %_{%},

? on H = 4\0 %

A _{ ⊕\@^4 %}∗A_{ where %}

3 is 0 class 5 operator on 4\0 %A and %?A= 0. Then
%A _{= %}3A ∑fghiV3 T3f T, T?iV3Vf

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **650 **

%∗A_{=} %3

∗A _{0}

∑_{fgh}iV3_{ T}

3f T, T?iV3Vf ∗ 0
%∗A_{ %}∗ 321_{%}321 _{− 1 + 0 %}∗_{% + 09 %}A

= %3∗A %3∗321%3321 − 1 + 0 %3∗%3+ 09 %3A k

X∗ _{m}

Where k = %3∗A( %3∗ 321 %3321− 1 + 0 %3∗%3+ 09)( AV3ngh %3n %, %?AV3Vn )

m = %_{3}n %, %?AV3Vn
AV3

ngh

∗

( %_{3}∗321%_{3}321 − 1 + 0 %∗_{3}_{%}

3 + 09) ( %3n %, %?AV3Vn AV3

ngh

)

We know that, " If Y is a matrix of the form Y "

"∗ _{o} ≥ 0 if and only if Y ≥ 0, o ≥ 0 and " = Y

S

TpoST $ for some

contraction p. Since %3 is 0-class 5 operator and m ≥ 0, then we have %∗A %∗321%321− 1 + 0 %∗% + 09 %A≥ 0. Hence T is (0, @) quasi class 5 operator.

**Theorem 2.13. **

Let B be a closed % -invariant subspace of !. Then the restriction %|q of is (0, @) quasi class 5 operator % to B is (0, @) quasi class 5 operator.

**Proof **

Let % = %_{0}3 %_{%},

? on ! = B ⊕ B

r_{. Since % is (0, @) quasi class 5 operator then by Theorem 2.12, we have %|}

q is also is (0, @) quasi class 5 operator.

**Theorem 2.14 **

Let % be a regular is (0, @) quasi class 5 operator, then the approximate point spectrum lies in the disc _st % ⊆ {Q ∈ o:

1 + 0 3,

%V A23 _{%}321 ,_{+ 0} 3,

≤ Q ≤ %

**Proof **

Suppose % is regular (0, @) quasi class 5 operator, then for every unit vector + in !, we have
+ ,_{≤ %}V A23 , _{%}A23 I ,_{ ≤} uR vwS T

321 ( %

321 , _{%}A_{+} ,_{ + 0 %}A_{+} ,_{). }

Hence %A_{+} ,_{≥} 321 IT
uR vwS T _{u}Swx T_{21} .

Now assume that Q ∈ _st(%). Then there exists a sequence { +y}, +y = 1 such that

% − Q +y → 0 when z → ∞. So we have %+y− Q+y ≥ %+y − Q +y ≥ 321

S T

uR vwS _{u}Swx T_{21}ST− Q

. Now,

when z → ∞, Q ≥ 321

S T

uR vwS _{u}Swx T_{21}ST

.

**III.**(E, F) QUASI CLASS G∗_{ OPERATORS }

In this section we define operators of (0, @) quasi class 5 and consider some basic properties and examples.
**Definition 3.1 **

An operator % is said to be (0, @) quasi class 5∗_{ ( quasi 0-class 5}∗_{)if }
%∗_{%}A_{+} ,_{≤} 1

1 + 0 %A2321+ ,+ 0 %A+ ,

for every + ∈ ! and every positive integer 0. When 0 = 1, it is of @ quasi class 5∗_{ ( @ quasi *-class 5)operator and when }
@ = 1, it is of quasi 0 class 5∗_{ operator. }

For example: let + = +3, +,, . . . ∈ K,, Define %: K,→ K, by %(+) = (0, +3, +,, . . . ), %∗(+) = (+,, +?, . . . ). Then
%∗A _{ %}∗321_{%}321_{− 1 + 0 %%}∗_{+ 09 %}A_{≥ 0. ie T is (0, @) quasi class 5}∗_{ operator. }

Using the definition of (0, @) quasi class 5∗_{ operator and by simple calculation we get the following theorem. }
**Theorem 3.2 **

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **651 **
From the definition of (0, @) quasi class 5∗_{ operator, we can easily say that every operator of 0-class 5^ ∗ and quasi 0-class 5}∗
is also an operator of (0, @) quasi class 5∗_{. Hence we have the following implications }

class 5∗_{ ⊂ 0 class 5}∗_{⊂ quasi 0-class 5}∗_{⊂ @ $ quasi 0 class 5}∗

Also every (0, @) quasi class 5∗_{ is (0 + 1, @) quasi class 5}∗_{ operator. Again, if % ∈ "(!) is (0, @) quasi class 5}∗_{ then P% is of }
(0, @) quasi class 5∗_{ operator for any complex number P. }

**Theorem 3.3 **

Let % ∈ "(!). If QVS_{T}_{% is an operator of (0, @) quasi class 5}∗_{, then % is @ quasi *- 0 -paranormal operator for all Q > 0. }
**Theorem 3.4 **

If (0, @) quasi class 5∗_{ operator % doubly commutes with an isometric operator X, then %X is an operator of (0, @) quasi class }
5∗_{. }

**Theorem 3.5 **

If (0, @) quasi class 5∗_{ operator % ∈ "(!) is unitarily equivalent to operator X, then X is an operator of (0, @) quasi class 5}∗_{. }

**Theorem 3.6 **

Let % ∈ "(!) be an invertible operator and * be an operator such that * commutes with %∗_{%. Then operator * is (0, @) quasi }
class 5∗_{ if and only if operator %*%}V3_{ is (0, @) quasi class 5}∗_{. }

**Corollary 3.7 **

Let X be (0, @) quasi class 5∗_{ operator and Y any positive operator such that Y}V3_{= Y}∗_{. Then % = Y}V3_{XY is (0, @) quasi class }
5∗_{ operator. }

**Thereom 3.8 **

Let % be (0, @) quasi class 5∗_{ operator. Then the tensor product % ⊗ 9 and 9 ⊗ % are both (0, @) quasi class 5}∗_{ operators. }
**Theorem 3.9 **

If % ∈ "(!) is of (0, @) quasi class 5∗_{ operator for any positive integer 0, a non zero complex number Q ∈ _}

t(%) and % is of

the form % = Q %_{0 %},

? on ! = @^4 % − Q ⊕ 4\0 % − Q

∗_{ , then }
1. %,= 0 and

2. %? is (0, @) quasi class 5∗operator.
**Proof **

Let % = _{0 %}Q %,

? on H = ker % − Q ⊕ 4\0 % − Q

∗_{ Without the loss of generality assume that Q = 1, then by Theorem }

3.2, %∗A _{%}∗321_{%}321_{− 1 + 0 %%}∗_{+ 09 %}A_{ ≥ 0 . Then, }

%A_{=} 1 ∑fghiV3T, T?iV3Vf

0 %?A

and
%∗ _{=} 1 0

∑fghiV3T, T?iV3Vf ∗ %?∗A
So, %∗A_{ %}∗321_{%}321_{− 1 + 0 %%}∗_{+ 09 %}A _{≥ 0 gives }

Y "
"∗ _{o} ≥ 0
Where

Y = 1 − 1 1 + 0 1 + %,%,∗ + 0,
" = ∑_{fgh}Ä _{T}

, T?

ÄVf_{− 1 + 0 %}

,%?∗ %?A− (1 + 0)(%,%,∗) AV3ngh%, %?AV3Vn and
o = "∗ _{∑}

fgh
iV3_{T}

, T?iV3Vf + ∑fghiV3T, T?iV3Vf ∗ ∑fgh1 T, T?ÄVf− 1 + 0 %,%?∗

+ %?A ∑fgh1 T, T?ÄVf ∗ ∑fgh1 T, T?ÄVf + %?∗A %?∗321%?321− 1 + 0 %?%?∗+ 09 %?A

Therefore 1 + 0 − (1 + 0)(1 + %,%,∗) + 0 ≥ 0, which implies that (1 + 0)(−%,%,∗) ≥ 0. This gives %,= 0, since n is a positive integer. Hence %? is k quasi n-class 5∗ operator.

