Aluthge Transformation Of quasi $n$-class $Q$ and quasi $n$-class $Q^*$ operators


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 4, 2018, 1108-1129
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Aluthge Transformation Of quasi n-class Q and quasi
n-class Q∗ operators

D.Senthilkumar1, S. Parvatham2,∗

1 Post Graduate and Research Department of Mathematics, Govt. Arts College, Coimbatore-
641018, Tamilnadu, India
2 Post Graduate and Research Department of Mathematics, Govt. Arts College, Coimbatore-
641018, Tamilnadu, India

Abstract. In this paper, a new class of operators called quasi n-class Q and quasi n-class Q∗

operators are introduced and studied some properties. Quasi n-class Q and quasi n-class Q∗

composition and weighted composition operators on L2(λ) and H2(β) are characterized. Also we
discuss quasi n-class Q and quasi n-class Q∗ composite multiplication operator on L2 space and
Aluthge transformation of these class of operators are obtained.

2010 Mathematics Subject Classifications: Primary 47B20; Secondary 47B33, 47B38.

Key Words and Phrases: Class Q Operators, Class Q∗ Operators, Composition Operators,
Weighted Composition Operators, Aluthge Transformation

1. Introduction

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

Recall that an operator T is said to be paranormal if ‖Tx‖2 ≤ ‖T 2x‖‖x‖ for every
x ∈ H [7]. An operator T is said to be n-paranormal if ‖Tx‖n+1 ≤ ‖Tn+1x‖‖x‖n for
every x ∈ H [16] and normaloid if r(T ) = ‖T‖, where r(T ) denotes the spectral radius
of T. An operator T is of class Q [3], if T ∗2T 2 − 2T ∗T + I ≥ 0. Equivalently T ∈ Q if

‖Tx‖2 ≤ 1

2
(‖T 2x‖2+‖x‖2) for every x ∈ H. Class Q operators are introduced and studied

by B. P. Duggal et al and it is well known that every class Q operator is not necessarily

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3329

Email addresses: senthilsenkumhari@gmail.com (D.Senthilkumar),
parvathasathish@gmail.com (S. Parvatham)

http://www.ejpam.com 1108 c© 2018 EJPAM All rights reserved.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1109

normaloid and every paranormal operator is a normaloid of class Q. ie P ⊆ Q∩N , where
P and N denotes the class of paranormal and normaloid operators respectively. Also he
showed that the restiction of T to an invariant subspace is again a class Q operator.

Devika, Suresh [5], introduced a new class of operators which we call the quasi class
Q operators and it is defined as, for T ∈ B(H)

‖T 2x‖2 ≤ 1

2
(‖T 3x‖2 + ‖Tx‖2) for every x ∈ H

In [8], A k-quasi class Q operator is defined as follows,
An operator T is of k-quasi class Q if

‖T k+1x‖2 ≤ 1

2
(‖T k+2x‖2 + ‖T kx‖2) for every x ∈ H

and k is a natural number. D. Senthil Kumar, Prasad. T in [11], has defined the new
class of operators which we call M -class Q operators. An operator T is of M class Q
if for a fixed real number M ≥ 1, T satisfies M2T ∗2T 2 − 2T ∗T + I ≥ 0 or equivalently

‖Tx‖2 ≤ 1

2
(M2‖T 2x‖2 + ‖x‖2) for every x ∈ H and a fixed real number M ≥ 1.

In [15], Youngoh Yang and Cheoul Jun Kim introduced a class Q∗ operators. If

T ∗2T 2 − 2TT ∗ + I ≥ 0,

then T is called class Q∗ operators. He also proved that if T is class Q∗ if and only if
‖T ∗x‖2 ≤ 1

2(‖T 2x‖2 + ‖x‖2) for every x ∈ H. In [4], D. Senthil Kumar et. al. introduced
quasi class Q∗ operators. If

T ∗3T 3 − 2(T ∗T )2 + T ∗T ≥ 0,

then T is called quasi class Q∗ operators. He also proved that if T is quasi class Q∗ if and
only if ‖T ∗Tx‖2 ≤ 1

2(‖T 3x‖2 + ‖Tx‖2) for every x ∈ H
In this paper, we study some properties of quasi n-class Q and quasi n-class Q∗ op-

erators and we derive conditions for composition and weighted composition operators to
be quasi n-class Q and quasi n-class Q∗. Aluthge transformation of quasi n-class Q and
quasi n-class Q∗ operators are derived. Conditions for Composite multiplication opera-
tors to be quasi n-class Q and quasi n-class Q∗ are also obtained. A characterization of
quasi n-class Q and quasi n-class Q∗ composition and weighted composition operators on
weighted Hardy space are obtained.

2. Quasi n class Q Operators

In this section, we define new class of operators called quasi n-class Q, which is a super
class of n-class Q operators and studied some properties of this class of operators.

Definition 1. An operator T ∈ B(H) is said to be quasi n-class Q if for every positive
integer n and for every x ∈ H



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1110

‖T 2x‖2 ≤ 1

1 + n
(‖T 2+nx‖2 + n‖Tx‖2)

when n = 1 it is of quasi class Q operators.

Theorem 1. An operator T is of quasi n-class Q if and only if T ∗2+nT 2+n−(1+n)T ∗2T 2+
nT ∗T ≥ 0 for every positive integer n.

Proof. Since T is quasi n class Q operator, we have

‖T 2x‖2 ≤ 1

1 + n
(‖T 2+nx‖2 + n‖Tx‖2)

⇔ ‖T 2+nx‖2 − (1 + n)‖T 2x‖2 + n‖Tx‖2) ≥ 0
⇔ 〈T 2+nx, T 2+nx〉 − (1 + n)〈T 2x, T 2x〉+ n〈Tx, Tx〉 ≥ 0
⇔ T ∗2+nT 2+n − (1 + n)T ∗2T 2 + nT ∗T ≥ 0

For example: let x = (x1, x2, ...) ∈ l2, Define T : l2 → l2 by T (x) = (0, x1, x2, ...),
T ∗(x) = (x2, x3, ...). Then T ∗2+nT 2+n − (1 + n)T ∗2T 2 + nT ∗T ≥ 0. ie T is quasi n-class
Q operators.

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

class Q ⊂ n class Q ⊂ quasi n class Q.

Theorem 2. Every quasi class Q operator is quasi n class Q operator.

Proof. By using induction principle and simple calculation we get the result.

Corollary 1. If T ∈ B(H) is of quasi n-class Q then T is of quasi n+ 1-class Q operator

Corollary 2. If T ∈ B(H) is of quasi n-class Q then αT is of quasi n-class Q operator
for any complex number α.

Theorem 3. Let T ∈ B(H). If λ
−1
2 T is an operator of quasi n class Q, then T is quasi

n paranormal operator for all λ > 0.

Proof. Since λ
−1
2 T is an operator of quasi n-class Q, then

(λ
−1
2 T )∗(2+n)(λ

−1
2 T )2+n − (1 + n)(λ

−1
2 T )∗2(λ

−1
2 T )2 + n((λ

−1
2 T )∗(λ

−1
2 T )) ≥ 0.

Hence |λ
−1
2 |2(2+n)T ∗2+nT 2+n − (1 + n)|λ

−1
2 |4T ∗2T 2 + n|λ

−1
2 |2T ∗T ≥ 0. By multiplying

|λ|2+n and let |λ| = µ, then

T ∗2+nT 2+n − (1 + n)µnT ∗2T 2 + nµ1+nT ∗T ≥ 0.

Hence T is quasi n-paranormal operator for all λ > 0.

Theorem 4. If quasi n-class Q operator T doubly commutes with an isometric operator
S, then TS is an operator of quasi n-class Q.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1111

Proof. Since T is quasi n-classQ operator, then T ∗(T ∗1+nT 1+n−(1+n)T ∗T+nI)T ≥ 0.
Suppose T doubly commutes with an isometric operator S, then TS = ST, S∗T = TS∗

and S∗S = I. Now let A = TS. So we get A∗(A∗(1+n)A(1+n) − (1 + n)A∗A + nI)A ≥ 0.
Therefore TS is a quasi n-class Q operator.

Theorem 5. If a quasi n-class Q operator T ∈ B(H) is unitarily equivalent to operator
S, then S is an operator of quasi n-class Q.

Proof. Assume T is unitarily equivalent to operator S. Then there exists an unitary
operator U such that S = U∗TU and T is quasi n-class Q operator, then S∗(S∗1+nS1+n−
(1 + n)S∗S + nI)S = (U∗TU)∗((U∗TU)∗1+n(U∗TU)1+n − (1 + n)(U∗TU)∗(U∗TU) +
nI)(U∗TU) ≥ 0. Therefore S is quasi n-class Q operator.

Theorem 6. Let T ∈ B(H) be an invertible operator and N be an operator such that N
commutes with T ∗T . Then operator N is quasi n class Q if and only if operator TNT−1

is of quasi n class Q.

