EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 830-840
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On M−quasi paranormal operators

Valdete Rexhëbeqaj Hamiti1, Qefsere Doko Gjonbalaj1,∗

1 Department of Mathematics, Faculty of Electrical and Computer Engineering,
University of Prishtina “Hasan Prishtina”, Prishtinë, 10000, Kosovë

Abstract. In this paper we introduce a new class of operators called M−quasi paranormal op-
erators. A bounded linear operator T in a complex Hilbert space H is said to be a M−quasi
paranormal operator if it satisfies

∥T 2x∥2 ≤ M∥T 3x∥ · ∥Tx∥,

∀x ∈ H, where M is a real positive number.
We prove basic properties, the structural and spectral properties of this class of operators.

2020 Mathematics Subject Classifications: 47B47, 47B20

Key Words and Phrases: M−quasi paranormal operator, M−quasi hyponormal operator, M−
paranormal operator

1. Introduction

Throughout this paper, let H be a complex Hilbert space with inner product ⟨·, ·⟩. Let
L(H) denote the C∗ algebra of all bounded operators on H. For T ∈ L(H), we denote by
kerT the null space, by T (H) the range of T. The null operator and the identity on H
will be denoted by O and I, respectively. If T is an operator, then T ∗ is its adjoint, and
∥T∥ = ∥T ∗∥. The closure of a set A will be denoted by A.

Recall that an operator T ∈ L(H) is said to be (see [6]):

• an isometry if ∥Tx∥ = ∥x∥, ∀x ∈ H;

• an unitary operator if T ∗T = TT ∗ = I;

• a positive operator T ≥ O, if ⟨Tx, x⟩ ≥ 0, ∀x ∈ H.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4392

Email addresses: valdete.rexhebeqaj@uni-pr.edu (V. R. Hamiti),
qefsere.gjonbalaj@uni-pr.edu (Q. D. Gjonbalaj)

https://www.ejpam.com 830 © 2022 EJPAM All rights reserved.



V. R. Hamiti, Q. D. Gjonbalaj / Eur. J. Pure Appl. Math, 15 (3) (2022), 830-840 831

By σ(T ) we write the spectrum of T, the r(T ) is the spectral radius of operator T which
is defined by r(T ) = sup{|λ| : λ ∈ σ(T )}. The σa(T ) is the approximate point spectrum of
operator T and it is proved that if λ ∈ σa(T ), then exists sequence (xn), where ∥xn∥ = 1
and ∥(T − λI)xn∥ → 0, n → +∞ (see [6]).

An operator T ∈ L(H) is said to be:

• a paranormal operator if ∥Tx∥2 ≤ ∥T 2x∥, ∀x ∈ H, ∥x∥ = 1, or equivalently if T ∗2T 2−
2kT ∗T + k2 ≥ O,∀k > 0 (see [2] [4], [5], [11]);

• a M−paranormal operators if M2T ∗2T 2 − 2kT ∗T + k2 ≥ O,∀k > 0 and for a fixed
real positive number M, or equivalently if ∥Tx∥2 ≤ M∥T 2x∥, ∀x ∈ H, ∥x∥ = 1 and
for a fixed real positive number M (see [1], [3], [8]);

• a M−quasi hyponormal if ∥T ∗Tx∥ ≤ M∥T 2x∥, ∀x ∈ H and for a fixed real positive
number M (see [9], [10]).

2. Main results

Analyzing the very good qualities of these classes of operators, M−paranormal and
M−quasi hyponormal operators, we came to the idea to introduce a new class of operators
M−quasi paranormal, which could include these classes of operators, to be their generality
and possibly to satisfy some of their properties.

Definition 1. An operator T ∈ L(H) is said to be a M−quasi paranormal operator, for
a fixed real positive number M if T satisfies

∥T 2x∥2 ≤ M∥T 3x∥ · ∥Tx∥,

∀x ∈ H.

From the definition we can prove the proper inclusions relation among the M−quasi
hyponormal, M−paranormal and M−quasi paranormal operators as follows:

Proposition 1. Let be T ∈ L(H).