**Corollary 3.10 **

If % ∈ "(!) is of (0, @) quasi class 5∗_{ operator for a positive integer }_{0, then }_{% is of the form }_{% =} Q 0

0 %? on ! =
@^4 % − Q ⊕ 4\0 % − Q ∗_{, where %}

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **652 **
If % ∈ "(!) is (0, @) quasi class 5∗_{ operator for a positive integer 0, % does not have dense range and % has the following 2×2 }
matrix representation % = %3 %,

0 %? on H = 4\0 %

A _{ ⊕ @^4 %}∗A_{ if and only if }_{%}

3∗321%3321− 1 + 0 %3%3∗+ %,%,∗ + 09 ≥ 0 $ and %?A= 0. Further more _ % = _ %3 ∪ { 0 }.

**Proof **

Let % ∈ "(!) be k quasi n class 5∗_{ operator and < be an orthogonal }
projection onto 4\0 %A _{. Then %}

3= %< = <%<. By Theorem 3.2 we have that
%∗A _{%}∗321_{%}321_{− 1 + 0 %}_{%}∗_{+ 09 %}A_{≥ 0 }
< %∗321_{%}321_{− 1 + 0 %}_{%}∗_{+ 09 < ≥ 0 }
%3∗321%3321− 1 + 0 %3%3∗+ %,%,∗ + 09 ≥ 0
Also for any + = (+3; +,) ∈ !,

%?A+,, +, = %A(9 − <)+, (9 − <)+
= (9 − <)+, %∗A_{(9 − <)+ = 0 }

This implies %?A= 0.

Since _ % ∪ c = _ %3 ∪ _(%? ) where c is the union of certain holes in _(%), which happens to be a subset of _ %3 ∩ _(%? ) [by corollary 7, [9] ]. _(%? ) = 0 and _ %3 ∩ _(%?) has no interior points we have _ % = _ %3 ∪ {0}.

Suppose that % = %3 %,

0 %? on H = 4\0 %

A _{ ⊕ ker %}∗A_{, %}

3∗321%3321− 1 + 0 %3%3∗+ %,%,∗ + 09 ≥ 0 and %?A = 0. Then we have

%∗A_{ %}∗321_{%}321_{− 1 + 0 %%}∗_{+ 09 %}A

= %3

∗A _{0}

∑fghiV3T3fT, T?iV3Vf ∗ %?∗A

%3∗321%3321− 1 + 0 %3%3∗+ %,%,∗ + 0 %3∗321 ∑fgh1 T3fT, T?ÄVf − 1 + 0 %,%?∗

∑_{fgh}1 _{T}

3fT, T?ÄVf ∗%3321− 1 + 0 %?%,∗ ∑fgh

1 _{T}

3fT, T?ÄVf ∗ ∑1fghT3fT, T?ÄVf − 1 + 0 %?%?∗+ 0

%3

A _{∑}

fgh
iV3_{T}

3fT, T?iV3Vf

0 %_{?}A

= Y_{"}_{∗} "_{o} ≥ 0.
Where

Y = %_{3}∗A _{%}

3∗321%3321− 1 + 0 %3%3∗+ %,%,∗ + 09 %3A,

" = %3∗A %3∗321%3321− 1 + 0 %3%3∗+ %,%,∗ + 09 AV3nghT3f%, %?AV3Vn and
o = ∑_{fgh}iV3_{T}

3fT,T?iV3Vf ∗ %3∗321%3321− 1 + 0 %3%3∗+ %,%,∗ + 09 (∑fghiV3T3fT,T?iV3Vf)
Hence % is k quasi n-class 5∗_{ operator. }

**Theorem 3.12 **

Let B be a closed % -invariant subspace of !. Then the restriction %|q of is (0, @) quasi class 5∗ operator % to B is (0, @)
quasi class 5∗_{ operator. }

**Proof **

By Theorem 3.11, %|q is also k quasi n class 5∗ operator.
**Theorem 3.13 **

Let % be a regular is (0, @) quasi class 5∗_{ operator, then the approximate point spectrum lies in the disc _}

st % ⊆ {Q ∈

o: 321

S T

uR v _{u}∗RS _{u}Swx T_{21}ST

≤ Q ≤ %

**Proof **

Suppose % is regular k quasi n class 5∗_{ operator, then for every unit vector + in !, we have }
%A_{+} ,_{≥} 1 + 0 + ,

%VA , _{%}∗V3 _{%}321 ,_{+ 0}

Now assume that Q ∈ _st(%). Then there exists a sequence +y +y = 1 such that % − Q +y → 0 when z → ∞ we have

%+y− Q+y ≥ %+y − Q +y ≥ % − Q

≥ 1 + 0 3 ,

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **653 **

Now when z → ∞, Q ≥ 321

S T

u∗RS _{u}Rv _{u}Swx T_{21}ST

.

**IV.**(E, F) QUASI CLASS G AND (E, F) QUASI CLASS G∗_{ COMPOSITION OPERATORS }

Let Ç,_{(Q) = Ç},_{(k, Σ, Q), where }_{(k, Σ, Q) be a sigma-finite measure space. A bounded linear operator }_{o}

uÑ = Ñ ∘ % on
Ç,_{(k, Σ, Q) is said to be a composition operator induced by %, a non-singular measurable transformation from k into itself, when }
the measure Q %V3_{ is absolutely continuous with respect to the measure }_{Q and the Radon-Nikodym derivative }ÜáuRS

Üá = Ñh is
essentially bounded. The Radon-Nikodym derivative of the measure Q %A V3_{ with respect to Q is denoted by Ñ}

hA , where %A is
obtained by composing %- @ times. Every essentially bounded complex-valued measurable function Ñh induces the bounded
operator B_{à}_{â} on Ç,_{(Q), which is defined by }_{B}

àâÑ = ÑhÑ for every Ñ ∈ Ç,(Q). Further ou∗ou= Bàâ, ou∗,ou,= Bà_{â}(T) and
ou∗321ou321= Bà_{â}Swx.

The following lemma due to Harrington and Whitley [9] is well known.
**Lemma 4.1 **

Let < denote the projection of Ç,_{ on ä o}
(1) o_{u}∗_{o}

uÑ = ÑhÑ and ouou∗Ñ = (Ñh∘ %)<Ñ for all Ñ ∈ Ç,, where < is the projection of Ç, onto ä o .
(2) ä o = {Ñ ∈ Ç,_{: Ñ ãå %}V3_{Σ z^\åç4\éK^ }. }

In this section @ quasi 0-class 5 and @ quasi 0-class 5∗_{ composition operator on Ç},_{ space are characterized as follows. }
**Theorem 4.2. **

Let o_{u} ∈ "(Ç,_{(Q)). Then o}

u is of @ quasi 0-class 5 if and only if Ñh(A2321)− 1 + 0 ÑhA23 + 0ÑhA ≥ 0 a.e.
**Proof **

Let ou ∈ "(Ç,(Q)) is of @ quasi 0-class 5 if and only if

ou∗A2321ouA2321− 1 + 0 ou∗A23ouA23+ 0ou∗AouA ≥ 0 By Theorem 2.2

Thus ou∗A2321ouA2321− 1 + 0 ou∗A23ouA23+ 0ou∗AouA èê, èê ≥ 0 for every characteristic function èê of ë in Σ such that
Q(ë) < ∞. Since o_{u}∗_{o}

u = Bàâ and ou∗A2321ouA2321= Bà_{â}(vwSwx) then Bà_{â}(vwSwx)− 1 + 0 Bà_{â}(vwS)+ 0Bà_{â}(v) èê, èê ≥ 0.
Hence _{ê} Ñ_{h}(A2321)− 1 + 0 Ñ_{h}(A23)+ 0Ñ_{h}(A) ìQ ≥ 0 for every ë in Σ.

Hence ou is of @ quasi 0-class 5 if and only if Ñh

(A2321)_{− 1 + 0 Ñ}

hA23 + 0ÑhA ≥ 0 a.e.

**Example 4.3 **

Let k = *, the set of all natural numbers and Q be the counting measure on it. Define %: * → * by %(1) = 1, %(4ï + ñ − 2) =
ï + 1 for ñ = 0,1,2,3 and ï ∈ *. We have Ñ_{h}t = Ñ_{h}, _{ï = ⋯ = Ñ}

h1(ï) = 1 for ï = 1. Ñh ï = 4, Ñh, ï = 16, … =
Ñ_{h}A2321(ï) = 4A2321_{ for ï ∈ * − {1}. Since Ñ}

hA2321(ï) − 1 + 0 ÑhA23(ï) + 0ÑhA (ï) ≥ 0 for every ï. Hence Hence ou is of @ quasi 0-class 5 operator.