Proof. Let N be quasi n class Q operator, then N∗(N∗1+nN1+n−(1+n)N∗N+nI)N ≥
0. Since operator N commutes with operator T ∗T , we have
(TNT−1)∗((TNT−1)∗1+n(TNT−1)1+n − (1 + n)(TNT−1)∗(TNT−1) + nI)(TNT−1) =
T (N∗(N∗1+nN1+n − (1 + n)N∗N + nI)N)T−1. Since N is quasi n class Q operator,
then T (N∗(N∗1+nN1+n − (1 + n)N∗N + nI)N)T ∗ ≥ 0. Which implies (TT ∗) commutes
with T (N∗(N∗1+nN1+n − (1 + n)N∗N + nI)N)T ∗. Also (TT ∗)−1 is also commutes with
TN∗((N∗1+nN1+n − (1 + n)N∗N + nI)N)T ∗. Then T (N∗(N∗1+nN1+n − (1 + n)N∗N +
nI)N)T−1 ≥ 0. Hence TNT−1 is quasi n class Q operator. Conversely suppose that
(TNT−1) is quasi n class Q operator, then N∗(N∗1+nN1+n − (1 + n)N∗N + nI)N ≥ 0.

Corollary 3. Let S be quasi n class Q operator and A any positive operator such that
A−1 = A∗. Then T = A−1SA is quasi n class Q operator.

Theorem 7. Let T be quasi n class Q operator. Then the tensor product T ⊗ I and I⊗T
are both quasi n class Q operators.

Proof. By the definition of quasi n class Q and tensor product and by the simple
calculation we get the result.

Theorem 8. If T ∈ B(H) is a quasi n-class Q operator for a positive integer n, the
range of T does not have dense range then T has the following 2×2 matrix representation

T =

(
T1 T2
0 T3

)
on H = ran(T )⊕ kerT ∗, if and only if T1 is also quasi n-class Q operator

on ran(T ) and T3 = 0. Further more σ(T ) = σ(T1)∪{0} where σ(T ) denotes the spectrum
of T .

Proof. Let P be an orthogonal projection of H onto ran(T ). Then T1 = TP = PTP .
By Theorem 1 we have that



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1112

P (T ∗2+nT 2+n − (1 + n)T ∗2T 2 + nT ∗T )P ≥ 0

Hence

(
T ∗2+n1 T 2+n

1 − (1 + n)T ∗21 T 2
1 + nT ∗1 T1 0

0 0

)
≥ 0

This implies T ∗2+n1 T 2+n
1 − (1 + n)T ∗21 T 2

1 + nT ∗1 T1 ≥ 0
So T1 is quasi n-class Q operator on ran(T ).
Also for any x = (x1, x2) ∈ H,

〈T k3 x2, x2〉 = 〈T k(I − P )x, (I − P )x〉
= 〈(I − P )x, T ∗k(I − P )x〉
= 0

This implies T3 = 0
Since σ(T ) ∪ τ = σ(T1) ∪ σ(T3) where τ is the union of certain holes in σ(T ), which

happens to be a subset of σ(T1)∩σ(T3) [by corollary 7, [10]]. σ(T3) = 0 and σ(T1)∩σ(T3)
has no interior points we have σ(T ) = σ(T1) ∪ {0}.

Suppose that T =

(
T1 T2
0 T3

)
on H = ran(T ) ⊕ kerT ∗ where T1 is quasi n-class Q

operator on ran(T ) and T3 = 0

T 2+n =

(
T 2+n
1

∑1+n
j=0 T

j
1T2T

n+1−j
3

0 T 2+n
3

)
and T ∗2+n =

(
T ∗2+n1 0

(
∑n+1

j=0 T
j
1T2T

n+1−j
3 )∗ T ∗2+n3

)
Since T3 = 0
T ∗2+nT 2+n − (1 + n)T ∗2T 2 + nT ∗T

=

(
T ∗2+n1 T 2+n

1 − (1 + n)T ∗21 T 2
1 + nT ∗1 T1 X

X∗ Y

)
≥ 0

Where X = T ∗2+n1 T 1+n
1 T2 − (1 + n)T ∗21 T1T2 + nT ∗1 T2

Y = (T ∗2 T
∗1+n
1 T 2+n

1 )(T ∗2+n1 T 1+n
1 T2)− (1 + n)T ∗2 T

∗
1 T1T2 + nT ∗2 T2

We know that, ” If A is a matrix of the form

(
A B
B∗ C

)
≥ 0 if and only if A ≥ 0, C ≥ 0

and B = A
1
2WC

1
2 for some contraction W . Since T1 is quasi n-class Q operator, then we

have T ∗2+nT 2+n − (1 + n)T ∗2T 2 + nT ∗T ≥ 0. Hence T is quasi n-class Q operator.

Theorem 9. Let M be a closed T -invariant subspace of H. Then the restriction T |M of
a quasi n-class Q operator T to M is quasi n-class Q operator.

Proof. Let T =

(
T1 T2
0 T3

)
on H = M ⊕M⊥. Since T is quasi n-class Q operator then

by Theorem 8, we have T |M is also quasi n-class Q operator.

Theorem 10. Let T be a regular quasi n class Q operator, then the approximate point
spectrum lies in the disc



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1113

σap(T ) ⊆ {λ ∈ C : (1+n)
1
2

‖T−2‖(‖T 1+n‖2+n)
1
2
≤ |λ| ≤ ‖T‖

Proof. Suppose T is regular quasi n class Q operator, then for every unit vector x in

H, we have ‖x‖2 ≤ ‖T−2‖2‖T 2x‖2 ≤ ‖T
−2‖2

1+n (‖T 1+n‖2‖Tx‖2 + n‖Tx‖2).

Hence ‖Tx‖2 ≥ (1+n)‖x‖2
‖T−2‖2(‖T 1+n‖2+n) .

Now assume that λ ∈ σap(T ). Then there exists a sequence { xm}, ‖xm‖ = 1 such
that ‖(T − λ)xm‖ → 0 when m → ∞ we have ‖Txm − λxm‖ ≥ ‖Txm‖ − |λ|‖xm‖ ≥

(1+n)1/2

‖T−2‖(‖T 1+n‖2+n)1/2 − |λ|. Now, when m→∞, |λ| ≥ (1+n)1/2

‖T−2‖(‖T 1+n‖2+n)1/2

3. Quasi n-class Q∗ Operators

In this section we define operators of quasi n-class Q∗ and consider some basic prop-
erties and examples.

Definition 2. An operator T is said to be quasi n-class Q∗ (quasi *- n-class Q)if

‖T ∗Tx‖2 ≤ 1

1 + n
(‖T 2+nx‖2 + n‖Tx‖2)

for every x ∈ H and every positive integer n. When n = 1, it is of quasi class Q∗ (quasi
*-class Q)operator.

Theorem 11. For each positive integer n, T is of quasi n-class Q∗ operator if and only
if T ∗(T ∗1+nT 1+n − (1 + n)TT ∗ + nI)T ≥ 0.

For example: let x = (x1, x2, ...) ∈ l2, Define T : l2 → l2 by T (x) = (0, x1, x2, ...),
T ∗(x) = (x2, x3, ...). Then T ∗2+nT 2+n − (1 + n)(T ∗T )2 + nT ∗T ≥ 0. ie T is quasi n-class
Q∗.

From the definition of n-class Q∗ operator, we can easily say that every operator of
n-class Q∗ is also an operator of quasi n-class Q∗. Hence we have the following implications

class Q∗ ⊂ n-class Q∗ ⊂ quasi n-class Q∗

Also every quasi class Q∗ is quasi n-class Q∗, but the converse is not true and every quasi
n-class Q∗ is quasi n+ 1-class Q∗ operator. Again, if T ∈ B(H) is quasi n-class Q∗ then
αT is of quasi n-class Q∗ operator for any complex number α.

Theorem 12. Let T ∈ B(H). If λ
−1
2 T is an operator of quasi n-class Q∗, then T is quasi

*-n-paranormal operator for all λ > 0.

Proof. Since λ
−1
2 T is an operator of quasi n-class Q∗ then

(λ
−1
2 T )∗(2+n)(λ

−1
2 T )2+n − (1 + n)((λ

−1
2 T )∗(λ

−1
2 T ))2 + n(λ

−1
2 T )∗(λ

−1
2 T ) ≥ 0

By multiplying |λ|2+n and letting |λ| = µ, we have T is quasi *-n-paranormal operator for
all λ > 0.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1114

Theorem 13. If quasi n-class Q∗ operator T doubly commutes with an isometric operator
S, then TS is an operator of quasi n-class Q∗.

Theorem 14. If a quasi n-class Q∗ operator T ∈ B(H) is unitarily equivalent to operator
S, then S is an operator of quasi n-class Q∗.

Theorem 15. Let T ∈ B(H) be an invertible operator and N be an operator such that N
commutes with T ∗T . Then operator N is quasi n class Q∗ if and only if operator TNT−1

is quasi of n class Q∗.

Corollary 4. Let S be quasi n class Q∗ operator and A any positive operator such that
A−1 = A∗. Then T = A−1SA is quasi n class Q∗ operator.

