International Journal of Analysis and Applications

Volume 18, Number 5 (2020), 774-783

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-774

CLASS OF (n, m)-POWER-D-HYPONORMAL OPERATORS IN HILBERT SPACE

CHERIFA CHELLALI1,∗, ABDELKADER BENALI2

1Higher School of Economics Oran, Algeria

2Faculty of The Exact Sciences and Computer, Mathematics Department, University of Hassiba Benbouali,

Chlef Algeria. B.P. 151 Hay Essalem, Chlef 02000, Algeria

∗Corresponding author: benali4848@gmail.com

Abstract. In this paper, we introduce a new classes of operators acting on a complex Hilbert space H, de-

noted by [(n,m)DH], called (n,m)-power-D-hyponormal associated with a Drazin inversible operator using

its Drazin inverse. Some proprieties of (n,m)-power-D-hyponormal, are investigated with some examples.

1. Introduction

Let H be a complex Hilbert space. Let B(H) be the algebra of all bounded linear operators defined in

H. Let T be an operator in B(H). The operator T is called normal if it satisfies the following condition

T ∗T = TT ∗ , i.e.,T commutes with T ∗. The class of quasi-normal operators was first introduced and studied

by A. Brown in [5] in 1953. The operator T is quasi-normal if T commutes with T ∗T , i.e. T (T ∗T ) = (T ∗T )T

and it is denoted by [QN]. A.A.S. Jibril [6, 7], in 2008 introduced the class of n power normal operators as a

generalization of normal operators. The operator T is called n power normal if T n commutes with T ∗ , i.e.,

T nT ∗ = T ∗T n and is denoted by [nN]. In the year 2011, O.A. Mahmoud Sid Ahmed introduced n power

quasi normal operators [14], as a generalization of quasi normal operators. The operator T is called n power

quasi normal if T n commutes with T ∗T , i.e., T n(T ∗T ) = (T ∗T )T n and it is denoted by [nQN].

Recently in [13], the authors introduced and studied the operator [(n, m)DN] and [(n, m)DQ].In this search,

Received April 11th, 2020; accepted May 26th, 2020; published July 14th, 2020.

2010 Mathematics Subject Classification. Primary 47A05, Secondary 47A10, 47B20, 47B25.

Key words and phrases. Hilbert space; (n,m)-power-D-hyponormal operators; D-idempotent; power-D-hyponormal opera-

tors; (2,2)-power-D-normal operators; Drazin inversible operator.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

774

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-774


Int. J. Anal. Appl. 18 (5) (2020) 775

we introduce a new class of operators T namely (n, m)-power-D-hyponormal operator for a positive integer

n, m if

T ∗m(T D)n ≥ (T D)nT ∗m, m = n = 1, 2, ...

denoted by [(n, m)DH]. And we in this work, we will try to apply the same results obtained in [8] for this

new classes.

Definition 1.1. An operator T ∈ B(H) be Drazin inversible operator. We said that T is (n, m)-power-D-

hyponormal operator for a positive integer n, m if

T ∗m(T D)n ≥ (T D)nT ∗m, m = n = 1, 2, ...

We denote the set of all (n, m)-Power-D-hyponormal operators by [(n, m)DH]

Remark 1.1. Clearly n = m = 1, then (1, 1)-Power-D-hyponormal operator is precisely Power-D-

hyponormal operator.

Definition 1.2. An operator T ∈B(H)D is said to be (n, m)-power-D-hyponormal if T ∗m(T D)n−(T D)nT ∗m

is positive i.e: T ∗m(T D)n − (T D)nT ∗m ≥ 0 or equivalently

〈
(
T ∗m(T D)n − (T D)nT ∗m

)
u | u〉≥ 0 for all u ∈H.

Example 1.1. Let T =


 3 −2

0 −3


 , S =


 1 1
−1 0


 ∈B(R2). A simple computation shows that

T D =
1

9


 3 −2

0 −3


 , SD =


 0 −1

1 1


 , S∗ =


 1 −1

1 0


 , T ∗ =


 3 0
−2 −3


 .