**Theorem 4.4 [17] **

If o_{u} ∈ "(Ç,_{(Q)) has dense range then Ñ}

h= öh∘ % a.e.

**Corollary 4.5 **

If ou is of @ quasi 0-class 5 with dense range on Ç,(Q) then öh∘ % (A2321)− 1 + 0 öh∘ % A23 + 0 öh∘ % A ≥ 0 a.e.
**Proof **

By Theorem 4.2 and Theorem 4.4, we obtain the result.
**Theorem 4.6 **

Let ou∈ "(Ç,(Q)). Then ou∗ is of @ quasi 0-class 5 operator if and only if ÑhA2321 ∘ %A2321 <A2321−
1 + 0 ÑhA23∘ %A23 <A23+ 0 ÑhA∘ %A <A ≥ 0 a.e, where <3, <,, <A2321 are the projections of Ç, onto
ä o , ä o, _{, … , ä(o}A2321_{) respectively. }

**Proof **

Suppose ou ∈ "(Ç,(Q)) and ou∗ is of @ quasi 0-class 5 operator if and only if

ouA2321ou∗A2321− 1 + 0 ouA23ou∗A23+ 0ouAou∗A ≥ 0 By Theorem 2.2. Then

ouA2321ou∗A2321− 1 + 0 ouA23ou∗A23+ 0ouAou∗A Ñ, Ñ ≥ 0 for every Ñ ∈ Ç,(Q).

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **654 **
Hence

ÑhA2321 ∘ %A2321 <A2321− 1 + 0 ÑhA23∘ %A23 <A23+ 0 ÑhA∘ %A <A Ñ, Ñ ≥ 0, ⇔ ÑhA2321 ∘ %A2321 <A2321− 1 + 0 ÑhA23∘ %A23 <A23+ 0 ÑhA∘ %A <A ≥ 0a.e.

**Corollary 4.7 **

Let ou ∈ "(Ç,(Q)) with dense range. Then ou∗ is of @ quasi 0-class 5 operator if and only if ÑhA2321 ∘ %A2321 − 1 + 0 ÑhA23∘ %A23 + 0 ÑhA∘ %A ≥ 0 a.e.

**Theorem 4.8 **

Let ou ∈ "(Ç,(Q)). Then ou is of @ quasi 0-class 5∗ if and only if Ñh(A2321)− 1 + 0 ÑhA ë(Ñh) ∘ %VA + 0ÑhA ≥ 0 a.e.
**Proof **

Let ou ∈ "(Ç,(Q)) is of @ quasi 0-class 5∗ if and only if
o_{u}∗A2321_{o}

uA2321− 1 + 0 ou∗A(ouou∗)ouA+ 0ou∗AouA≥ 0

Thus ou∗A2321ouA2321− 1 + 0 ou∗A(ouou∗)ouA+ 0ou∗AouA èê, èê ≥ 0 for every characteristic function èê of ë in Σ such that
Q(ë) < ∞. Since o_{u}∗_{o}

u = Bàâ and ou∗A2321ouA2321= Bà_{â}(vwSwx) then ê Ñh(A2321)− 1 + 0 Ñh(A)ë(Ñh) ∘ %VA+ 0Ñh(A) ìQ ≥ 0
for every ë in Σ.

Hence ou is of @ quasi 0-class 5∗ if and only if Ñh

(A2321)_{− 1 + 0 Ñ}

hA ë(Ñh) ∘ %VA+ 0ÑhA ≥ 0 a.e.

**Example 4.9 **

Let k = *, the set of all natural numbers and Q be the counting measure on it. Define %: * → * by %(1) = %(2) = %(3) = 1,
%(4ï + ñ) = ï + 1 for ñ = 0,1,2,3 and ï ∈ *. Since Ñ_{h}A2321 − 1 + 0 Ñ_{h}A ë(Ñh) ∘ %VA+ 0ÑhA ≥ 0 for every ï. Hence
Hence o_{u} is of @ quasi 0-class 5∗_{ operator. }

**Corollary 4.10 **

If ou is @ quasi 0-class 5∗ with dense range on Ç,(Q) if and only if ÑhA2321 − 1 + 0 ÑhA23+ 0ÑhA≥ 0 a.e.
**Theorem 4.11 **

Let ou∈ "(Ç,(Q)). Then ou∗ is of @ quasi 0-class 5 operator if and only if ÑhA2321 ∘ %A2321 <A2321−
1 + 0 Ñ_{h}A23_{∘ %}A23 _{<}

A23+ 0 ÑhA∘ %A <A ≥ 0 a.e, where <õ’s are the projections of Ç, onto ä oú respectively.
**Proof **

Let o_{u}∗ _{∈ "(Ç},_{(Q)) is of @ quasi 0-class 5 operator if and only if }
o_{u}A2321_{o}

u∗A2321− 1 + 0 ouA(ou∗ou)ou∗A+ 0ouAou∗A≥ 0 Thus

ouA2321ou∗A2321− 1 + 0 ouA(ou∗ou)ou∗A+ 0ouAou∗A Ñ, Ñ ≥ 0 for every characteristic function èê of ë in Σ such that
Q(ë) < ∞. Since o_{u}∗_{o}

u = Bàâ, ou∗321ou321= Bà_{â}(Swx) and ouou∗= Ñh∘ % < then ê ÑhA2321 ∘ %A2321 <A2321−
1 + 0 ÑhA Ñh ∘ %AV3 + 0 ÑhA∘ %A <A ìQ ≥ 0 for every ë in Σ.

Hence ou is of @ quasi 0-class 5∗ if and only if ÑhA2321 ∘ %A2321 <A2321− 1 + 0 ÑhA Ñh ∘ %AV3 + 0 ÑhA∘ %A <A ≥ 0 a.e.
**Corollary 4.12 **

Let ou ∈ "(Ç,(Q)) with dense range. Then ou∗ is of @ quasi 0-class 5∗ if and only if ÑhA2321 ∘ %A2321 − 1 + 0 ÑhA Ñh ∘ %AV3 + 0 ÑhA∘ %A ≥ 0 a.e.

**V.**F QUASI E-CLASS G AND F QUASI E-CLASS G∗_{ WEIGHTED COMPOSITION OPERATORS }

A weighted composition operator is a linear transformation acting on the set of complex valued Σ measurable functions Ñ of
the form puÑ = ù(Ñ ∘ %), where ù is a complex valued measurable function. In the case that ù = 1 a.e., we say that pu is a
composition operator. Let ù_{A} denote ù ù_{u} ù_{u}, _{, … , (ù}

uAV3) so that pAuÑ = ùA Ñ ∘ % A [12].

To examine the weighted composition operators efficiently, Alan Lambert [11], associated conditional expectation operator
ë with each transformation % as ë(⦁|%3_{Σ ) = ë(⦁). }

ë(Ñ) is defined for each non-negative measurable function Ñ ∈ Çt_{(1 ≤ ï) and is uniquely determined by the conditions }
(i) ë(Ñ) is %V3_{Σ measurable and }

(ii) If " is any %3_{Σ measurable set for which } _{Ñ}

ü ìá converges, then we have üÑìá= üë(Ñ)ìá .
As an operator on Çt_{, ë is the projection onto the closure range of o. ë}

1 the identity on Çt if and only if %V3_ = _. Now we are
ready to derive the characterization of @ quasi 0-class 5 and of @ quasi 0-class 5∗_{ weighted composition operator as follows. }
**Theorem 5.1 **

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **655 **
**Proof **

Let pu ∈ "(Ç,(Q)) is of @ quasi 0-class 5 if and only if
p_{u}∗A2321_{p}

uA2321− 1 + 0 pu∗A23puA23+ 0pu∗ApuA ≥ 0 By Theorem 2.2

Thus pu∗A2321puA2321− 1 + 0 pu∗A23puA23+ 0pu∗ApuA èê, èê ≥ 0 for every characteristic function èê of ë in Σ such
that Q(ë) < ∞. Since pu∗pu = Ñhë ù, ∘ %V3 , puAÑ = ùA Ñ ∘ % A, pu∗AÑ = ÑhAë ùAÑ ∘ %VA and pu∗ApuAÑ = ÑhAë ùA, ∘
%VA_{Ñ. Then } _{Ñ}

hA2321ë ùA2321, ∘ %V(A2321)− 1 + 0 Ñh

(A23)_{ë ù}

A23, ∘ %V(A23) + 0 ÑhAë ùA, ∘ %VA èê, èê ≥ 0.
Which implies _{ê} ÑhA2321ë ùA2321, ∘ %V A2321 − 1 + 0 ÑhA23ë ùA23, ∘ %V A23 + 0 ÑhAë ùA, ∘ %VA ìQ ≥ 0 for
every ë in Σ.