Theorem 16. Let T be quasi n class Q∗ operator. Then the tensor product T ⊗ I and
I ⊗ T are both quasi n class Q∗ operators.

Theorem 17. If T ∈ B(H) is of quasi n class Q∗ operator for any positive integer n,

a non zero complex number λ ∈ σp(T ) and T is of the form T =

(
λ T2
0 T3

)
on H =

ker(T − λ)⊕ ran(T − λ)
∗
, then

1. T2 = 0 and
2. T3 is quasi n-class Q∗ operator.

Proof. Let T =

(
λ T2
0 T3

)
onH = ker(T−λ)⊕ran(T − λ)

∗
. Without the loss of general-

ity assume that λ = 1, then by Theorem 11, T ∗2+nT 2+n−(1+n)(T ∗T )2+nT ∗T ≥ 0. Now,

T 2+n =

(
1
∑1+n

j=0 T2T
n+1−j
3

0 T 2+n
3

)
and

T ∗2+n =

(
1 0

(
∑2+n

j=0 T2T
n+1−j
3 )∗ T ∗2+n3

)

T ∗2+nT 2+n =

(
1

∑1+n
j=0 T2T

1+n−j
3

(
∑1+n

j=0 T2T
1+n−j
3 )∗ 2(

∑1+n
j=0 T2T

1+n−j
3 )∗(

∑1+n
j=0 T2T

1+n−j
3 ) + T 2+n

3 T ∗1+n3

)
So, T ∗2+nT 2+n − (1 + n)(T ∗T )2 + nT ∗T ≥ 0 gives

(
A B
B∗ C

)
≥ 0

Where A = 1 − (1 + n)(1 + T2T
∗
2 ) + n, B =

∑1+n
j=0 T2T

1+n−j
3 − (1 + n)[T2 + T2(T

∗
2 T2 +

T ∗3 T3)] + nT2 and C = (
∑1+n

j=0 T2T
1+n−j
3 )∗

∑1+n
j=0 T2T

1+n−j
3 + T ∗2+n3 T 2+n

3 − (1 + n)T ∗2 T2 +

T ∗2 T2 + T ∗3 T3
2 + n(T ∗2 T2 + T ∗3 T3)

But,we know that, ” If A is a matrix of the form

(
A B
B∗ C

)
≥ 0 if and only if A ≥ 0,



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1115

C ≥ 0 and B = A
1
2WC

1
2 for some contraction W .

Therefore 1 + n− (1 + n)(1 + T2T
∗
2 ) + n ≥ 0, which implies that (1 + n)(−T2T ∗2 ) ≥ 0.

This gives T2 = 0, since n is a positive integer. Also T3 is quasi n-class Q∗ operator.

Corollary 5. If T ∈ B(H) is of quasi n class Q∗ operator for a positive integer n, then T

is of the form T =

(
λ 0
0 T3

)
on H = ker(T − λ)⊕ ran(T − λ)

∗
, where T3 is quasi n-class

Q∗ operator and ker(T − λ) = {0}.

Theorem 18. If T ∈ B(H) is a quasi n-class Q∗ operator for a positive integer n, T does
not have dense range and T has the following 2× 2 matrix representation

T =

(
T1 T2
0 T3

)
on H = ran(T )⊕ kerT ∗,

if and only if T ∗1+n1 T 1+n
1 − (1 + n)(T1T

∗
1 + T2T

∗
2 ) + nI ≥ 0 and T3 = 0.Further more

σ(T ) = σ(T1) ∪ {0} where σ(T ) denotes the spectrum of T .

Proof. Let T ∈ B(H) be quasi n class Q∗ operator and P be an orthogonal projection
onto ran(T ). Then T1 = TP = PTP . By Theorem 11 we have that

P (T ∗1+nT 1+n − (1 + n)(TT ∗) + nI)P ≥ 0

T ∗1+n1 T 1+n
1 − (1 + n)(T1T

∗
1 + T2T

∗
2 ) + nI ≥ 0

Also for any x = (x1, x2) ∈ H,

〈T3x2, x2〉 = 〈T (I − P )x, (I − P )x〉
= 〈(I − P )x, T ∗(I − P )x〉 = 0

This implies T3 = 0
Since σ(T )∪τ = σ(T1)∪σ(T3) where τ is the union of the holes in σ(T ), which happens

to be a subset of σ(T1) ∩ σ(T3) [by corollary 7, [10]]. σ(T3) = 0 and σ(T1) ∩ σ(T3) has no
interior points we have σ(T ) = σ(T1) ∪ {0}.

Suppose that T =

(
T1 T2
0 T3

)
on H = ran(T ) ⊕ kerT ∗, T ∗1+n1 T 1+n

1 − (1 + n)(T1T
∗
1 +

T2T
∗
2 ) + nI ≥ 0 and T3 = 0. Then we have

T ∗2+nT 2+n−(1 + n)(T ∗T )2 + nT ∗T

=

(
T ∗2+n1 0

(T ∗2 T
∗1+n
1 ) 0

)(
T 2+n
1 T 1+n

1 T2
0 0

)
− (1 + n)

(
(T ∗1 T1)

2 + T ∗1 T2T
∗
2 T1 T ∗1 T1T

∗
1 T2 + T ∗1 T2T

∗
2 T2

T ∗2 T1T
∗
1 T1 + T ∗2 T2T

∗
2 T1 T ∗2 T1T

∗
1 T2 + (T ∗2 T2)

2

)



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1116

+ n

(
T ∗1 T1 T ∗1 T2
T ∗2 T1 T ∗2 T2

)
=

(
A B
B∗ C

)
≥ 0

Where A = T ∗1 (T ∗1+n1 T 1+n
1 − (1 + n)(T1T

∗
1 + T2T

∗
2 ) + nI)T1

B = T ∗1 (T ∗1+n1 T 1+n
1 − (1 + n)(T1T

∗
1 + T2T

∗
2 ) + nI)T2

C = T ∗2 (T ∗1+n1 T 1+n
1 − (1 + n)(T1T

∗
1 + T2T

∗
2 ) + nI)T2.

Hence T is quasi n class Q∗ operator.

Theorem 19. Let M be a closed T -invariant subspace of H. Then the restriction T |M of
a quasi n class Q∗ operator T to M is quasi n class Q∗ operator.

Proof. By Theorem 18, T |M is also quasi n class Q∗ operator.

Theorem 20. Let T be a regular quasi n class Q∗ operator, then the approximate point
spectrum lies in the disc

σap(T ) ⊆ {λ ∈ C :
(1+n)( 1

2
)

‖T−1‖‖T ∗−1‖(‖T 1+n‖2+n)
1
2
≤ |λ| ≤ ‖T‖

Proof. Suppose T is regular quasi n class Q∗ operator, then for every unit vector x in
H, we have

‖Tx‖2 ≥ (1 + n)‖x‖2

‖T−1‖2‖T ∗−1‖2(‖T 1+n‖2 + n)

Now assume that λ ∈ σap(T ). Then there exists a sequence { xm}, ‖xm‖ = 1 such that
‖(T − λ)xm‖ → 0 when m→∞ we have

‖Txm − λxm‖ ≥ ‖Txm‖ − |λ|‖xm‖
≥ ‖T‖ − |λ|

≥ (1 + n)1/2

‖T ∗−1‖‖T−1‖(‖T 1+n‖2 + n)1/2
− |λ|

Now when m→∞, |λ| ≥ (1+n)1/2

‖T−1‖‖T ∗−1‖(‖T 1+n‖2+n)1/2

4. Quasi n-class Q and Quasi n-class Q∗ Composition Operators

Let L2(λ) = L2(X,Σ, λ), where (X,Σ, λ) be a sigma-finite measure space. A bounded
linear operator CT f = f ◦ T on L2(X,Σ, λ) is said to be a composition operator induced
by T , a non-singular measurable transformation from X into itself, when the measure
λT−1 is absolutely continuous with respect to the measure λ and the Radon-Nikodym



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1117

derivative
dλT−1

dλ
= f0 is essentially bounded. The Radon-Nikodym derivative of the

measure λ(T k)−1 with respect to λ is denoted by f
(k)
0 , where T k is obtained by composing

T - k times. Every essentially bounded complex-valued measurable function f0 induces the
bounded operator Mf0 on L2(λ), which is defined by Mf0f = f0f for every f ∈ L2(λ).
Further C∗TCT = Mf0 , C∗2T C

2
T = Mf0

(2) and C∗1+nT C1+n2T = Mf0
(1+n) .

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

Lemma 1. Let P denote the projection of L2 on R(C)

(i) C∗TCT f = f0f and CTC
∗
T f = (f0 ◦T )Pf for all f ∈ L2, where P is the projection of

L2 onto R(C).

(ii) R(C) = {f ∈ L2 : f is T−1Σ measurable}.

In this section quasi n-class Q and quasi n-class Q∗ composition operator on L2 space are
characterized as follows.

Theorem 21. Let CT ∈ B(L2(λ)). Then CT is of quasi n-class Q if and only if f
(2+n)
0 −

(1 + n)f
(2)
0 + nf0 ≥ 0 a.e.