Then T ∈ [(2, 2)DH], but T /∈ [(3, 3)DH] and S ∈ [(3, 2)DH], but S /∈ [(2, 2)DH]

Proposition 1.1. If S, T ∈B(H)D are unitarily equivalent and if T is (n, m)-Power-D-hyponormal opera-

tors then so is S

Proof. Let T be an (n, m)-Power-D-hyponormal operator and S be unitary equivalent of T. Then there exists

unitary operator U such that S = UTU∗ so Sn = UT nU∗



Int. J. Anal. Appl. 18 (5) (2020) 776

We have

S∗m(SD)n = (UT mU∗)
∗
U(T D)nU∗

= UT ∗mU∗U(T D)nU∗

= UT ∗m(T D)nU∗

≥ U(T D)nT ∗mU∗

≥ U(T D)nU∗UT ∗mU∗

= (SD)nS∗m

Hence, S∗m(SD)n − (SD)nS∗m ≥ 0 �

Proposition 1.2. Let T ∈ B(H)D be an (n, m)-Power-D-hyponormal operator. Then T ∗ is (n, m)-Power-

D-co-hyponormal operator

Proof. Since T is (n, m)-Power-D-hyponormal operator. We have

T ∗m(T D)n ≥ (T D)nT ∗m ⇒
(
T ∗m(T D)n)∗ ≥

(
(T D)nT ∗m

)∗
⇒ (T D)∗nT m ≥ T m(T D)∗n.

Hence, T ∗ is (n, m)-Power-D-co-hyponormal operator. �

Theorem 1.1. If T, T ∗ are two (n, m)-Power-D-hyponormal operator, then T is an (n, m)-Power-D-normal

operator.

Proposition 1.3. If T is (2, 2)-power-D-hyponormal operator and T DT ∗ = −T ∗T D. Tthen T is (2, 2)-

Power-D-normal operator.

Proof. Since (T D)2T ∗2 = T DT DT ∗T ∗ = −T DT ∗T DT ∗ = T DT ∗T ∗T D = −T ∗T DT ∗T D = T ∗2(T D)2

And

T ∗2(T D)2 = T ∗T ∗T DT D = −T ∗T DT ∗T D = T DT ∗T ∗T D = −T DT ∗T DT ∗ = (T D)2T ∗2

So



Int. J. Anal. Appl. 18 (5) (2020) 777

T is (2, 2)-Power-D-hyponormal, then

(T D)2T ∗2 ≤ T ∗2(T D)2 ⇒ T DT DT ∗T ∗ ≤ T ∗T ∗T DT D

⇒ −T DT ∗T DT ∗ ≤−T ∗T DT ∗T D

⇒ T DT ∗T DT ∗ ≥ T ∗T DT ∗T D

⇒ T DT ∗T ∗T D ≥ T DT ∗T ∗T D

⇒ −T ∗T DT ∗T D ≥−T DT ∗T DT ∗

⇒ T ∗T DT ∗T D ≤ T DT ∗T DT ∗

⇒ T ∗2(T D)2 ≥ (T D)2T ∗2.

Hence T ∗2(T D)2 = (T D)2T ∗2. �

Example 1.2. Let T =




1 0 −1

0 0 0

1 0 −1


 ∈ B(C3). A simple computation, shows that ; T∗ =




1 0 1

0 0 0

−1 0 −1


 ,TD =




0 0 0

0 0 0

0 0 0


 .

Then power-D-hyponormal operator, but T∗T 6= TT ∗and ‖Tu‖ 6≥ ‖T ∗u‖.

Lemma 1.1. Let Tk, Sk ∈B(H)D, k = 1, 2 such that T1 ≥ T2 ≥ 0 and S1 ≥ S2 ≥ 0, then

(
T1 ⊗S1

)
≥
(
T2 ⊗S2

)
≥ 0.

Theorem 1.2. . Let T, S ∈B(H)D, such that (SD)nS∗ ≥ 0 and (T D)nT ∗ ≥ 0, then .

T ⊗S is (n, 1)-Power-D-hyponormal if and only if T and S are (n, 1)-Power-D-hyponormal operators

Proof. Assume that T, S are (n, 1)-power-D-hyponormal operators. Then

(
(
T ⊗S

)D
)n
(
T ⊗S

)∗
=

(
T D ⊗SD

)n(
T ∗ ⊗S∗

)
= (T D)nT ∗ ⊗ (SD)nS∗

≤ T ∗(T D)n ⊗S∗(SD)n

=
(
T ⊗S

)∗
(
(
T ⊗S

)D
)n.

Which implies that T ⊗S is (n, 1)-power-D-hyponormal operator.