Hence pu is of @ quasi 0-class 5 if and only if ÑhA2321ë ùA2321, ∘ %V(A2321) − 1 + 0 Ñh

(A23)_{ë ù}

A23, ∘ %V(A23) + 0 ÑhAë ùA, ∘ %VA ≥ 0 a.e.

**Corollary 5.2 **

Let pu be a weighted composition operator on "(Ç,(Q)) and assume that %V3Σ = Σ . Then pu is of @ quasi 0-class 5 if and only if ÑhA2321 ùA2321, ∘ %V(A2321) − 1 + 0 Ñh

(A23) _{ù}

A23, ∘ %V(A23) + 0 ÑhA ùA, ∘ %VA ≥ 0 a.e.

**Theorem 5.3 **

Let pu be a weighted composition operator on "(Ç,(Q)). Then pu∗ is of @ quasi 0-class 5 if and only if
ù_{A2321} Ñ_{h}A2321_{∘ %}V(A2321) _{ë ù}

A2321 − 1 + 0 ùA23 Ñh

(A23)_{∘ %}_{V(A23)} _{ë ù}

A23 + 0ùA ÑhA∘ %VA ë ùA ≥ 0 a.e.

**Proof **

Let p_{u}∗_{∈ "(Ç},_{(Q)) is of @ quasi 0-class 5 if and only if }
p_{u}A2321_{p}

u∗A2321− 1 + 0 puA23pu∗A23+ 0puApu∗A ≥ 0 By Theorem 2.2

Thus puA2321pu∗A2321− 1 + 0 puA23pu∗A23+ 0puApu∗A èê, èê ≥ 0 for every characteristic function èê of ë in Σ such
that Q(ë) < ∞. Since pupu∗ = ù(Ñh∘ %)ë ùÑ , puAÑ = ùA Ñ ∘ % A, pu∗AÑ = ÑhAë ùAÑ ∘ %VA and pu∗ApuAÑ = ùA(Ñh A∘
% A_{)ë ù}

AÑ . Then ùA2321 ÑhA2321∘ %V(A2321) ë ùA2321 − 1 + 0 ùA23 Ñh

(A23)_{∘ %}_{V(A23)} _{ë ù}

A23 + 0ùA ÑhA∘
%VA _{ë ù}

A èê, èê ≥ 0.

Which implies _{ê}ùA2321 ÑhA2321∘ %V(A2321) ë ùA2321 − 1 + 0 ùA23 Ñh(A23)∘ %V(A23) ë ùA23 + 0ùA ÑhA∘
%VA _{ë ù}

A ìQ ≥ 0 for every ë in Σ.

Hence p_{u}∗_{ is of }_{@ quasi }_{0-class }_{5 if and only if }_{ù}

A2321 ÑhA2321∘ %V(A2321) ë ùA2321 − 1 + 0 ùA23 Ñh(A23)∘

%V(A23) _{ë ù}

A23 + 0ùA ÑhA∘ %VA ë ùA ≥ 0 a.e.

** Corollary 5.4 **

Let pu be a weighted composition operator on "(Ç,(Q)) and %V3Σ = Σ. Then pu∗ is of @ quasi 0-class 5 if and only if ùA2321, ÑhA2321∘ %V(A2321) − 1 + 0 ùA23 , Ñh(A23)∘ %V(A23) + 0ùA , ÑhA∘ %VA ≥ 0 a.e.

**Theorem 5.5 **

Let pu be a weighted composition operator on "(Ç,(Q)). Then pu is of @ quasi 0-class 5∗ if and only if ÑhA2321ë ùA2321, ∘ %V(A2321) − 1 + 0 ÑhA ë ùA23, ë(Ñh) ∘ %VA + 0 ÑhAë ùA, ∘ %VA ≥ 0 a.e.

**Proof **

Let pu ∈ "(Ç,(Q)) is of @ quasi 0-class 5 if and only if

pu∗A2321puA2321− 1 + 0 pu∗Apupu∗puA+ 0pu∗ApuA≥ 0 By Theorem 2.2

Thus p_{u}∗A2321_{p}

uA2321− 1 + 0 pu∗Apupu∗puA+ 0pu∗ApuA èê, èê ≥ 0 for every characteristic function èê of ë in Σ such
that Q(ë) < ∞. Since p_{u}∗_{p}

u = Ñhë ù, ∘ %V3Ñ , puAÑ = ùA Ñ ∘ % A, pu∗AÑ = ÑhAë ùAÑ ∘ %VA and pu∗ApuAÑ =
ÑhAë ùA, ∘ %VAÑ. Then ÑhA2321ë ùA2321, ∘ %V(A2321)− 1 + 0 Ñh(A)ë ùA23, ë(Ñh) ∘ %VA + 0 ÑhAë ùA, ∘
%VA _{è}

ê, èê ≥ 0.

Which implies Ñ_{h}A2321_{ë ù}

A2321, ∘ %V A2321

ê − 1 + 0 Ñh

A23 _{ë ù}

A23, ë(Ñh) ∘ %VA + 0 ÑhAë ùA, ∘ %VA ìQ ≥ 0 for every ë in Σ.

Hence pu is of @ quasi 0-class 5∗ if and only if ÑhA2321ë ùA2321, ∘ %V(A2321) − 1 + 0 Ñh(A23)ë ùA23, ë(Ñh) ∘ %VA +
0 Ñ_{h}A_{ë ù}

A, ∘ %VA ≥ 0 a.e.

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **656 **
Let pu be a weighted composition operator on "(Ç,(Q)) and assume that %V3Σ = Σ . Then pu is of @ quasi 0-class 5∗ if and
only if Ñ_{h}A2321 _{ù}

A2321, ∘ %V(A2321) − 1 + 0 Ñh(A23) ùA23, ∘ %VA + 0 ÑhA ùA, ∘ %VA ≥ 0 a.e.

**Theorem 5.7 **

Let p_{u} be a weighted composition operator on "(Ç,_{(Q)). Then } _{p}

u∗ is of @ quasi 0-class 5∗ if and only if
ù_{A2321} Ñ_{h}A2321_{∘ %}A2321 _{ë ù}

A2321 − 1 + 0 ùAë ùA2, Ñhë(ÑhA) ∘ %A + 0ùA ÑhA∘ %A ë ùA ≥ 0 a.e.

**Corollary 5.8 **

Let pu be a weighted composition operator on "(Ç,(Q)) and %V3Σ = Σ. Then pu∗ is of @ quasi 0-class 5∗ if and only if
ù_{A2321}, _{Ñ}

hA2321∘ %A2321 − 1 + 0 ùAùA2, ÑhA23∘ %A + 0ùA , ÑhA∘ %A ≥ 0 a.e.

The Aluthge transform of T is the operator % given by % = |%|ST'|%|ST was introduced in [1] by Aluthge is the. The idea behind

the Aluthge transform is to convert an operator into another operator which shares with the first one some spectral properties
but it is closed to being a normal operator. More generally we may have family of operators %_{†}∶ 0 < 4 ≤ 1 where %_{†} =
|%|†_{'|%|}3V†_{[2. For a composition operator o, the polar decomposition is given by o = '|o| where |o|Ñ = Ñ}

hÑ and 'Ñ = 3

àâ∘ uÑ ∘ %.

In [11] Lambert has given general Aluthge transformation for composition operator as o† = |o|†'|o|3V† and o†Ñ =
à_{â}

àâ∘ u ¢

T_{Ñ ∘ %. That is }_{o}

† is the weighted composition operators with weights £ = _{à}àâ

â∘ u ¢ T

where 0 < 4 < 1. Since o† is weighted composition operator it is easy to show that o† Ñ = Ñh. ë £ ,%V3 Ñ and |o†∗ |Ñ = §ë[§. Ñ] where £ =

ß à_{â}∘ u

ê(ß à_{â}∘ u)TS®

. Also we have
o†A Ñ = £A(Ñ ∘ %A)
o†∗A = ÑhAë(£A. Ñ) ∘ %VA
o_{†}∗A_{o}

†AÑ = ÑhAë(£A,) ∘ %VAÑ

**Theorem 5.9 **

Let o†∈ "(Ç,(Q)). Then o† is of @ quasi 0-class 5 if and only if ÑhA2321ë £A2321, ∘ %V(A2321) − 1 + 0 Ñh(A23)ë £A23, ∘

%V(A23) _{+ 0 Ñ}

hAë £A, ∘ %VA ≥ 0 a.e.