Proof. Let CT ∈ B(L2(λ)) is of quasi n-class Q if and only if C∗2+nT C2+n
T − (1 +

n)C∗2T C
2
T + nC∗TCT ≥ 0. By Theorem 1

Thus 〈(C∗2+nT C2+n
T − (1 + n)C∗2T C

2
T + nC∗TCT )χE , χE〉 ≥ 0 for every characteristic

function χE of E in Σ such that λ(E) <∞. Since C∗TCT = Mf0 and C∗2+nT C2+n
T = M

f
(2+n)
0

,

then 〈(M
f
(2+n)
0

−(1+n)M
f
(2)
0

+nMf0)χE , χE〉 ≥ 0. Hence
∫
E(f

(2+n)
0 −(1+n)f

(2)
0 +nf0)dλ ≥

0 for every E in Σ. Hence CT is of quasi n class Q if and only if f
(2+n)
0 −(1+n)f

(2)
0 +nf0 ≥ 0

a.e.

Example 1. Let X = N , the set of all natural numbers and λ be the counting measure
on it. Define T : N → N by T (1) = 1, T (4p+ q − 2) = p+ 1 for q = 0, 1, 2, 3 and p ∈ N .

We have f0(p) = f
(2)
0 (p) = ... = f

(n)
0 (p) = 1 for p = 1. f0(p) = 4, f

(2)
0 (p) = 16, ... =

f
(2+n)
0 (p) = 42+n for p ∈ N −{1}. Since f

(2+n)
0 (p)− (1 + n)f

(2)
0 (p) + nf0(p) ≥ 0 for every

p, Hence CT is of quasi n class Q operator.

Theorem 22. [14] If CT ∈ B(L2(λ)) has dense range then f0 = g0 ◦ T a.e.

Corollary 6. If CT is quasi n-class Q with dense range on L2(λ) then (g0 ◦ T )(2+n) −
(1 + n)(g0 ◦ T )(2) + n(g0 ◦ T ) ≥ 0 a.e.

Proof. By Theorem 21 and Theorem 22, we obtain the result.

Theorem 23. Let CT ∈ B(L2(λ)). Then C∗T is of quasi n-class Q operator if and only if

(f
(2+n)
0 ◦ T 2+n)P2+n − (1 + n)(f

(2)
0 ◦ T (2))P2 + n(f0 ◦ T )P1 ≥ 0 a.e, where P1, P2, ..., P2+n

are the projections of L2 onto R(C), R(C2), ..., R(C2+n) respectively.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1118

Proof. Suppose CT ∈ B(L2(λ)) and C∗T is of quasi n-class Q if and only if C2+n
T C∗2+nT −

(1 + n)C2
TC
∗2
T + nCTC

∗
T ≥ 0. By Theorem 1. then 〈(C2+n

T C∗2+nT − (1 + n)C2
TC
∗2
T +

nCTC
∗
T )f, f〉 ≥ 0 for every f ∈ L2(λ). Since 〈CC∗f, f〉 = 〈(f0 ◦ T )P1f, f〉 By [9]. Hence

〈(f (2+n)0 ◦ T 2+n)P2+nf, f〉 − (1 + n)〈(f (2)0 ◦ T 2)P2f, f〉 + n〈(f0 ◦ T )P1f, f〉 ≥ 0 for every

f ∈ L2(λ). Hence 〈((f (2+n)0 ◦ T 2+n)P2+n − (1 + n)(f
(2)
0 ◦ T 2)P2 + n(f0 ◦ T )P1)f, f〉 ≥ 0⇔

(f
(2+n)
0 ◦ T 2+n)P2+n − (1 + n)(f

(2)
0 ◦ T 2)P2 + n(f0 ◦ T )P1 ≥ 0 a.e.

Corollary 7. Let CT ∈ B(L2(λ)) with dense range. Then C∗T is of quasi n-class Q

operator if and only if (f
(2+n)
0 ◦ T 2+n)− (1 + n)(f

(2)
0 ◦ T 2) + n(f0 ◦ T ) ≥ 0 a.e.

Theorem 24. Let CT ∈ B(L2(λ)). Then CT is of quasi n-class Q∗ if and only if f
(2+n)
0 −

(1 + n)(f0)
2P + nf0 ≥ 0 a.e.

Proof. Let CT ∈ B(L2(λ)) is of quasi n-class Q∗ if and only if C∗2+nT C2+n
T − (1 +

n)(C∗TCT )2 + nC∗TCT ≥ 0.
Thus 〈(C∗2+nT C2+n

T − (1 + n)(C∗TCT )2 + nC∗TCT )χE , χE〉 ≥ 0 for every characteristic
function χE of E in Σ such that λ(E) < ∞. Since C∗TCT = Mf0 , C∗2+nT C2+n

T = M
f
(2+n)
0

,

then 〈(M
f
(2+n)
0

− (1 + n)(Mf0)2 + nMf0)χE , χE〉 ≥ 0. Hence
∫
E(f

(2+n)
0 − (1 + n)(f0)

2 +

nf0)dλ ≥ 0 for every E in Σ. Hence CT is quasi n class Q∗ if and only if f
(2+n)
0 − (1 +

n)(f0)
2 + nf0 ≥ 0 a.e.

Example 2. Let X = N , the set of all natural numbers and λ be the counting measure
on it. Define T : N → N by T (1) = T (2) = T (3) = 1, T (4p+ q) = p+ 1 for q = 0, 1, 2, 3

and p ∈ N . Since f
(2+n)
0 − (1 + n)(f0)

2 + nf0 ≥ 0 for every p, Hence CT is of quasi n
class Q∗ operator.

Corollary 8. If CT is quasi n-class Q∗ with dense range on L2(λ) if and only if f
(2+n)
0 −

(1 + n)(f0)
2 + nf0 ≥ 0 a.e.

Theorem 25. Let CT ∈ B(L2(λ)). Then C∗T is quasi n-class Q∗ if and only if (f
(2+n)
0 ◦

T 2+n)P2+n − (1 + n)(f0 ◦ T )2P1 + n(f0 ◦ T )P1 ≥ 0 a.e. where Pi’s are the projections of
L2 onto R(Ci), respectively.

Proof. Let C∗T ∈ B(L2(λ)) is of quasi n-class Q∗ if and only if C2+n
T C∗2+nT − (1 +

n)(CTC
∗
T )2 + nCTC

∗
T ≥ 0.

Thus 〈(C2+n
T C∗2+nT − (1 + n)(CTC

∗
T )2 + nCTC

∗
T )χE , χE〉 ≥ 0 for every characteristic

function χE of E in Σ such that λ(E) < ∞. Since C∗TCT = Mf0 , C∗1+nT C1+n
T = M

f
(1+n)
0

and CTC
∗
T = (f0◦T )P , then

∫
E((f

(2+n)
0 ◦T 2+n)P2+n−(1+n)(f0◦T )2P1+n(f0◦T )P1)dλ ≥ 0

for every E in Σ. Hence CT is of quasi n class Q∗ if and only if (f
(2+n)
0 ◦ T 2+n)P2+n −

(1 + n)(f0 ◦ T )2P1 + n(f0 ◦ T )P1 ≥ 0 a.e.

Corollary 9. Let CT ∈ B(L2(λ)) with dense range. Then C∗T is quasi n-class Q∗ if and

only if (f
(2+n)
0 ◦ T 2+n)− (1 + n)(f0 ◦ T )2 + n(f0 ◦ T ) ≥ 0 a.e.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1119

5. Quasi n-class Q and quasi n-class Q∗ Weighted Composition
Operators

A weighted composition operator is a linear transformation acting on the set of complex
valued Σ measurable functions f of the form WT f = w(f ◦T ), where w is a complex valued
Σ measurable function. In the case that w = 1 a.e., we say that WT is a composition
operator. Let wk denote w(w ◦ T )(w ◦ T 2)...(w ◦ T k−1) so that W k

T f = wk(f ◦ T )k [13].
To examine the weighted composition operators efficiently, Alan Lambert [12], associ-

ated conditional expectation operator E with each transformation T as E(•|T 1Σ) = E(•).
E(f) is defined for each non-negative measurable function f ∈ Lp(1 ≤ p) and is

uniquely determined by the conditions
(i) E(f) is T−1Σ measurable and
(ii) If B is any T−1Σ measurable set for which

∫
B fdλ converges, then we have

∫
B fdλ =∫

B E(f)dλ.
As an operator on Lp, E is the projection onto the closure range of C. En the identity

on Lp if and only if T−1σ = σ. Now we are ready to derive the characterization of quasi
n-class Q and quasi n-class Q∗ weighted composition operator as follows.

Theorem 26. Let WT be a weighted composition operator on B(L2(λ)). Then

WT is of quasi n-class Q if and only if (f
(2+n)
0 E(w2

2+n) ◦ T−(2+n))− (1 + n)(f
(2)
0 E(w2

2) ◦
T−2) + nf0E(w2) ◦ T−1 ≥ 0 a.e.