1. Every M−quasi hyponormal operator is M−paranormal operator.

2. Every M−paranormal operator is M−quasi paranormal operator.

Proof. Let be T ∈ L(H) and x ∈ H, ∥x∥ = 1. We may assume that Tx ̸= 0.

1. Since T is a M−quasi hyponormal operator we have

∥T ∗Tx∥ ≤ M∥T 2x∥,

∀x ∈ H, ∥x∥ = 1 and for a fixed real positive number M.



V. R. Hamiti, Q. D. Gjonbalaj / Eur. J. Pure Appl. Math, 15 (3) (2022), 830-840 832

We know that for any bounded linear operator T on H and ∀x ∈ H, ∥x∥ = 1 it is valid:

∥Tx∥2 ≤ ∥T ∗Tx∥

Therefore
M∥T 2x∥ ≥ ∥T ∗Tx∥ ≥ ∥Tx∥2

∀x ∈ H, ∥x∥ = 1. This prove that operator T is also M−paranormal.

2. Now let’s suppose that T is a M−paranormal operator. Then we have:

M∥T 3x∥ = M
∥∥T 2 Tx

∥Tx∥
∥∥ · ∥Tx∥ ≥∥∥T Tx

∥Tx∥
∥∥2 · ∥Tx∥ =

∥T 2x∥2

∥Tx∥

Therefore,
M∥T 3x∥ · ∥Tx∥ ≥ ∥T 2x∥2.

Which prove that operator T is M−quasi paranormal operator.

From this proposition we have the inclusion:
M−quasi hyponormal ⊆ M−paranormal ⊆ M−quasi paranormal
In the following proposition we give the necessary and sufficient conditions under which

an operator T is a M−quasi paranormal operator.

Proposition 2. An operator T ∈ L(H) is M−quasi paranormal operator if and only if

M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T ≥ 0

∀k > 0.

Proof. Since T is a M−quasi paranormal operator, for a fixed real positive number
M , then

∥T 2x∥2 ≤ M∥T 3x∥ · ∥Tx∥,

∀x ∈ H. Then,

∥T 2x∥2 −M∥T 3x∥ · ∥Tx∥ ≤ 0,

i.e.,
4(∥T 2x∥2)2 − 4M2∥T 3x∥2 · ∥Tx∥2 ≤ 0.



V. R. Hamiti, Q. D. Gjonbalaj / Eur. J. Pure Appl. Math, 15 (3) (2022), 830-840 833

By elementary properties of real quadratic forms, this gives

k2∥Tx∥2 − 2k∥T 2x∥2 +M2∥T 3x∥2 ≥ 0,∀x ∈ H,∀k > 0;

k2⟨T ∗Tx|x⟩ − 2k⟨T ∗2T 2x|x⟩+M2⟨T ∗3T 3x|x⟩ ≥ 0, ∀x ∈ H,∀k > 0;

⟨(M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T )x|x⟩ ≥ 0, ∀x ∈ H,∀k > 0.

Hence,
M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T ≥ 0,∀x ∈ H,∀k > 0.

The reverse implication follows by retracing the steps back.

In following we give an example of M−quasi paranormal operator.

Example 1. Let T =

(
1 0
1 0

)
∈ L(l2 ⊕ l2). Then T is M−quasi paranormal operator

∀M ≥ 1.
By simple calculation we have that:

T ∗ =

(
1 1
0 0

)
,

T ∗2 = T ∗3 =

(
1 1
0 0

)
,

T 2 = T 3 =

(
1 0
1 0

)
,

T ∗T = T ∗2T 2 = T ∗3T 3 =

(
2 0
0 0

)
M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T

=

(
2M2 − 4k + 2k2 0

0 0

)
=

(
2[(1− k)2 +M2 − 1] 0

0 0

)
.

Therefore T is M−quasi paranormal operator ∀M ≥ 1, ∀k > 0.

The next proposition give the necessary and sufficient conditions for a weighted shift
operator T with decreasing weighted sequence (αn) to be a M−quasi paranormal operator.