**Proof **

Since o† is a weighted composition operator with weight = _{à}àâ

â∘ u ¢

T_{ , it follows from Theorem 5.1, that o}

† is of @ quasi 0-class 5 if and only if ÑhA2321ë £A2321, ∘ %V(A2321) − 1 + 0 Ñh(A23)ë £A23, ∘ %V(A23) + 0 ÑhAë £A, ∘ %VA ≥ 0 a.e.

**Corollary 5.10 **
If %V3_{Σ = Σ and } _{o}

†∈ "(Ç,(Q)). Then o† is of @ quasi 0-class 5 if and only if ÑhA2321£A2321, ∘ %V(A2321) −
1 + 0 Ñ_{h}(A23)£_{A23}, _{∘ %}V(A23) _{+ 0 Ñ}

hA£A,∘ %VA ≥ 0 a.e.

**Theorem 5.11 **

Let o†∈ "(Ç,(Q)). Then o†∗ is of @ quasi 0-class 5 if and only if £A2321 ÑhA2321∘ %V(A2321) ë £A2321 −
1 + 0 £_{A23} Ñ_{h}(A23)∘ %V(A23) _{ë £}

A23 + 0£A ÑhA∘ %VA ë £A ≥ 0 a.e.

**Proof **

Since o†∗ is a weighted composition operator with weight = _{à}àâ

â∘ u ¢

T_{ , it follows from Theorem 5.3, that o}

†∗ is of @ quasi 0-class
5 if and only if £_{A2321} Ñ_{h}A2321_{∘ %}V(A2321) _{ë £}

A2321 − 1 + 0 £A23 Ñh

(A23)_{∘ %}_{V(A23)} _{ë £}

A23 + 0£A ÑhA∘ %VA ë £A ≥ 0 a.e.

**Corollary 5.12 **

Let o†∈ "(Ç,(Q)) and %V3Σ = Σ. Then o†∗ is of @ quasi 0-class 5 if and only if £A2321, ÑhA2321∘ %V(A2321) − 1 + 0 £A23, Ñh(A23)∘ %V(A23) + 0£A, ÑhA∘ %VA ≥ 0 a.e.

**Theorem 5.13 **

Let o†∈ "(Ç,(Q)). Then o† is of @ quasi 0-class 5∗ if and only if ÑhA2321ë £A2321, ∘ %V(A2321) − 1 +
0 Ñ_{h}(A23)ë £A23, ë(Ñh) ∘ %V(A23) + 0 ÑhAë £A, ∘ %VA ≥ 0 a.e.

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **657 **
Since o† is a weighted composition operator with weight = _{à}àâ

â∘ u ¢

T_{ , it follows from Theorem 7.2, that o}

† is of @ quasi 0-class
5∗_{ if and only if } _{Ñ}

hA2321ë £A2321, ∘ %V(A2321) − 1 + 0 Ñh(A23)ë £A23, ë(Ñh) ∘ %V(A23) + 0 ÑhAë £A, ∘ %VA ≥ 0 a.e.

**Corollary 5.14 **
If %V3_{Σ = Σ and } _{o}

†∈ "(Ç,(Q)). Then o† is of @ quasi 0-class 5∗ if and only if ÑhA2321£A2321, ∘ %V(A2321) −
1 + 0 Ñ_{h}(A23)£A23, ∘ %V(A23) + 0 ÑhA£A,∘ %VA ≥ 0 a.e.

**Theorem 5.15 **

Let o†∈ "(Ç,(Q)). Then o†∗ is of @ quasi 0-class 5∗ if and only if £A2321 ÑhA2321∘ %(A2321) ë £A2321 −
1 + 0 £_{A23}ë £_{A23} Ñ_{h}(A23)ë(Ñh) ∘ %(A23) + 0£A ÑhA∘ %A ë £A ≥ 0 a.e.

**Corollary 5.16 **
If %V3_{Σ = Σ and } _{o}

†∗∈ "(Ç,(Q)). Then o†∗ is of @ quasi 0-class 5∗ if and only if £A2321, ÑhA2321∘ %(A2321) −
1 + 0 £_{A23}, _{Ñ}

h(A23)∘ %(A23) + 0£A, ÑhA∘ %A ≥ 0 a.e.

B. P Duggal [5] described the second Aluthge Transformation of % by % = |%|ST©|%| S

T, where % = ©|%| is the polar

decomposition of %. Now we consider o = |o†|

S

T©|o_{†}|ST, where o_{†}= © o_{†} is the polar decomposition of the generalized

Aluthge transformation is o†: 0 < 4 < 1. We have |o†|Ñ = ™Ñ, where ™ = Ñh. ë(£,) ∘ %V3. o = |o†|

S
T©|o_{†}|

S T= ™

S

T_{© ™}ST_{ Ñ = ™}ST_{£} ´ ¨≠Æ Ø

Ø ™

S

®Ñ ∘ % = ™ S

®£ ´ ¨≠Æ Ø

ØS®

∘ % Ñ ∘ % . We see then that o is a weighted composition operator with weight w′ = ™S®£ ´ ¨≠Æ Ø

ØS®

∘ % .

**Theorem 5.17 **

If o is of @ quasi 0-class 5 if and only if Ñ_{h}A2321_{ë ù′}
A2321

, _{∘ %}V(A2321) _{− 1 + 0 Ñ}

h(A23)ë ù′A23, ∘ %V(A23) +
0 Ñ_{h}A_{ë ù′}

A

, _{∘ %}VA _{≥ 0 a.e. }

**Proof **

Since o is a weighted composition operator with weight w′ = ™S®£ ´ ¨≠Æ Ø

ØS®

∘ % , then by Theorem 5.1 we obtain the result.

**Corollary 5.18 **

If %V3_{Σ = Σ and o ∈ "(Ç},_{(Q)) is of @ quasi 0-class 5 if and only if } _{Ñ}

hA2321ù′A2321, ∘ %V(A2321) − 1 + 0 Ñh
(A23)_{ù′}

A23

, _{∘}

%V(A23) _{+ 0 Ñ}

hAù′A,∘ %VA ≥ 0 a.e.

**Theorem 5.19 **

Let o ∈ "(Ç,_{(Q)). Then } _{o}∗_{ is of } _{@ quasi } _{0-class } _{5 if and only if } _{ù′}

A2321 ÑhA2321ë ù′A2321, ∘ %V(A2321) − 1 + 0 ù≤

A23 ÑhA23 ë ù≤A23, ∘ %V A23 + 0 ù′A ÑhAë ù′A, ∘ %VA ≥ 0 a.e.

**Proof **

Since o∗_{ is a weighted composition operator with weight w′ = ™}S_{®}_{£} ´ ¨≠Æ Ø
ØS® ∘ %

, then by Theorem 5.3 we obtain the result.

**Corollary 5.20 **

Let o ∈ "(Ç,_{(Q)) and } _{%}V3_{Σ = Σ. Then } _{o}∗_{ is of } _{@ quasi } _{0-class } _{5 if and only if }_{ù′}
A2321

, _{Ñ}

hA2321∘ %V(A2321) − 1 + 0 ù≤

A23

, _{Ñ}

hA23 ∘ %V A23 + 0 ù′A , ÑhA∘ %VA ≥ 0 a.e.

**Theorem 5.21 **

If o is of @ quasi 0-class 5∗_{ if and only if } _{Ñ}

hA2321ë ù′A2321, ∘ %V(A2321) − 1 + 0 Ñh(A23)ë ù′A23, ë(Ñh) ∘ %V(A23) + 0 ÑhAë ù′A, ∘ %VA ≥ 0 a.e.

**Corollary 5.22 **

If %V3_{Σ = Σ and o ∈ "(Ç},_{(Q)) is of @ quasi 0-class 5}∗_{ if and only if } _{Ñ}

hA2321ù′A2321, ∘ %V(A2321) − 1 + 0 Ñh
(A23)_{ù′}

A23

, _{∘}

%V(A23) _{+ 0 Ñ}

hAù′A,∘ %VA ≥ 0 a.e.