Proof. Since WT ∈ B(L2(λ)) is of quasi n-class Q if and only if

W ∗2+nT W 2+n
T − (1 + n)W ∗2T W 2

T + nW ∗TWT ≥ 0. By Theorem 1.

Thus 〈(W ∗2+nT W 2+n
T − (1 + n)W ∗2T W 2

T + nW ∗TWT )χE , χE〉 ≥ 0 for every character-
istic function χE of E in Σ such that λ(E) < ∞. Since W ∗TWT = f0E(w2) ◦ T−1f ,

W k
T f = wk(f ◦ T )k,W ∗kT f = f

(k)
0 E(wkf) ◦ T−k and W ∗kT W k

T f = f
(k)
0 E(w2

k) ◦ T−kf . Hence

〈(f (2+n)0 E(w2
2+n)◦T−(2+n)−(1+n)f

(2)
0 E(w2

2)◦T−2 +nf0E(w2)◦T−1)χE , χE〉 ≥ 0. Hence∫
E(f

(2+n)
0 E(w2

2+n)◦T−(2+n)− (1 +n)f
(2)
0 E(w2)2 ◦T−2 +nf0E(w2)◦T−1)dλ ≥ 0 for every

E in Σ. Hence W is of quasi n class Q if and only if f
(2+n)
0 E(w2

2+n) ◦ T−(2+n) − (1 +

n)f
(2)
0 E(w2

2) ◦ T−2 + nf0E(w2) ◦ T−1 ≥ 0 a.e.

Corollary 10. Let WT be a weighted composition operator in B(L2(λ)) and assume that
T−1Σ = Σ. Then
WT is of quasi n-class Q if and only if f

(2+n)
0 w2

2+n ◦ T−(2+n) − (1 + n)f
(2)
0 w2

2 ◦ T−2 +
nf0(w

2) ◦ T−1 ≥ 0 a.e.

Theorem 27. Let WT be a weighted composition operator in B(L2(λ)). Then W ∗T is of
quasi n-class Q if and only if

w2+n(f
(2+n)
0 ◦ T 2+n)E(w2+n)− (1 + n)w2(f

(2)
0 ◦ T 2)E(w2) + nw(f0 ◦ T )E(w) ≥ 0 a.e.

Proof. Since W ∗T ∈ B(L2(λ)) is of quasi n-class Q if and only if



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1120

W 2+n
T W ∗2+nT − (1 + n)W 2

TW
∗2
T + nWTW

∗
T ≥ 0. By Theorem 1

Thus 〈(W 2+n
T W ∗2+nT − (1 + n)W 2

TW
∗2
T + nWTW

∗
T )χE , χE〉 ≥ 0 for every characteristic

function χE of E in Σ such that λ(E) < ∞. Since WTW
∗
T f = w(f0 ◦ T )E(wf), W k

T f =

wk(f ◦ T )k,W ∗kT f = fk0E(wkf) ◦ T−k and W k
TW

∗k
T f = wk(f

(k)
0 ◦ T (k))E(wkf). Then

〈(w2+n(f
(2+n)
0 ◦ T 2+n)E(w2+n)− (1 + n)w2(f

(2)
0 ◦ T 2)E(w2) + nw(f0 ◦ T )E(w))χE , χE〉 ≥

0. which gives
∫
E(w2+n(f

(2+n)
0 ◦ T 2+n)E(w2+n) − (1 + n)w2(f

(2)
0 ◦ T 2)E(w2) + nw(f0 ◦

T )E(w))dλ ≥ 0 for every E in Σ. Hence W ∗T is quasi n class Q if and only if (w2+n(f
(2+n)
0 ◦

T 2+n)E(w2+n)− (1 + n)w2(f
(2)
0 ◦ T 2)E(w2) + nw(f0 ◦ T )E(w)) ≥ 0 a.e.

Corollary 11. Let WT be a weighted composition operator in B(L2(λ)) and T−1(Σ) = Σ.
Then W ∗T is of n-class Q if and only if

w2
2+n(f

(2+n)
0 ◦ T 2+n)− (1 + n)w2

2(f
(2)
0 ◦ T 2) + nw2(f0 ◦ T ) ≥ 0 a.e.

Theorem 28. Let WT be a weighted composition operator on B(L2(λ)). Then

WT is quasi n-class Q∗ if and only if (f
(2+n)
0 E(w2

2+n) ◦ T−(2+n)) − (1 + n)w(f0E(w2) ◦
T−1)2 + nf0E(w2) ◦ T−1 ≥ 0 a.e.

Proof. Since WT ∈ B(L2(λ)) is quasi n-class Q∗ if and only if

W ∗2+nT W 2+n
T − (1 + n)(W ∗TWT )2 + nW ∗TWT ≥ 0 a.e.

Thus 〈(W ∗2+nT W 2+n
T − (1 + n)(W ∗TWT )2 + nW ∗TWT )χE , χE〉 ≥ 0 for every characteris-

tic function χE of E in Σ such that λ(E) < ∞. Since W ∗TWT = f0E(w2) ◦ T−1f ,

W k
T f = wk(f ◦ T )k,W ∗kT f = f

(k)
0 E(wkf) ◦ T−k and W ∗kT W k

T f = f
(k)
0 E(w2

k) ◦ T−kf . Then

〈(f (2+n)0 E(w2
2+n)◦T−(2+n)−(1+n)(f0E(w2)◦T−1)2+nf0E(w2)◦T−1)χE , χE〉 ≥ 0. Which

implies
∫
E(f

(2+n)
0 E(w2

2+n) ◦ T−(2+n) − (1 + n)(f0E(w2) ◦ T−1)2 + nf0E(w2) ◦ T−1)dλ ≥ 0

for every E in Σ. Hence W is quasi n class Q∗ if and only if (f
(2+n)
0 E(w2

2+n) ◦ T−(2+n) −
(1 + n)(f0E(w2) ◦ T−1)2 + nf0E(w2) ◦ T−1) ≥ 0 a.e.

Corollary 12. Let WT be a weighted composition operator in B(L2(λ)) and assume that
T−1Σ = Σ. Then WT is quasi n-class Q∗ if and only if

(f
(2+n)
0 (w2

2+n) ◦ T−(2+n) − (1 + n)(f0(w
2) ◦ T−1)2 + nf0(w

2) ◦ T−1) ≥ 0 a.e.

Theorem 29. Let WT be a weighted composition operator on B(L2(λ)). Then W ∗T is
quasi n-class Q∗ if and only if

w2+n(f
(2+n)
0 ◦ T 2+n)E(w2+n)− (1 + n)[w(f0 ◦ T )E(w)]2 + nw(f0 ◦ T )E(w) ≥ 0 a.e.

Corollary 13. If WT is a weighted composition operator in B(L2(λ)) and assume that
T−1Σ = Σ. Then W ∗T is quasi n-class Q∗ if and only if

w2
2+n(f

(2+n)
0 ◦ T 2+n)− (1 + n)w4(f0 ◦ T )2 + nw2f0 ◦ T ≥ 0 a.e.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1121

The Aluthge transform of T is the operator T̃ given by T̃ = |T |
1
2U |T |

1
2 was intro-

duced in [1] by Aluthge. 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 form the family of opera-
tors Tr : 0 < r ≤ 1 where Tr = |T |rU |T |1−r [2]. For a composition operator C, the polar
decomposition is given by C = U |C| where |C|f =

√
f0f and Uf = 1√

f0◦T
f ◦ T .

In [12] Lambert has given more general Aluthge transformation for composition oper-
ators as Cr = |C|rU |C|1−r and Crf = ( f0

f0◦T )
r
2 f ◦ T . That is Cr is weighted composition

operator with weight π = ( f0
f0◦T )

r
2 where 0 < r < 1. Since Cr is a weighted composition

operator it is easy to show that |Cr|f =
√
f0(E(π)2 ◦ T−1)f and |C∗r |f = vE(vf) where

v = π
√
f0◦T

(E(π
√
f0◦T )2)

1
4

. Also we have

Ckr f = πk(f ◦ T )k

C∗kr f = fk0E(πkf) ◦ T−k
C∗kr C

k
r f = fk0E(π2k) ◦ T−kf

Theorem 30. Let Cr ∈ B(L2(λ)). Then Cr is of quasi n-class Q if and only if (f
(2+n)
0 E(π22+n)◦

T−(2+n))− (1 + n)(f
(2)
0 E(π22) ◦ T−2) + n(f0E(π2) ◦ T−1) ≥ 0 a.e.

Proof. Since Cr is a weighted composition operator with weight π = ( f0
f0◦T )

r
2 , it follows

from Theorem 26 that Cr is quasi n-class Q if and only if (f
(2+n)
0 E(π22+n) ◦ T−(2+n)) −

(1 + n)(f
(2)
0 E(π22) ◦ T−2) + n(f0E(π2) ◦ T−1) ≥ 0 a.e.

Corollary 14. If T−1Σ = Σ and Cr ∈ B(L2(λ)). Then Cr is of quasi n-class Q if and

only if (f
(2+n)
0 (π22+n) ◦ T−(2+n))− (1 + n)(f

(2)
0 (π22) ◦ T−2) + n(f0(π

2) ◦ T−1) ≥ 0 a.e.