Int. J. Anal. Appl. 18 (5) (2020) 778

Conversely, assume that T ⊗S is (n, 1)-power-D-hyponormal operator.We aim to show that T, S are (n, 1)-

power-D-hyponormal. Since T ⊗S is a (n, 1)-power-D-hyponormal operator, we obtain

(T ⊗S) is (n, 1)-power-D-hyponormal ⇐⇒ ((
(
T ⊗S

)D
)n
(
T ⊗S

)∗ ≤ (T ⊗S)∗((T ⊗S)D)n
⇐⇒ (T D)nT ∗ ⊗ (SD)nS∗ ≤ T ∗(T D)n ⊗S∗(SD)n.

Then, there exists d > 0 such that




d (T D)nT ∗ ≤ T ∗(T D)n.

and

d−1(SD)nS∗ ≤ S∗(SD)n

a simple computation shows that d = 1 and hence

(T D)nT ∗ ≤ T ∗(T D)n and (SD)nS∗ ≤ S∗(SD)n.

Therefore, T, S are (n, 1)-power-D-hyponormal.

�

Proposition 1.4. If T, S ∈ B(H)D are (n, 1)-D-power-hyponormal operators commuting, such that such

that S∗(SD)nT ∗(T D)n ≥ (SD)nS∗(T D)nT ∗ ≥ 0 and (T D)nT ∗ ≥ 0, then TS ⊗ T, TS ⊗ S, ST ⊗ T and

ST ⊗S ∈B(H⊗H)D are (n, 1)-power-D-power-D-hyponormal if the following assertions hold:

(1) S∗(T D)n = (T D)nS∗.

(2) T ∗(SD)n = (SD)nT ∗.

Proof. Assume that the conditions (1) and (2) are hold. Since T and S are (n, 1)-power-D-hyponormal, we

have

(
(
TS ⊗T

)D
)n
(
TS ⊗T

)∗
=

(
(TS)D ⊗T D

)n(
(TS)∗ ⊗T ∗

)
=

(
((TS)D)n(TS)∗ ⊗ (T D)nT ∗

)
=

(
((SD)n(T D)n)S∗T ∗ ⊗ (T D)nT ∗

)
=

(
(SD)nS∗(T D)nT ∗ ⊗ (T D)nT ∗

)



Int. J. Anal. Appl. 18 (5) (2020) 779

≤
(
S∗(SD)nT ∗(T D)n ⊗T ∗(T D)n

)
=

(
S∗T ∗(SD)n(T D)n ⊗T ∗(T D)n

)
=

(
(TS)∗((TS)D)n ⊗T ∗(T D)n

)
=

(
(TS)∗ ⊗T ∗

)(
((TS)D)n ⊗ (T D)n

)
=

(
TS ⊗T

)∗
(
(
TS ⊗T

)D
)n

Then TS ⊗S is (n, 1)-power-D-hyponormal operator.

In the same way, we may deduce the (n, 1)-power-D-hyponormal operator of TS⊗S, ST ⊗T and ST ⊗S. �

Theorem 1.3. If T, S ∈B(H)D two operators commuting. Then :(
I ⊗S

)
,
(
T ⊗ I

)
are (n, 1)-power-D-hyponormal then T � S is (n, 1)-power-D-hyponormal.

Proof. Firstly, observe that if
(
I ⊗ S

)
,
(
T ⊗ I

)
are (n, 1)-power-D-hyponormal, then we have following

inequalities (
(T ⊗ I

)D
)n
(
T ⊗ I

)∗ ≤ (T ⊗ I)∗((T ⊗ I)D)n
and (

(S ⊗ I
)D

)n
(
S ⊗ I

)∗ ≤ (S ⊗ I)∗((S ⊗ I)D)n.
Then

((T � S)D)n(T � S)∗

=
(
(T ⊗ I + I ⊗S)D

)n(
T ⊗ I + I ⊗S

)∗
=

(
(T ⊗ I)D + (I ⊗S)D

)n(
(T ⊗ I)∗ + (I ⊗S

)∗
≤ ((T ⊗ I)D)n

(
T ⊗ I)∗ + ((T ⊗ I)D)n(I ⊗S

)∗
+ ((I ⊗S)D)n(T ⊗ I)∗ + ((I ⊗S)D)n(I ⊗S

)∗
≤

(
T ⊗ I)∗((T ⊗ I)D)n + (I ⊗S

)∗
((T ⊗ I)D)n

+ (T ⊗ I)∗((I ⊗S)D)n + (I ⊗S
)∗

((I ⊗S)D)n

= (T � S)∗((T � S)D)n.