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **658 **
Let o ∈ "(Ç,_{(Q)). Then } _{o}∗_{ is of } _{@ quasi } _{0-class } _{5}∗_{ if and only if } _{ù′}

A2321 ÑhA2321∘ %(A2321) ë ù′A2321 −

1 + 0 ù′_{A23}ë ù′_{A23} Ñ_{h}(A23)ë(Ñh) ∘ %(A23) + 0ù′A ÑhA∘ %A ë ù′A ≥ 0 a.e.

**Corollary 5.24 **

Let o ∈ "(Ç,_{(Q)) and }_{%}V3_{Σ = Σ. Then }_{o}∗_{ is of }_{@ quasi }_{0-class }_{5}∗_{ if and only if }_{ù′}
A2321

, _{Ñ}

hA2321∘ %(A2321) −
1 + 0 ù′_{A23}, _{Ñ}

h(A23)∘ %(A23) + 0ù′A, ÑhA∘ %A ≥ 0 a.e.

**VI.**F QUASI E-CLASS G OF F QUASI E-CLASS G∗** _{ WEIGHTED COMPOSITION OPERATORS ON WEIGHTED HARDY SPACE. }**
The set !,

_{(≥) of formal complex power series }

_{Ñ(¥) = }

_{\}

y¥y µ

ygh such that Ñ ∂, = µygh\y ,≥y, < ∞ is a Hilbert space of functions analytic in the unit disc with the inner product.

Ñ, ö ∂ = yghµ \yéy≥y, for an analytic map f on the open unit disc ∑ and ö(¥) = µyghéy¥y.

Let ∏: ∑ → ∑ be an analytic self map of the unit disc and consider the corresponding composition operator oπ acting on !,(≥).
That is oπ(Ñ) = Ñ ∘ ∏ for Ñ ∈ !,(≥). The operators oπ are not necessarily defined on all of !,(≥). They are everywhere
defined in some special cases on the classical Hardy Space !,_{ (the case when ≥}

1 = 1 for all n) and on a general space !,(≥)
if the function ∏ is analytic on some open set containing the closed unit disc having supremum norm strictly smaller than one.
The weighted composition operator pπ is defined as (pπÑ)(¥) = £Ñ(∏(¥)) and (pπ∗Ñ)(¥) = £Ñ(∏(¥)) for every ¥ ∈ ∑.
Let ù be a point on the open disc. Define @_{∫}∂(¥) = ªº∫Rº

∂_{º}T
µ

ygh . Then the function @∫∂ is a point evaluation for !,(≥).Then @∫∂
is in !,_{(≥) and } _{@}

∫∂ ,=

∫Tº

∂_{º}T
µ

ygh . Thus @∫ is an increasing function ofù. If Ñ(¥) = µygh\y¥y then Ñ, @∫∂ = Ñ(ù)
for all Ñ and @_{∫}∂. Hence we can easily seen that o_{π}∗_{@}

∫∂ = @π(∫)

∂ _{, p}

π∗ ∫∂ = £@∫∂ and @h∂= 1 (the function identically equal to 1).
Now we characterize @ quasi 0-class 5 and @ quasi 0-class 5∗_{ composition operators on this space as follows. }

**Theorem 6.1 **

If o_{π} is of @ quasi 0-class 5 operator in !,_{(≥), then o}

π∗A2321oπA2321− 1 + 0 oπ∗A23oπA23+ 0oπ∗AoπA ≥ 0.

**Proof **

For Ñ ∈ !,_{(≥), consider }
o_{π}∗A2321_{o}

πA2321− 1 + 0 oπ∗A23oπA23+ 0oπ∗AoπA Ñ, Ñ

= oπ∗A2321oπA2321 Ñ, Ñ − 1 + 0 oπ∗A23oπA23 Ñ, Ñ + 0 oπ∗AoπA Ñ, Ñ
= o_{π}∗A2321_{Ñ, o}

πA2321Ñ − 1 + 0 oπ∗A23Ñ, oπA23Ñ + 0 oπ∗AÑ, oπAÑ

= o_{π}A2321_{Ñ} ,_{− 1 + 0 o}

πA23Ñ ,

+ 0 o_{π}A_{Ñ} ,

Let Ñ = @_{h}∂ then
o_{π}∗A2321_{o}

πA2321− 1 + 0 oπ∗A23oπA23+ 0oπ∗AoπA Ñ, Ñ

= oπA2321@h∂ ,− 1 + 0 oπA23@h∂ ,+ 0 oπA@h∂ ,
= @_{h}∂ ,− 1 + 0 @_{h}∂ ,+ 0 @_{h}∂ ,

= 0

If oπ is of @ quasi 0-class 5 operator.
**Theorem 6.2 **

If o_{π}∗_{ is of @ quasi 0-class 5 operator in !},_{(≥), then o}

πA2321oπ∗A2321− 1 + 0 oπA23oπ∗A23+ 0oπAoπ∗A ≥ 0.

**Proof **

For Ñ ∈ !,_{(≥), consider }

oπA2321oπ∗A2321− 1 + 0 oπA23oπ∗A23+ 0oπAoπ∗A Ñ, Ñ

= o_{π}∗A2321_{Ñ} ,_{− 1 + 0 o}

π∗A23Ñ ,

+ 0 o_{π}∗A_{Ñ} ,

Let Ñ = @_{h}∂ and ∏ 0 = 0 then we have

oπA2321oπ∗A2321− 1 + 0 oπA23oπ∗A23+ 0oπAoπ∗A Ñ, Ñ
= o_{π}∗A2321_{@}

h∂ ,− 1 + 0 oπ∗A23@h∂ ,+ 0 oπ∗A@h∂ ,
= @_{h}∂ ,− 1 + 0 @_{h}∂ ,+ 0 @_{h}∂ ,

= 0

Hence o_{π}∗_{ is of @ quasi 0-class 5 operator. }
**Theorem 6.3 **

If o_{π} is of @ quasi 0-class 5∗_{ operator in !},_{(≥) if and only } _{@}

h∂ ,≥ @π(h)

∂ ,

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **659 **
**Theorem 6.4 **

If o_{π}∗_{ is of @ quasi 0-class 5}∗_{ operator in !},_{(≥) if and only } _{@}

πvwSwx_{(h)}

∂ ,_{≥ @}

πv_{(h)}

∂ ,

.

Next we characterize the @ quasi 0-class 5 and @ quasi 0-class 5∗_{ weighted composition operator on weighted hardy space as }
follows

**Theorem 6.5 **

An operator p_{π}∈ !,_{(≥) is @ quasi 0-class 5 if and only if } _{£}A2321 ,_{− 1 + 0 £}A23 ,_{+ 0 £}A ,_{≥ 0. }
**Proof **

Since p_{π} is @ quasi 0-class 5 operator, then for any Ñ ∈ !,_{(≥), we have }
p_{π}∗A2321_{p}

πA2321− 1 + 0 pπ∗A23pπA23+ 0pπ∗ApπA Ñ, Ñ ≥ 0
⇔ p_{π}A2321_{Ñ} ,_{− 1 + 0 p}

πA23Ñ ,

+ 0 p_{π}A_{Ñ} ,_{≥ 0 }
⇔ p_{π}A2321_{@}

h∂ ,− 1 + 0 pπA23@h∂ ,+ 0 pπA@h∂ ,≥ 0 when Ñ = @h∂
⇔ £A2321_{@}

h∂ ,− 1 + 0 £A23@h∂ ,+ 0 £A@h∂ ,≥ 0
⇔ £A2321 , _{@}

h∂ ,− 1 + 0 £A23 , @h∂ ,+ 0 £A , @h∂ ,≥ 0
⇔ £A2321 ,_{− 1 + 0 £}A23 ,_{+ 0 £}A , _{≥ 0. }
**Theorem 6.6 **

An operator pπ∗∈ !,(≥) is @ quasi 0-class 5 if and only if £A2321 ,− 1 + 0 £A23 ,+ 0 £A ,≥ 0.
**Proof **

Since p_{π}∗ _{ is @ quasi 0-class 5 operator, we have }
p_{π}A2321_{p}

π∗A2321− 1 + 0 pπA23pπ∗A23+ 0pπApπ∗A Ñ, Ñ ≥ 0 for any Ñ ∈ !,(≥) pπA2321pπ∗A2321− 1 + 0 pπA23pπ∗A23+ 0pπApπ∗A Ñ, Ñ ≥ 0