Theorem 31. Let Cr ∈ B(L2(λ)). Then C∗r is of quasi n-class Q if and only if π2+n(f
(2+n)
0 ◦

T 2+n)E(π2+n)− (1 + n)π2(f
(2)
0 ◦ T 2)E(π2) + nπ(f0 ◦ T )E(π) ≥ 0 a.e.

Proof. Since C∗r is a weighted composition operator with weight π = ( f0
f0◦T )

r
2 , it follows

from Theorem 27 that C∗r is of quasi n-class Q if and only if π2+n(f
(2+n)
0 ◦T 2+n)E(π2+n)−

(1 + n)π2(f
(2)
0 ◦ T 2)E(π2) + nπ(f0 ◦ T )E(π) ≥ 0 a.e.

Corollary 15. Let Cr ∈ B(L2(λ)) and T−1Σ = Σ. Then C∗r is of quasi n-class Q if and

only if π22+n(f
(2+n)
0 ◦ T 2+n)− (1 + n)π22(f

(2)
0 ◦ T 2) + nπ2(f0 ◦ T ) ≥ 0 a.e.

Theorem 32. Let Cr ∈ B(L2(λ)). Then Cr is of quasi n-class Q∗ if and only if

(f
(2+n)
0 E(π21+n) ◦ T−(2+n))− (1 + n)(f0E(π2) ◦ T−1)2 + n(f0E(π2) ◦ T−1) ≥ 0 a.e.

Proof. Since Cr is a weighted composition operator with weight π = ( f0
f0◦T )

r
2 , it follows

from Theorem 46 that Cr is of quasi n-class Q∗ if and only if (f
(2+n)
0 E(π21+n) ◦T−(2+n))−

(1 + n)(f0E(π2) ◦ T−1)2 + n(f0E(π2) ◦ T−1) ≥ 0 a.e.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1122

Corollary 16. If T−1Σ = Σ and Cr ∈ B(L2(λ)). Then Cr is of quasi n-class Q∗ if and

only if (f
(2+n)
0 (π21+n) ◦ T−(2+n))− (1 + n)(f0(π

2) ◦ T−1)2 + n(f0(π
2) ◦ T−1) ≥ 0 a.e.

Theorem 33. Let Cr ∈ B(L2(λ)). Then C∗r is of quasi n-class Q∗ if and only if

π2+n(f
(2+n)
0 ◦ T 2+n)E(π2+n)− (1 + n)(π(f0 ◦ T )E(π))2 + nπ(f0 ◦ T )E(π) ≥ 0 a.e.

Corollary 17. If T−1Σ = Σ and C∗r ∈ B(L2(λ)) is quasi n-class Q∗ if and only if

π22+n(f
(2+n)
0 ◦ T 2+n)− (1 + n)(π2(f0 ◦ T ))2 + nπ2(f0 ◦ T ) ≥ 0 a.e.

B. P Duggal [6] described the second Aluthge Transformation of T by T̃ = |T̂ |
1
2V |T̂ |

1
2 ,

where T̂ = V |T̂ | is the polar decomposition of T̂ . Now we consider C̃ = |Cr|
1
2V |Cr|

1
2 ,

where Cr = V |Cr| is the polar decomposition of the generalized Aluthge transformation
Cr : 0 < r < 1. We have |Cr|f =

√
Jf , where J = f0E(π2) ◦ T−1.

C̃ = |Cr|
1
2V |Cr|

1
2 =
√
J

1
2V (
√
J

1
2 f)|=

√
J

1
2π(

χsupJ√
J
J

1
4 f)◦T = J

1
4π((

χsupJ

J
1
4

)◦T )(f ◦T ).

We see then that C̃ is a weighted composition operator with weight w′ = J
1
4π((

χsupJ

J
1
4

)◦T ).

Theorem 34. If C̃ is of quasi n-class Q if and only if f
(2+n)
0 E(w

′2
2+n) ◦ T−(2+n) − (1 +

n)(f
(2)
0 E(w

′2
2 ) ◦ T−2) + n(f0E(w′2) ◦ T−1) ≥ 0 a.e.

Proof. Since C̃ is a weighted composition operator with weight w′ = J
1
4π((

χsupJ

J
1
4

)◦T ),

then by Theorem 26 we obtain the result.

Corollary 18. If T−1Σ = Σ and C̃ ∈ B(L2(λ)) is of quasi n-class Q if and only if

f
(2+n)
0 (w

′2
2+n) ◦ T−(2+n) − (1 + n)(f

(2)
0 (w

′2
2 ) ◦ T−2) + n(f0(w

′2) ◦ T−1) ≥ 0 a.e.

Theorem 35. Let C̃ ∈ B(L2(λ)). Then C̃∗ is of quasi n-class Q if and only if w′2+n(f
(2+n)
0 ◦

T 2+n)E(w′2+n)− (1 + n)w′2(f
(2)
0 ◦ T 2)E(w′2) + nw′(f0 ◦ T )E(w′) ≥ 0 a.e.

Proof. Since C̃∗ is a weighted composition operator with weight w′ = J
1
4π((

χsupJ

J
1
4

)◦T ),

then by from Theorem 27 we obtain the result.

Corollary 19. Let C̃ ∈ B(L2(λ)) and T−1Σ = Σ. Then C̃∗ is quasi n-class Q if and only

if w
′2
2+n(f

(2+n)
0 ◦ T 2+n)− (1 + n)w

′2
2 (f

(2)
0 ◦ T 2) + nw

′2(f0 ◦ T ) ≥ 0 a.e.

Theorem 36. If C̃ is quasi n-class Q∗ if and only if f
(2+n)
0 E(w

′2
2+n) ◦ T−(2+n) − (1 +

n)(f0E(w
′2) ◦ T−1)2 + n(f0E(w

′2) ◦ T−1) ≥ 0 a.e.

Corollary 20. If T−1Σ = Σ and C̃ ∈ B(L2(λ)) is of n-class Q∗ if and only if f
(2+n)
0 (w

′2
2+n)◦

T−(2+n) − (1 + n)(f0(w
′2) ◦ T−1)2 + n(f0(w

′2) ◦ T−1) ≥ 0 a.e.

Theorem 37. Let C̃ ∈ B(L2(λ)). Then C̃∗ is of quasi n-class Q∗ if and only if w′2+n(f
(2+n)
0 ◦

T 2+n)E(w′2+n)− (1 + n)(w′(f0 ◦ T )E(w′))2 + n(w′(f0 ◦ T )E(w′)) ≥ 0 a.e.

Corollary 21. Let C̃ ∈ B(L2(λ)) and T−1Σ = Σ. Then C̃∗ is of n-class Q∗ if and only

if w
′2
2+n(f

(2+n)
0 ◦ T 2+n)− (1 + n)(w

′2(f0 ◦ T ))2 + n(w
′2(f0 ◦ T )) ≥ 0 a.e.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1123

6. Quasi n-class Q and Quasi n-class Q∗ Weighted Composition
Operators on Weighted Hardy Space

The set H2(β) of formal complex power series f(z) =
∑∞

m=0 amz
m such that ‖f‖2β =∑∞

m=0|am|2β2m <∞ is a Hilbert space of functions analytic in the unit disc with the inner
product.
〈f, g〉β =

∑∞
m=0 ambmβ

2
m for an analytic map f on the open unit disc D and g(z) =∑∞

m=0 bmz
m.

Let φ : D → D be an analytic self map of the unit disc and consider the corresponding
composition operator Cφ acting on H2(β). That is Cφ(f) = f ◦ φ for f ∈ H2(β). The
operators Cφ are not necessarily defined on all of H2(β). They are everywhere defined in
some special cases on the classical Hardy Space H2 (the case when βn = 1 for all n) and on
a general space H2(β) if the function φ is analytic on some open set containing the closed
unit disc having supremum norm strictly smaller than one. The weighted composition
operator Wφ is defined as (Wφf)(z) = πf(φ(z)) and (W ∗φf)(z) = π̄f(φ(z)) for every
z ∈ D

Let w be a point on the open disc. Define kβw(z) =
∑∞

m=0
zmw−m

β2
m

. Then the function

kβw is a point evaluation for H2(β).Then kβw is in H2(β) and ‖kβw‖2 =
∑∞

m=0
|w|2m
β2
m

. Thus

‖kw‖ is an increasing function of |w|. If f(z) =
∑∞

m=0 amz
m then 〈f, kβw〉 = f(w) for all

f and kβw. Hence we can easily seen that C∗φk
β
w = kβφ(w), W

∗
φk

β
w = π̄kβφ(w) and kβ0 = 1 (the

function identically equal to 1).
Now we characterize quasi n class Q and quasi n-class Q∗ composition operators on

this space as follows.