Then T � S is (n, 1)-power-D-hyponormal.

�

Theorem 1.4. Let T1, T2, ...., Tm are (n, 1)-power-D-hyponormal operator in B(H)D, such that

(T Dk )
nT ∗k ≥ 0, ∀k ∈ {1, 2...m} . Then (T1 ⊕T2 ⊕ ....⊕Tm) is (n, 1)-power-D-hyponormal operators and

(T1 ⊗T2 ⊗ ....⊗Tm) is (n, 1)-power-D-hyponormal operators.



Int. J. Anal. Appl. 18 (5) (2020) 780

Proof. Since

(
(T1 ⊕T2 ⊕ ...⊕Tm)D

)n
(T1 ⊕T2 ⊕ ...⊕Tm)

∗
=

(
(T D1 )

n ⊕ (T D2 )
n ⊕ ...⊕ (T Dm )

n
)

(T ∗1 ⊕T
∗
2 ⊕ ...⊕T

∗
m)

= ((T D1 )
nT ∗1 ⊕ (T

D
2 )

nT ∗2 ⊕ ...⊕ (T
D
m )

nT ∗m)

≤ (T ∗1 (T
D
1 )

n ⊕T ∗(T D2 )
n
2 ⊕ ...⊕T

∗
m(T

D
m )

n)

= (T ∗1 ⊕T
∗
2 ⊕ ...⊕T

∗
m)
(
(T D1 )

n ⊕ (T D2 )
n ⊕ ...⊕ (T Dm )

n
)

= (T1 ⊕T2 ⊕ ...⊕Tm)
∗ (

(T1 ⊕T2 ⊕ ...⊕Tm)D
)n

.

Then (T1 ⊕T2 ⊕ ....⊕Tm) is (n, 1) -power-D-hyponormal operators.

Now,

(
(T1 ⊗T2 ⊗ ...⊗Tm)D

)n
(T1 ⊗T2 ⊗ ...⊗Tm)

∗
=

(
(T D1 )

n ⊗ (T D2 )
n ⊗ ...⊗ (T Dm )

n
)

(T ∗1 ⊗T
∗
2 ⊗ ...⊗T

∗
m)

= ((T D1 )
nT ∗1 ⊗ (T

D
2 )

nT ∗2 ⊗ ...⊗ (T
D
m )

nT ∗m)

≤ (T ∗1 (T
D
1 )

n ⊗T ∗(T D2 )
n
2 ⊗ ...⊗T

∗
m(T

D
m )

n)

= (T ∗1 ⊗T
∗
2 ⊗ ...⊗T

∗
m)
(
(T D1 )

n ⊗ (T D2 )
n ⊗ ...⊗ (T Dm )

n
)

= (T1 ⊗T2 ⊗ ...⊗Tm)
∗ (

(T1 ⊗T2 ⊗ ...⊗Tm)D
)n

.

Then (T1 ⊗T2 ⊗ ....⊗Tm) is (n, 1) -power-D-hyponormal operators. �

Proposition 1.5. If T is (2, 1)-power-D-hyponormal and T is D-idempotent. Then T is power-D-

hyponormal operator

Proof. Since T is (2, 1)-power-D-hyponormal operator, then

(T D)2T ∗ ≤ T ∗(T D)2

since T is D-idempotent (T D)2 = T D, wich implies

T DT ∗ ≤ T ∗T D

Thus T is is power-D-hyponormal operator �

Proposition 1.6. If T is (3, 1)-power-D-hyponormal and T is D-idempotent. Then T is power-D-

hyponormal operator

Proof. Since T is (3, 1)-power-D-hyponormal operator, then

(T D)3T ∗ ≤ T ∗(T D)3

since T is D-idempotent (T D)2 = T D, wich implies

(T D)T ∗ ≤ T ∗T D

Then T is power-D-hyponormal operator �



Int. J. Anal. Appl. 18 (5) (2020) 781

Proposition 1.7. If T, S are (2, 1)-power-D-hyponormal operators commuting, such that T DS∗ = S∗T D

and T DS −ST D = 0, then S + T is (2, 1)-power-D-hyponormal operator.

Proof. Since T DS −ST D = 0, hence (T D)2S2 + S2(T D)2 = 0, so
(
SD + T D

)2
= (SD)2 + (T D)2.