Proposition 3. Let T be a weighted shift operator with decreasing weighted sequence (αn).
Then T is a M−quasi paranormal operator if and only if

|αn+1| ≤ M |αn+2|

∀n ∈ N.



V. R. Hamiti, Q. D. Gjonbalaj / Eur. J. Pure Appl. Math, 15 (3) (2022), 830-840 834

Proof. Since T is a weighted shift, its adjoint T ∗ is also a weighted shift and defined
by T (en) = |αn|en+1 we have:

T ∗(en) = |αn−1|en−1,

(T ∗T )(en) = |αn|2en,
(T ∗2T 2)(en) = |αn|2|αn+1|2en,
(T ∗3T 3)(en) = |αn|2|αn+1|2|αn+2|2en.

Now, since T is a M−quasi paranormal operator then,

M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T ≥ 0, ∀k > 0

⇔M2|αn|2|αn+1|2|αn+2|2 − 2k|αn|2|αn+1|2 + k2|αn|2 ≥ 0,∀k > 0

⇔M2|αn+1|2|αn+2|2 − 2k|αn+1|2 + k2 ≥ 0, ∀k > 0.

By elementary properties of real quadratic forms, this gives

4|αn+1|4 − 4M2|αn+1|2|αn+2|2 ≤ 0

|αn+1| ≤ M |αn+2|

Example 2. A weighted shift operator T with decreasing weighted sequence αn = 2n, n ∈
N is a M−quasi paranormal operator for every fixed real number M ≥ 1

2
(it is clear from

Proposition 3).

Proposition 4. Let T be a non singular weighted shift operator with decreasing weighted
sequence (αn). Then T−1 is a M−quasi paranormal operator if and only if

|αn−3| ≤ M |αn−2|

∀n ≥ 3.

In the next propositions we will prove some properties of M−quasi paranormal oper-
ators.

Proposition 5. Let T ∈ L(H) be a M−quasi paranormal operator.
a) If T double commutes with an isometric operator S, then TS is a M−quasi paranormal
operator.
b) If S is unitarily equivalent to operator T, then S is a M−quasi paranormal operator.
c) If A is a closed T invariant subset of H, then, the restriction T|A is a M−quasi para-
normal operator.



V. R. Hamiti, Q. D. Gjonbalaj / Eur. J. Pure Appl. Math, 15 (3) (2022), 830-840 835

Proof. Let be T ∈ L(H) a M−quasi paranormal operator.
a) Let be S an isometric operator and let be B = TS. Since operator T double commutes
with operator S we have TS = ST, S∗T = TS∗ and S∗S = I. Now,

M2B∗3B3 − 2kB∗2B2 + k2B∗B

= M2(TS)∗3(TS)3 − 2k(TS)∗2(TS)2 + k2(TS)∗(TS)

= M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T ≥ 0, ∀k > 0

so TS is a M−quasi paranormal operator.

b) Since operator S is unitarly equivalent to operator T, then there exists an unitary
operator U such that S = U∗TU. Since T is a M−quasi paranormal operator then

M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T ≥ 0,∀k > 0.

Hence,
M2S∗3S3 − 2kS∗2S2 + k2S∗S

= M2(U∗TU)∗3(U∗TU)3 − 2k(U∗TU)∗2(U∗TU)2 + k2(U∗TU)∗(U∗TU)

= U∗(M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T )U ≥ 0,∀k > 0

so S is a M−quasi paranormal operator.

c)

∥(T |A)2u∥2

= ∥T 2u∥2 ≤ M(∥T 3u∥ · ∥Tu∥)
= M(∥(T |A)3u∥ · ∥Tu∥).

This implies that T |A is a M−quasi paranormal operator.

Proposition 6. If T ∈ L(H) is a invertible M−quasi paranormal operator then T−1 is
also M−quasi paranormal operator.