⇔ p_{π}∗A2321_{Ñ} ,_{− 1 + 0 p}
π∗A23Ñ

,

+ 0 p_{π}∗A_{Ñ} ,_{≥ 0 }
⇔ £A2321_{@}

h∂ ,− 1 + 0 £A23@h∂ ,+ 0 £A@h∂ ,≥ 0 for Ñ = @h∂ and ∏ 0 = 0
⇔ £A2321 ,_{− 1 + 0} A23 ,_{+ 0 £}A ,_{≥ 0. }

Hence the theorem.
**Theorem 6.7 **

An operator p_{π} is of @ quasi 0-class 5∗_{ operator in !}, _{≥} _{ if and only if } _{£}A2321 ,_{− 1 + 0 £}, _{£}AV3 ,_{+ 0 £}A ,_{≥ 0. }
**Theorem 6.8 **

An operator p_{π}∗_{∈ !},_{(≥) is of @ quasi 0-class 5}∗_{ if and only if } _{£}A2321 ,_{− 1 + 0 £}, _{£}AV3 ,_{+ 0 £}A ,_{≥ 0. }
**VII.**(E, F) QUASI CLASS G AND (E, F) QUASI CLASS G∗_{ OPERATORS }

As composite multiplication operator to a linear transformation acting on a set of complex value Σ measurable functions Ñ of the form BΩ,u Ñ = ouBΩÑ = ç ∘ %Ñ ∘ % where ç is a complex valued Σ measurable function. In the case ç = 1a.e, BΩ,u becomes a composition operator denoted by ou.

**Proposition 7.1 **

Let the composite multiplication operator BΩ,u Ñ ∈ " Ç, Q then for ç ≥ 0
(1) B_{Ω,u}∗ _{B}

Ω,u Ñ = ç,ÑhÑ.
(2) B_{Ω,u}B_{Ω,u}∗ _{ Ñ = ç},_{∘ % Ñ}

h∘ % . ë(Ñ).

Since BΩ,u Ñ = ouBΩÑ = ç ∘ %Ñ ∘ %, BΩ,u1 Ñ = ouBΩ 1Ñ = ç1 Ñ ∘ % ,, BΩ,u∗ Ñ = çÑh. ë(Ñ) ∘ %V3 and BΩ,u∗1 Ñ =
çÑh. ë çÑh ∘ %V(1V3). ë(Ñ) ∘ %V1 where ë çÑh ∘ %V 1V3 = ë çÑh ∘ %V3, ë çÑh ∘ %V,, … , ë çÑh ∘ %V(1V3), ë çÑh ∘
%1V3_{= ë çÑ}

h ∘ %3, ë çÑh ∘ %,, … , ë çÑh ∘ %1V3.

In this section, we study @ quasi 0-class 5 and @ quasi 0-class 5∗_{ composite }
multiplication operator as follows.

**Theorem 7.2 **

Let the composite multiplication operator BΩ,u Ñ ∈ " Ç, Q . Then BΩ,u is @ quasi 0-class 5 if and only if çÑh. ë çÑh ∘

%V A21_{ë ç}

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **660 **
**Proof **

Suppose BΩ,u is @ quasi 0-class 5 operator, then

BΩ,u∗A2321BΩ,uA2321− 1 + 0 BΩ,u∗A23BΩ,uA23+ 0BΩ,u∗ABΩ,uA ≥ 0. Then for any Ñ ∈ Ç, Q , we have BΩ,u∗A2321BΩ,uA2321− 1 + 0 BΩ,u∗A23BΩ,uA23+ 0BΩ,u∗ABΩ,uA Ñ, Ñ ≥ 0 BΩ,u∗A2321BΩ,uA2321Ñ, Ñ − 1 + 0 BΩ,u∗A23BΩ,uA23Ñ, Ñ + 0 BΩ,u∗ABΩ,uA Ñ, Ñ ≥ 0

Since BΩ,u∗ABΩ,uA = çÑh. ë çÑh ∘ %V(AV3). ë(Ñ) ∘ %V1, BΩ,uA BΩ,u∗A = çAç ∘ %A. Ñh∘ %A. ë çÑh ∘ %AV3. ë(Ñ). where çA = ç ∘
%. ç ∘ %,_{… ç ∘ %}A

⇔ çÑh. ë çÑh ∘ %V A21ë çA2321 ∘ %V A2321 − 1 + 0 çÑh. ë çÑh ∘ %VA ë çA23 ∘ %V A23 + 0çÑh. ë çÑh ∘ %V AV3ë çA ∘ %VA≥ 0

**Corollary 7.3 **

If the composition operator ou∈ " Ç, Q then ou is @ quasi 0-class 5 if and only if Ñh. ë Ñh ∘ %V A21 − (1 + 0)Ñh. ë Ñh ∘
%VA_{+ 0Ñ}

h. ë Ñh ∘ %V AV3 ≥ 0. a.e.

**Proof **

By putting ç = 1 in Theorem 7.2, we get the result.
**Theorem 7.4 **

Let the composite multiplication operator BΩ,u Ñ ∈ " Ç, Q . Then BΩ,u∗ is @ quasi 0-class 5 if and only if
çA2321ç ∘ %A2321Ñh∘ %A2321. ë çÑh ∘ %A21 − (1 + 0)çA23 ç ∘ %A23 Ñh∘ %A23 . ë çÑh ∘ %A + 0 çAç ∘ %A Ñh∘
%A _{. ë Ñ}

h ∘ %AV3≥ 0. a.e.

**Corollary 7.5 **

If the composition operator ou ∈ " Ç, Q then ou∗ is @ quasi 0-class 5 if and only if Ñh∘ %A2321. ë Ñh ∘ %A21− (1 + 0)Ñh∘
%A23_{. ë Ñ}

h ∘ %A+ 0Ñh∘ %A. ë Ñh ∘ %AV3≥ 0. a.e.

**Theorem 7.6 **

Let the composite multiplication operator B_{Ω,u} Ñ ∈ " Ç, _{Q} _{. Then B}

Ω,u is @ quasi 0-class 5∗ if and only if çÑh. ë çÑh ∘

%V A21_{ë ç}

A2321 ∘ %V A2321 − (1 + 0)çÑh. ë çÑh ∘ %V(AV3)ë çA2, ∘ %VA+ 0çÑh. ë çÑh ∘ %V AV3ë çA ∘ %VA≥ 0. a.e.

**Corollary 7.7 **

If the composition operator o_{u}∈ " Ç, _{Q} _{ then o}

u is @ quasi 0-class 5∗ if and only if Ñh. ë Ñh ∘ %V A21 − (1 + 0)Ñhë Ñh ∘
%V(AV3)_{. ë Ñ}

h ∘ %VA+ 0Ñh. ë Ñh ∘ %V AV3 ≥ 0. a.e.

**Theorem 7.8 **

Let the composite multiplication operator BΩ,u Ñ ∈ " Ç, Q . Then BΩ,u∗ is @ quasi 0-class 5∗ if and only if çA2321çÑh∘
%A2321_{. ë çÑ}

h ∘ %V A21 − (1 + 0)çAçÑh. ë ç?Ñh, ∘ % + 0 çAçÑh∘ %A . ë çÑh ∘ %V(AV3)≥ 0. a.e.

**Corollary 7.9 **

If the composition operator ou ∈ " Ç, Q then ou∗ is @ quasi 0-class 5∗ if and only if Ñh∘ %A2321. ë Ñh ∘ %V(A21)− (1 + 0)Ñh. ë Ñh, ∘ % + 0Ñh∘ %A. ë Ñh ∘ %V(AV3)≥ 0. a.e.

**VIII.ALUTHGE TRANSFORMATION OF F QUASI E-CLASS G OF F QUASI E-CLASS G**∗_{ OPERATOR }

Let % = ' % be the polar decomposition of %. Then the Aluthge transformation % = % ST' %ST was introduced by

Aluthge[1]. An operator % is called ù hyponormal if % ≥ % ≥ %∗_{ and he defined % = %} ST_{' %}ST_{ where % = ' %}_{. Also the }

adjoint of aluthge transformation is defined %∗_{= %} S_{T}_{'}∗ _{%}S_{T}_{, ∗-Aluthge transformation is %}∗_{= %}∗ S_{T}_{' %}∗S_{T}_{, and adjoint of }
∗-Aluthge transformation is given by %∗∗_{= %}∗S_{T}_{'}∗_{%}∗ S_{T}_{. }

**Theorem 8.1 **

An operator % is @ quasi n class 5 if and only if 1 + 0 %∗A _{%},_{%}A_{≤ %}∗A _{%} 321 ,_{%}A_{ + 0%}∗A_{%}A_{ for all + ∈ ! and for every }
positive integer 0.