Theorem 38. If Cφ is of quasi n-class Q operator in H2(β), then C∗2+nφ C2+n
φ − (1 +

n)C∗2φ C
2
φ + nC∗φCφ ≥ 0

Proof. For f ∈ H2(β), consider

〈(C∗2+nφ C2+n
φ −(1 + n)C∗2φ C

2
φ + nC∗φCφ)f, f〉

= 〈C∗2+nφ C2+n
φ f, f〉 − (1 + n)〈C∗2φ C2

φf, f〉+ n〈C∗φCφf, f〉

= 〈C2+n
φ f, C2+n

φ f〉 − (1 + n)〈C2
φf, C

2
φf〉+ n〈Cφf, Cφf〉

= ‖C2+n
φ f‖2 − (1 + n)‖C2

φf‖2 + n‖Cφf‖2

Let f = kβ0 then

〈(C∗2+nφ C2+n
φ − (1 + n)C∗2φ C

2
φ + nC∗φCφ)f, f〉

= ‖C2+n
φ kβ0 ‖

2 − (1 + n)‖C2
φk

β
0 ‖

2 + n‖Cφkβ0 ‖
2

= ‖kβ0 ‖
2 − (1 + n)‖kβ0 ‖

2 + n‖kβ0 ‖
2 = 0

Hence Cφ is quasi n-class Q operator.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1124

Theorem 39. If C∗φ is quasi n-class Q operator in H2(β), then C2+n
φ C∗2+nφ − (1 +

n)C2
φC
∗2
φ + nCφC

∗
φ ≥ 0

Proof. For f ∈ H2(β), consider

〈(C2+n
φ C∗2+nφ −(1 + n)C2

φC
∗2
φ + nCφC

∗
φ)f, f〉

= 〈C2+n
φ C∗2+nφ f, f〉 − (1 + n)〈C2

φC
∗2
φ f, f〉+ n〈CφC∗φf, f〉

= 〈C∗2+nφ f, C∗2+nφ f〉 − (1 + n)〈C∗2φ f, C∗2φ f〉+ n〈C∗φf, C∗φf〉

= ‖C∗2+nφ f‖2 − (1 + n)‖C∗2φ f‖2 + n‖C∗φf‖2

Let f = kβ0 and φ(0) = 0 then we have

〈(C2+n
φ C∗2+nφ −(1 + n)C2

φC
∗2
φ + nCφC

∗
φ)f, f〉

= ‖C∗2+nφ kβ0 ‖
2 − (1 + n)‖C∗2φ k

β
0 ‖

2 + n‖C∗φk
β
0 ‖

2

= ‖kβ0 ‖
2 − (1 + n)‖kβ0 ‖

2 + n‖kβ0 ‖
2 = 0

Hence C∗φ is quasi n-class Q operator.

Theorem 40. If Cφ is quasi n-class Q∗ operator in H2(β) if and only if ‖kβ0 ‖2 ≥ ‖k
β
φ(0)‖

2.

Theorem 41. If C∗φ is of quasi n-class Q∗ operator in H2(β) if and only if ‖kβ
φ2+n(0)

‖2 ≥
‖kβφ(0)‖

2.

Next we characterize the quasi n class Q and quasi n class Q∗ weighted composition
operator on weighted hardy space as follows

Theorem 42. An operator Wφ ∈ H2(β) is quasi n class Q if and only if ‖π2+n‖2 − (1 +
n)‖π2‖2 + n‖π‖2 ≥ 0.

Proof. Since Wφ is quasi n class Q operator, then for any f ∈ H2(β), we have

〈(W ∗2+nφ W 2+n
φ − (1 + n)W ∗2φ W 2

φ + nW ∗φWφ)f, f〉 ≥ 0

⇔ ‖W 2+n
φ f‖2 − (1 + n)‖W 2

φf‖2 + n‖Wφf‖2 ≥ 0

⇔ ‖W 2+n
φ kβ0 ‖

2 − (1 + n)‖W 2
φk

β
0 ‖

2 + n‖Wφk
β
0 ‖

2 ≥ 0 when f = kβ0

⇔ ‖π2+nkβ0 ‖
2 − (1 + n)‖π2kβ0 ‖

2 + n‖πkβ0 ‖
2 ≥ 0

⇔ ‖π2+n‖2‖kβ0 ‖
2 − (1 + n)‖π2‖2‖kβ0 ‖

2 + n‖π‖2‖kβ0 ‖
2 ≥ 0

⇔ ‖π2+n‖2 − (1 + n)‖π2‖2 + n‖π‖2 ≥ 0

Theorem 43. An operator W ∗φ ∈ H2(β) is quasi n class Q if and only if ‖π2+n‖2 − (1 +

n)‖π2‖2 + n‖π‖2 ≥ 0.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1125

Proof. Since W ∗φ is quasi n class Q operator, we have

〈(W 2+n
φ W ∗2+nφ − (1 + n)(WφW

∗
φ)2 + nWφW

∗
φ)f, f〉 ≥ 0 for any f ∈ H2(β)

〈(W 2+n
φ W ∗2+nφ − (1 + n)(WφW

∗
φ)2 + nWφW

∗
φ)f, f〉 ≥ 0

⇔ ‖W ∗2+nφ f‖2 − (1 + n)‖W ∗2φ f‖2 + n‖W ∗φf‖2 ≥ 0

⇔ ‖π2+nkβ0 ‖
2 − (1 + n)‖π2kβ0 ‖

2 + n‖πkβ0 ‖
2 ≥ 0 for f = kβ0 andφ(0) = 0

⇔ ‖π2+n‖2 − (1 + n)‖π2‖2 + n‖π‖2 ≥ 0

Hence the theorem.

Theorem 44. An operator Wφ is of quasi n-class Q∗ operator in H2(β) if and only if

(‖π2+n‖2 + n‖π‖2)‖kβ0 ‖2 ≥ (1 + n)|π|2‖kβφ(0)‖
2.

Theorem 45. An operator W ∗φ ∈ H2(β) is of quasi n-class Q∗ if and only if

‖π2+n‖2 ≥ (1 + n)|π|2 − n‖π‖2.

7. quasi n-class Q and quasi n-class Q∗ Composite Multiplication
operator

As composite multiplication operator to a linear transformation acting on a set of
complex value Σ measurable functions f of the form Mu,T (f) = CTMuf = u ◦ Tf ◦ T
where u is a complex valued Σ measurable function. In the case u = 1 a.e, Mu,T becomes
a composition operator denoted by CT .

Proposition 1. Let the composite multiplication operator Mu,T (f) ∈ B(L2(λ)) then for
u ≥ 0

(i) M∗u,TMu,T f = u2f0f .

(ii) Mu,TM
∗
u,T f = (u2 ◦ T )(f0 ◦ T ).E(f).

Since Mu,T (f) = CTMuf = u◦Tf◦T Mn
u,T (f) = (CTMu)n(f) = un(f◦T )2 and M∗u,T (f) =

uf0.E(f) ◦ T−1 M∗nu,T (f) = uf0.E(uf0) ◦ T−(n−1).E(f) ◦ T−n where E(uf0) ◦ T−(n−1) =

E(uf0) ◦ T−1, E(uf0) ◦ T−2, ..., E(uf0) ◦ T−(n−1)
E(uf0) ◦ Tn−1 = E(uf0) ◦ T 1, E(uf0) ◦ T 2, ..., E(uf0) ◦ Tn−1

In this section, we study quasi n-class Q and quasi n-class Q∗ composite multiplication
operator as follows.

Theorem 46. Let the composite multiplication operator Mu,T ∈ B(L2(λ)). Then Mu,T is
quasi n class Q if and only if uf0.E(uf0)◦T−(1+n).E(u2+n)◦T−(2+n)−(1+n)uf0.E(uf0)◦
T−1.E(u2) ◦ T−2 + nu2f0 ≥ 0. a.e.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1126

Proof. Suppose Mu,T is quasi n class Q operator, then
M∗2+nu,T M2+n

u,T − (1 + n)M∗2u,TM
2
u,T + nM∗u,TMu,T ≥ 0 Then for any f ∈ L2(λ), we have

〈(M∗2+nu,T M2+n
u,T − (1 + n)M∗2u,TM

2
u,T + nM∗u,TMu,T )f, f〉 ≥ 0

〈M∗2+nu,T M2+n
u,T f, f〉 − (1 + n)〈M∗2u,TM2

u,T f, f〉+ n〈M∗u,TMu,T f, f〉 ≥ 0

Since M∗ku,TM
k
u,T = uf0.E(uf0) ◦ T−(k−1).E(f) ◦ T−n

Mk
u,TM

∗k
u,T = uk.u ◦ T k.f0 ◦ T k.E(uf0) ◦ T k−1.E(f)

where uk = u ◦ T.u ◦ T 2...u ◦ T k

⇔ 〈(uf0.E(uf0) ◦ T−(1+n).E(u2+n) ◦ T−(2+n))f, f〉−
(1 + n)〈(uf0.E(uf0) ◦ T−1.E(u2) ◦ T−2)f, f〉+ n〈(u2f0)f, f〉 ≥ 0

⇔
∫
E

(uf0.E(uf0) ◦ T−(1+n).E(u2+n) ◦ T−(2+n) − (1 + n)uf0.E(uf0) ◦ T−1

.E(u2) ◦ T−2 + nu2f0)dλ ≥ 0

⇔ uf0.E(uf0) ◦ T−(1+n).E(u2+n) ◦ T−(2+n) − (1 + n)uf0.E(uf0) ◦ T−1

.E(u2) ◦ T−2 + nu2f0 ≥ 0 a.e

Corollary 22. If the composition operator CT ∈ B(L2(λ)) then CT is quasi n class Q if
and only if f0.E(f0) ◦ T−(1+n) − (1 + n)f0.E(f0) ◦ T−1 + nf0 ≥ 0. a.e.