Proof. Since T is a M−quasi paranormal operator, for a fixed real positive number
M , then

∥T 2x∥2 ≤ M∥T 3x∥ · ∥Tx∥,

∀x ∈ H. Then,

∥T 2x∥
∥T 3x∥

≤ M∥Tx∥
∥T 2x∥

∀x ∈ H. Now replacing x by T−4x, we have



V. R. Hamiti, Q. D. Gjonbalaj / Eur. J. Pure Appl. Math, 15 (3) (2022), 830-840 836

∥T 2T−4x∥
∥T 3T−4x∥

≤ M∥TT−4x∥
∥T 2T−4x∥

∥T−2x∥
∥T−1x∥

≤ M∥T−3x∥
∥T−2x∥

∥T−2x∥2 ≤ M∥T−3x∥ · ∥T−1x∥

∀x ∈ H. This shows that T−1 is a M−quasi paranormal operator.

Proposition 7. Let T ∈ L(H) be a M−quasi paranormal operator. If T k has dense range,
then T is a M−paranormal operator.

Proof. Since T k has dense range, T k(H) = H. Let y ∈ H. Then there exists a sequence
{xn}+∞

n=1 in H such that T k(xn) → y, n → +∞. Since T is a M−quasi paranormal
operator, then 〈

(M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T )xn, xn
〉
≥ 0, ∀k > 0;〈

(T ∗(M2T ∗2T 2 − 2kT ∗T + k2)T )xn, xn
〉
≥ 0,∀k > 0;〈

(M2T ∗2T 2 − 2kT ∗T + k2)Txn, Txn
〉
≥ 0, ∀k > 0.

By the continuity of the inner product, we have

⟨(M2T ∗2T 2 − 2kT ∗T + k2)y, y⟩ ≥ 0,∀y ∈ H, ∀k > 0.

Therefore T is a M−paranormal operator.

In following we give the inclusion of approximate point spectrum of this class of oper-
ators.

Proposition 8. Let T ∈ L(H) be a regular M−quasi paranormal operator. Then the
approximate point spectrum of operator T lies in the disc

σa(T ) ⊆ {λ ∈ C :
1√

M∥T−2∥ ·
√
∥T 2∥

≤ |λ| ≤ ∥T∥}.

Proof. Let T be a regular M−quasi paranormal operator, ∀x ∈ H, ∥x∥ = 1 we have:

∥x∥2 = ∥T−2 · T 2x∥2

≤∥T−2∥2 · ∥T 2x∥2

≤∥T−2∥2 ·M · ∥T 3x∥ · ∥Tx∥



V. R. Hamiti, Q. D. Gjonbalaj / Eur. J. Pure Appl. Math, 15 (3) (2022), 830-840 837

≤M · ∥T−2∥2 · ∥T 2∥ · ∥Tx∥ · ∥Tx∥
=M · ∥T−2∥2 · ∥T 2∥ · ∥Tx∥2.

So,
1 ≤ M · ∥T−2∥2 · ∥T 2∥ · ∥Tx∥2,

where we have

∥Tx∥ ≥ 1√
M∥T−2∥ ·

√
∥T 2∥

.

Now, assume that λ ∈ σa(T ), then ∃(xn), ∥xn∥ = 1 and ∥(T − λI)xn∥ → 0, n → +∞.
From the last inequation we have:

∥Txn − λxn∥ ≥ ∥Txn∥ − |λ| · ∥xn∥ ≥ 1√
M∥T−2∥ ·

√
∥T 2∥

− |λ|.

Now, when n → +∞ we have

|λ| ≥ 1√
M∥T−2∥ ·

√
∥T 2∥

.

So, we have

σa(T ) ⊆ {λ ∈ C :
1√

M∥T−2∥ ·
√
∥T 2∥

≤ |λ| ≤ ∥T∥}.

Therefore the proof is completed.

Now we will give some results for the matrix representation of M−quasi paranormal
operators.

Proposition 9. Let T ∈ L(H⊕H) be the operator defined as

T =

(
A B
0 0

)
.

If A is a M−paranormal operator, then T is a M−quasi paranormal operator.