**Proof **

Since % is @ quasi n class 5 operator, then %∗A _{%}∗ 321_{%} 321 _{− 1 + 0 %}∗_{% + 09 %}A_{≥ 0 for every positive integer 0. By }
simple calculation we get the result.

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **661 **
If % = ' %is the polar decomposition of @ quasi n class 5 operator %, then % is @ quasi n class 5 operator.

**Theorem 8.3 **

If % is @ quasi n class 5 operator % and X is unitary such that %X = X% then Y = %X is also @ quasi n class 5 operator.
**Theorem 8.4 **

Let % = ' % be the polar decomposition of @ quasi n class 5 operator %, where ' is unitary if and only if % is @ quasi n class 5 operator.

**Proof **

Suppose we assume that % is @ quasi n class 5 operator and % = ' % is the polar decomposition of %, then we have that
%∗A _{%}∗ 321_{%} 321 _{− 1 + 0 %}∗_{% + 09 %}A_{≥ 0 for every positive integer 0. }

⇔ ' % ∗A _{' %} ∗ 321 _{' %} 321 _{− 1 + 0 ' %} ∗ _{' % + 09 ' %} A_{≥ 0 }

⇔ %A 3_{,}_{'}∗ _{%}A 3_{,}_{( %}321 3,_{'}∗ 321 _{%}∗ 321 _{'}321 _{%} 321 3,_{− 1 + 0}
%3,_{'}∗_{%}∗ _{' %}3,_{+ 09) %}A 3,_{'}A _{%}A 3,_{≥ 0 }
⇔ %∗A _{%}∗ 321_{%}321 _{− 1 + 0 %}∗_{% + 09 %}A _{≥ 0 }
for every positive integer 0. Hence % is @ quasi n class 5 operator.

**Theorem 8.5 **

Let % = ' % be the polar decomposition of @ quasi n class 5 operator %and ' is unitary, then % is @ quasi n class 5 if and only
if %∗_{ is @ quasi n class 5 operator. }

**Proof **

Suppose we assume that % is @ quasi n class 5 operator and % = ' % is the polar decomposition of %, then we have that
%∗A _{%}∗ 321_{%} 321 _{− 1 + 0 %}∗_{% + 09 %}A_{≥ 0 for every positive integer 0. }

⇔ ' % ∗A _{' %} ∗ 321 _{' %} 321 _{− 1 + 0 ' %} ∗ _{' % + 09 ' %} A_{≥ 0 }

⇔ %A 3_{,}_{'}∗ _{%}A 3_{,}_{( %}321 3,_{'}∗ 321 _{%}∗ 321 _{'}321 _{%} 321 3,_{− 1 + 0}
%3,_{'}∗_{%}∗ _{' %}3,_{+ 09) %}A 3,_{'}A _{%}A 3,_{≥ 0 }
⇔ %∗A _{%} 321_{%}∗ 321 _{− 1 + 0 %}_{%}∗_{+ 09 %}A_{≥ 0 }
for every positive integer 0. Hence %∗_{ is @ quasi n class 5 operator. }

**Corollary 8.6 **

If that % is @ quasi n class 5 if and only if that %∗_{ is @ quasi n class 5 operator. }
**Theorem 8.7 **

Let % = ' % be the polar decomposition of @ quasi n class 5 operator % and ' is unitary, then that % is @ quasi n class 5 if and
only if that %∗∗_{ is @ quasi n class 5 operator. }

**Theorem 8.8 **

Let % = ' % be the polar decomposition of @ quasi n class 5 operator T and U is unitary, then that %∗_{ is @ quasi n class 5 if }
and only if that %∗∗_{ is @ quasi n class 5 operator. }

**Theorem 8.9 **

An operator that % is @ quasi n class 5∗_{ if and only if } _{1 + 0 %}∗A _{%}∗ ,_{%}A_{≤ %}∗A _{%}321 ,_{%}A_{ + 0%}∗A_{%}A_{ for all + ∈ ! and for }
every positive integer 0.

**Theorem 8.10 **

If % = ' % is the polar decomposition of @ quasi n class 5∗_{ operator %, then that % is @ quasi n class 5}∗_{ operator. }
**Theorem 8.11 **

If that % is @ quasi n class 5∗_{ operator % and X is unitary such that %X = X% then Y = %X is also @ quasi n class 5}∗_{ operator. }
**Theorem 8.12 **

If that % is @ quasi n class 5∗_{ if and only if that %}∗_{ is @ quasi n class 5}∗_{ operator. }
**Theorem 8.13 **

If that %∗_{ is @ quasi n class 5}∗_{ if and only if that %}∗∗_{ is @ quasi n class 5}∗_{ operator. }
**IX.ACKNOWLEDGMENT **

**IJEDR1903114 ** International Journal of Engineering Development and Research (www.ijedr.org) **662 **
**REFERENCES **

1. A. Aluthge, On p-hyponormal operators for 0 < ï < 1, Integr. Equat. Oper. Th. (13) (1990), 307-315. 2. A. Aluthge, Some generalized theorems on p-hyponormal operators for 0 < ï < 1, Integral equations operator

theory. 24 (1996), 497-502.

3. C. Burnap, I.B. Jung and A. Lambert, Separating Partial Normality classes with Composition operators, J. Operator Theory. 53 (2005), No. 2, 381-387.

4. A. Devika and G. Suresh, Some properties of quasi class 5 operators, International Journal of Applied Mathematics and Statistical Sciences (IJAMSS). 2 (2013), No. 1, 63-68.

5. B.P. Duggal, Quasi Similar p - hyponormal operators, Integr. Equat.oper. 26 (1996), 338-345.

6. B.P. Duggal, C. S Kubrusly and N. Levan, Contractions of class 5 and invariant subspaces, Bull. Korean Mathematical Society. (42) (2005), 169-177.

7. T. Furuta, On the class of paranormal operators, Proc. Japan. Acad. (43) (1967), 594-598.

8. V.R. Hamiti, on @-quasi class 5 operators, Bulletin of Mathematical Analysis and Applications. (6) (2014), No. 3, 31-37.

9. J. K. Han, H. Y. Lee, and W. Y. Lee, Invertible completions of 2×2 upper triangular operator matrices, Proc. Amer. Math. Soc., 128 (2000), 119-123.

10. D. J. Harrington and R. Whitley, Seminormal composition operators, J. Operator Theory.(11) (1984), 125-135. 11. A Lambert, Hyponormal composition operators, Bull. London. Math. Soc. (18) (1986), 395-400.

12. S. Panayappan, Non-hyponormal weighted composition operators, Indian J. Pure Appl.Math. (27) (1996), 979-983.
13. D. Senthilkumar, P. Maheswari Naik And D. Kiruthika, Quasi class 5∗_{ composition operators, International J. of }

Math. Sci. and Engg. Appls. (IJMSEA),5 (2011), no. 4, 1-9.

14. D.Senthilkumar and S. Parvathaam, some properties of n class 5 operators, international Journal of Pure and Applied Mathematics, 117,(2017), no. 11, 53-59.

15. D.Senthilkumar and S. Parvathaam, some properties of ∗-0 class 5 operators, international Journal of Pure and Applied Mathematics, 117,(2017), no. 11, 129-135.

16. D. Senthil Kumar and T. Prasad, M class 5 composition operators, Scientia Magna. (6) (2010), no. 1, 25-30. 17. T. Veluchamy and S.Panayappan, paranormal composition operators, indian journal of pure and applied math. 24

(1993) 257-262.

18. Y. Yang and Cheoul Jun Kim, Contractions of class 5∗_{, Far East. J.Math.Sci.(FJMS), 27(2007), no. 3, 649-657. }
19. J. Yuan and Z. Gao, Weyl spectrum of class Y(0) and 0-paranormal operators, Integr. Equ. Oper. Theory., 60 (2008)

no. 2,289-298.