Proof. By putting u = 1 in Theorem 46, we get the result.

Theorem 47. Let the composite multiplication operator Mu,T ∈ B(L2(λ)). Then M∗u,T is

quasi n class Q if and only if u2+n.u ◦ T 2+n.f0 ◦ T 2+n.E(uf0) ◦ T 1+n.E(f)− (1 +n)u2(u ◦
T 2)(f0 ◦ T 2).E(uf0) ◦ TE(f) + n(u2 ◦ T )(f0 ◦ T ).E(f) ≥ 0. a.e.

Proof. SupposeM∗u,T is quasi n classQ operator, thenM2+n
u,T M∗2+nu,T −(1+n)M2

u,TM
∗2
u,T+

nMu,TM
∗
u,T ≥ 0 Then for any f ∈ L2(λ), we have

〈(M2+n
u,T M∗2+nu,T − (1 + n)M2

u,TM
∗2
u,T + nMu,TM

∗
u,T )f, f〉 ≥ 0

⇔
∫
E

(u2+n.u ◦ T 2+n.f0 ◦ T 2+n.E(uf0) ◦ T 1+n.E(f)− (1 + n)

u2(u ◦ T 2)(f0 ◦ T 2).E(uf0) ◦ TE(f) + n(u2 ◦ T )(f0 ◦ T ).E(f))dλ ≥ 0

⇔ u2+n.u ◦ T 2+n.f0 ◦ T 2+n.E(uf0) ◦ T 1+n.E(f)− (1 + n)

u2(u ◦ T 2)(f0 ◦ T 2).E(uf0) ◦ TE(f) + n(u2 ◦ T )(f0 ◦ T ).E(f) ≥ 0 a.e

Corollary 23. If the composition operator CT ∈ B(L2(λ)) then C∗T is quasi n class Q if
and only if f0◦T 2+n.E(f0)◦T 1+n.E(f)−(1+n)(f0◦T 2).E(f0)◦TE(f)+n(f0◦T ).E(f) ≥ 0.
a.e.



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1127

Theorem 48. Let the composite multiplication operator Mu,T ∈ B(L2(λ)). Then Mu,T is
quasi n class Q∗ if and only if uf0.E(uf0)◦T−(1+n).E(u2+n)◦T−(2+n)− (1+n)(u4)(f20 )+
n(u2)(f0) ≥ 0. a.e.

Corollary 24. If the composition operator CT ∈ B(L2(λ)). Then CT is quasi n class Q∗

if and only if f0.E(f0) ◦ T−(1+n) − (1 + n)(f20 ) + n(f0) ≥ 0. a.e.

Theorem 49. Let the composite multiplication operator Mu,T ∈ B(L2(λ)). Then M∗u,T is

quasi n class Q∗ if and only if u2+nu ◦T 2+nf0 ◦T 2+n.E(uf0) ◦T 1+n.E(f)− (1 +n)(u2f0 ◦
TE(f))2 + n(u2f0 ◦ T )E(f) ≥ 0. a.e.

Corollary 25. If the composition operator CT ∈ B(L2(λ)). Then C∗T is quasi n class Q∗

if and only if f0 ◦ T 2+n.E(f0) ◦ T 1+n.E(f) − (1 + n)(f0 ◦ TE(f))2 + n(f0 ◦ T )E(f) ≥ 0.
a.e.

8. Aluthge transformation of quasi n-class Q and quasi n class Q∗

operator

Let T = U |T | be the polar decomposition of T. Then the Aluthge transformation

T̃ = |T |
1
2U |T |

1
2 was introduced by Aluthge[1]. An operator T is called w hyponormal if

|T̃ | ≥ |T | ≥ |T̃ ∗| and he defined ˜̃T = |T̃ |
1
2 T̃ |T̃ |

1
2 where T̃ = Ũ |T̃ |. Also the adjoint of

aluthge transformation is defined as T̃ ∗ = |T |
1
2U∗|T |

1
2 , *-Aluthge transformation is T̃ ∗ =

|T ∗|
1
2U |T ∗|

1
2 and adjoint of *-Aluthge transformation is given by T̃ ∗

∗
= |T ∗|

1
2U∗|T ∗|

1
2 .

Theorem 50. An operator T is quasi n class Q if and only if (1+n)T ∗|T |2T ≤ T ∗|T (1+n)|2T+
nT ∗T for all x ∈ H and for every positive integer n.

Proof. Since T is quasi n class Q operator, then T ∗(T ∗1+nT 1+n−(1+n)T ∗T+nI)T ≥ 0
for every positive integer n. By simple calculation we get the result.

Theorem 51. If T = U |T | is the polar decomposition of quasi n class Q operator T , then
T is quasi n class Q operator.

Theorem 52. If T is quasi n class Q operator T and S is unitary such that TS = ST
then A = TS is also quasi n class Q operator.

Theorem 53. Let T = U |T | be the polar decomposition of quasi n class Q operator T ,
where U is unitary if and only if T̃ is quasi n class Q operator.

Proof. Suppose we assume that T is quasi n class Q operator and T = U |T | is the
polar decomposition of T , then we have that
T ∗(T ∗1+nT 1+n − (1 + n)T ∗T + nI)T ≥ 0 for every positive integer n.

⇔ (U |T |)∗((U |T |)∗1+n(U |T |)1+n − (1 + n)(U |T |)∗(U |T |) + nI)(U |T |) ≥ 0.

⇔ |T |
1
2U∗|T |

1
2 (|T (1+n)|

1
2U∗(1+n)|T ∗(1+n)|U (1+n)|T (1+n)|

1
2 − (1 + n)



D. Senthilkumar, S. Parvatham / Eur. J. Pure Appl. Math, 11 (4) (2018), 1108-1129 1128

|T |
1
2U∗|T ∗|U |T |

1
2 + nI)|T |

1
2U |T |

1
2 ≥ 0.

⇔ T̃ ∗(T̃ ∗1+nT̃ 1+n − (1 + n)T̃ ∗T̃ + nI)T̃ ≥ 0

for every positive integer n. Hence T̃ is quasi n class Q operator.

Theorem 54. Let T = U |T | be the polar decomposition of quasi n class Q operator T
and U is unitary, then T is quasi n class Q if and only if T̃ ∗ is quasi n class Q operator.

Proof. Suppose we assume that T is quasi n class Q operator and T = U |T | is the
polar decomposition of T , then we have that
T ∗(T ∗1+nT 1+n − (1 + n)T ∗T + nI)T ≥ 0 for every positive integer n.

⇔ (U |T |)∗[(U |T |)∗1+n(U |T |)1+n − (1 + n)(U |T |)∗(U |T |) + nI](U |T |) ≥ 0.

⇔ |T |
1
2U∗|T |

1
2 (|T (1+n)|

1
2U (1+n)|T ∗(1+n)|U∗(1+n)|T (1+n)|

1
2 − (1 + n)

|T |
1
2U |T ∗|U∗|T |

1
2 + nI)|T |

1
2U |T |

1
2 ≥ 0.

⇔ T̃ ∗(T̃ 1+nT̃ ∗1+n − (1 + n)T̃ T̃ ∗ + nI)T̃ ≥ 0

for every positive integer n. Hence T̃ ∗ is quasi n class Q operator.

Corollary 26. If T̃ is quasi n class Q if and only if T̃ ∗ is quasi n class Q operator.

Theorem 55. Let T = U |T | be the polar decomposition of quasi n class Q operator T
and U is unitary, then T is quasi n class Q if and only if T̃ ∗

∗
is quasi n class Q operator.

Theorem 56. Let T = U |T | be the polar decomposition of quasi n class Q operator T and
U is unitary, then T̃ ∗ is quasi n class Q if and only if T̃ ∗

∗
is quasi n class Q operator.

Theorem 57. An operator T is quasi n class Q∗ if and only if (1 + n)T ∗|T ∗|2T ≤
T ∗|T (1+n)|2T + nT ∗T for all x ∈ H and for every positive integer n.

Theorem 58. If T = U |T | is the polar decomposition of quasi n class Q∗ operator T ,
then T is quasi n class Q∗ operator.

Theorem 59. If T is quasi n class Q∗ operator T and S is unitary such that TS = ST
then A = TS is also quasi n class Q∗ operator.

Theorem 60. If T̃ is quasi n class Q∗ if and only if T̃ ∗ is quasi n class Q∗ operator.

Theorem 61. If T̃ ∗ is quasi n class Q∗ if and only if T̃ ∗
∗

is quasi n class Q∗ operator.



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