(
(T + S)D

)2
(S + T )

∗
=

(
(SD)2 + (T D)2

)
(S∗ + T ∗)

= (SD)2S∗ + (SD)2T ∗ + (T D)2S∗ + (T D)2T ∗

= (SD)2S∗ + T ∗(SD)2 + S∗(T D)2 + (T D)2T ∗

≤ S∗(SD)2 + T ∗(SD)2 + S∗(T D)2 + T ∗(T D)2

= (S + T )
∗ (

(T + S)D
)2

Then S + T is (2, 1)-power-D-hyponormal operator.

�

Proposition 1.8. If T, S are (2, 1)-power-D-hyponormal operators commuting, such that T DS∗ = S∗T D

and T DS −ST D = 0, TS = ST = S + T then ST is (2, 1)-power-D-hyponormal operator.

Proof. Since T DS −ST D = 0, hence (T D)2S2 + S2(T D)2 = 0, so
(
SD + T D

)2
= (SD)2 + (T D)2.

Since,

(
(ST )D

)2
(ST )

∗
=

(
(T + S)D

)2
(S + T )

∗

= (SD)2S∗ + (SD)2T ∗ + (T D)2S∗ + (T D)2T ∗

= (SD)2S∗ + T ∗(SD)2 + S∗(T D)2 + (T D)2T ∗

≤ S∗(SD)2 + T ∗(SD)2 + S∗(T D)2 + T ∗(T D)2

= (S + T )
∗ (

(T + S)D
)2

= (ST )
∗ (

(TS)D
)2

Hence(
(ST )D

)2
(ST )

∗ ≥ (ST )∗
(
(ST )D

)2
.

Then ST is (2, 1)-power-D-hyponormal operator. �

Example 1.3. Let T =


 1 1

1 −1


 , S =


 −1 1

1 1


 ∈B(C2). A simple computation shows that

T ∗ =


 1 1

1 −1


 , S∗ =


 −1 1

1 1


 , T D = 1

2


 1 1

1 −1


 , SD = 1

2


 −1 1

1 1


 .



Int. J. Anal. Appl. 18 (5) (2020) 782

Then T is (2, 1)-power-D-hyponormal operator, but〈(
(T D)2T ∗ −T ∗(T D)2

) u
v


 |


 u

v


 〉 = 0.

For all (u, v) ∈ (C2)

and S is (2, 1)-power-D-hyponormal operator, but〈(
(SD)2S∗ −S∗(SD)2

) u
v


 |


 u

v


 〉 = 0.

For all (u, v) ∈ (C2)

Such that TS + ST = 0 and T DS∗ 6= S∗T D

but S + T and ST are (2, 1)-power-D-hyponormal operator

the following example shows that proposition (1.7) is not necessarily true if T DS∗ 6= S∗T D

Proposition 1.9. Let T, S ∈ B(H)D are commuting and are (n, 1)-power-D-hyponormal operators, such

that T DS∗ = S∗T D and (T + S)
∗
is commutes with

∑
1≤p≤n−1

(
n

p

)(
(T D)p(SD)n−p

)
.

Then (T + S) is an (n, 1)-power-D-hyponormal operator.

Proof. Since

(
(T + S)D

)n(
T + S

)∗
=


 ∑
0≤p≤n

(
n

p

)(
(T D)p(SD)n−p

)(T + S)∗
= (SD)nS∗ +

∑
1≤p≤n−1

(
n

p

)(
(T D)p(SD)n−p

)(
T + S

)∗
+ (T D)nS∗ + (SD)nT ∗

+ (T D)nT ∗

= (SD)nS∗ +
∑

1≤p≤n−1

(
n

p

)(
(T D)p(SD)n−p

)(
T + S

)∗
+ S∗(T D)n + T ∗(SD)n

+ (T D)nT ∗

≤ S∗(SD)n +
(
T + S

)∗ ∑
1≤p≤n−1

(
n

p

)(
(T D)p(SD)n−p

)
+ S∗(T D)n + T ∗(SD)n

+ T ∗(T D)n

≤
(
T + S

)∗  ∑
0≤p≤n

(
n

p

)(
(T D)p(SD)n−p

)
=

(
T + S

)∗(
(T + S)D

)n
.

Then (T + S) is an (n, 1)-power-D-hyponormal operator. �



Int. J. Anal. Appl. 18 (5) (2020) 783

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	References