Proof. A simple calculation shows that:

T ∗ =

(
A∗ 0
B∗ 0

)
,

T ∗2 =

(
A∗2 0
B∗A∗ 0

)
,

T 2 =

(
A2 AB
0 0

)
,

T ∗3 =

(
A∗3 0

B∗A∗2 0

)
,



V. R. Hamiti, Q. D. Gjonbalaj / Eur. J. Pure Appl. Math, 15 (3) (2022), 830-840 838

T 3 =

(
A3 A2B
0 0

)
,

T ∗3T 3 =

(
A∗3A3 A∗3A2B

B∗A∗2A3 B∗A∗2A2B

)
.

M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T

=

(
A∗(M2A∗2A2 − 2kA∗A+ k2)A A∗(M2A∗2A2 − 2kA∗A+ k2)B
B∗(M2A∗2A2 − 2kA∗A+ k2)A B∗(M2A∗2A2 − 2kA∗A+ k2)B

)
,∀k > 0.

Let u = x⊕ y ∈ H ⊕H. Then,

⟨(M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗Tu, u⟩
= ⟨A∗(M2A∗2A2 − 2kA∗A+ k2)Ax, x⟩+ ⟨A∗(M2A∗2A2 − 2kA∗A+ k2)By, x⟩
+ ⟨B∗(M2A∗2A2 − 2kA∗A+ k2)Ax, y⟩+ ⟨B∗(M2A∗2A2 − 2kA∗A+ k2)By, y⟩
= ⟨(M2A∗2A2 − 2kA∗A+ k2)Ax,Ax⟩+ ⟨(M2A∗2A2 − 2kA∗A+ k2)By,Ax⟩
+ ⟨(M2A∗2A2 − 2kA∗A+ k2)Ax,By⟩+ ⟨(M2A∗2A2 − 2kA∗A+ k2)By,By⟩
= ⟨(M2A∗2A2 − 2kA∗A+ k2)(Ax+By), (Ax+By)⟩ ≥ 0,∀k > 0

because A is a M−paranormal operator this prove that T is a M−quasi paranormal
operator.

Proposition 10. Let T be a M−quasi paranormal operator, the range of T not to be
dense, and

T =

(
A B
O C

)
on H = T (H)⊕ kerT ∗.

Then, A is a M−paranormal operator on T (H), C = O and σ(T ) = σ(A) ∪ {0}.

Proof.
Suppose that T is a M−quasi paranormal operator. Since that T does not have dense

range, we can represent T as the upper triangular matrix:

T =

(
A B
0 C

)
on H = T (H)⊕ kerT ∗.

Since T is a M−quasi paranormal operator, we have

M2T ∗3T 3 − 2kT ∗2T 2 + k2T ∗T ≥ 0,∀k > 0

⇒ T ∗(M2T ∗2T 2 − 2kT ∗T + k2)T ≥ 0,∀k > 0.

Therefore, after some calculation similar as in Proposition 9 we get:

⟨(M2T ∗2T 2 − 2kT ∗T + k2)x, x⟩ = ⟨(M2A∗2A2 − 2kA∗A+ k2)y, y⟩ ≥ 0,

∀y ∈ T (H),∀k > 0.



REFERENCES 839

Hence
M2A∗2A2 − 2kA∗A+ k2 ≥ 0, ∀k > 0.

This shows that A is a M−paranormal operator, on T (H).
Let P be the orthogonal projection of H onto T (H). For any

x =

(
x1
x2

)
∈ H = T (H)⊕ kerT ∗.

Then
⟨Cx2, x2⟩ = ⟨T (I − P )x, (I − P )x⟩ = ⟨(I − P )x, T ∗(I − P )x⟩ = 0.

Thus T ∗ = 0.
Since σ(A)∪σ(C) = σ(T )∪ϑ, where ϑ is the union of the holes in σ(T ), which happen

to be a subset of σ(A)∩σ(C) by [7, Corollary 7]. Since σ(A)∩σ(C) has no interior points,
then σ(T ) = σ(A) ∪ σ(C) = σ(A) ∪ {0} and Ck = 0.

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