EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1836-1853
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Structural and Spectral Properties of k−Quasi Class
Q(N) and k−Quasi Class Q∗(N) Operators

Shqipe Lohaj

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

Abstract. Let T be a bounded linear operator on a complex Hilbert space H. In this paper we
introduce two new classes of operators: k−quasi class Q(N) and k−quasi class Q∗(N). An operator
T ∈ L(H) is of k−quasi class Q(N) for a fixed real number N ≥ 1 and k a natural number, if T
satisfies

N∥T k+1x∥2 ≤ ∥T k+2x∥2 + ∥T kx∥2,

for all x ∈ H. An operator T ∈ L(H) is of k−quasi class Q∗(N) for a fixed real number N ≥ 1 and
k a natural number, if T satisfies

N∥T ∗T kx∥2 ≤ ∥T k+2x∥2 + ∥T kx∥2,

for all x ∈ H. We study structural and spectral properties of these classes of operators. Also we
compare this new classes of operators with other known classes of operators.
2020 Mathematics Subject Classifications: 47B47, 47B20
Key Words and Phrases: k−quasi class Q(N) operator, k−quasi class Q∗(N) operator, unitary
operator, approximate point spectrum

1. Introduction

In this paper let L(H) stand for the C∗ algebra of all bounded linear operators on
an infinite dimensional complex Hilbert space H. For T ∈ L(H), we denote by kerT the
null space, by T (H) the range of T. 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 there exist the sequence (xn), such as ∥xn∥ = 1 and ∥(T − λI)xn∥ → 0, n → ∞.
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 operator T is an isometry if
∥Tx∥ = ∥x∥, for all x ∈ H. The operator T is called unitary operator if T ∗T = TT ∗ = I.
The operator T is normaloid if r(T ) = ∥T∥ and it is quasinilpotent if r(T ) = 0.

Recall that an operator T ∈ L(H) is said to be:

DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4580

Email address: shqipe.lohaj@uni-pr.edu (Sh. Lohaj)

https://www.ejpam.com 1836 © 2022 EJPAM All rights reserved.



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1837

• k−quasi−paranormal (see [9]) if

∥T k+1x∥2 ≤ ∥T k+2x∥∥T kx∥,

for all x ∈ H, where k is a natural number.

• k−quasi− ∗ −paranormal (see [5]) if

∥T ∗T kx∥2 ≤ ∥T k+2x∥∥T kx∥,

for all x ∈ H, where k is a natural number.

• k−quasi−class A operator (see [2]) if

T ∗k|T 2|T k ≥ T ∗k|T |2T k.

• k−quasi− ∗ −class A operator (see [10]), if

T ∗k|T 2|T k ≥ T ∗k|T ∗|2T k.

• k−quasi class Q (see [4]) if

∥T k+1x∥2 ≤ 1

2
(∥T k+2x∥2 + ∥T kx∥2),

for all x ∈ H, where k is a natural number. It is proved that an operator T ∈ L(H)
is of the k−quasi class Q if

T ∗k(T ∗2T 2 − 2T ∗T + I)T k ≥ O.

• k−quasi class Q∗ (see [6]) if

∥T ∗T kx∥2 ≤ 1

2

(
∥T k+2x∥2 + ∥T kx∥2

)
,

for all x ∈ H, where k is a natural number. It is proved that an operator T ∈ L(H)
is of the k−quasi class Q∗ if

T ∗k(T ∗2T 2 − 2TT ∗ + I)T k ≥ O.

• class Q(N) (see [7]) if
N∥Tx∥2 ≤ ∥T 2x∥2 + ∥x∥2,

for all x ∈ H. It is proved that an operator T ∈ L(H) is of the class Q(N) if

T ∗2T 2 −NT ∗T + I ≥ O.



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1838

• Class Q∗(N) (see [7]) if

N∥T ∗x∥2 ≤ ∥T 2x∥2 + ∥x∥2,

for all x ∈ H. It is proved that an operator T ∈ L(H) is of the class Q∗(N) if

T ∗2T 2 −NTT ∗ + I ≥ O.

The following definitions describes the classes of operators we will study in this paper.

Definition 1. An operator T ∈ L(H) is of k−quasi class Q(N), for a fixed real number
N ≥ 1 and k a natural number, if T satisfies

N∥T k+1x∥2 ≤ ∥T k+2x∥2 + ∥T kx∥2,

for all x ∈ H.

Definition 2. An operator T ∈ L(H) is of quasi class Q∗(N), for a fixed real number
N ≥ 1 and k a natural number, if T satisfies

N∥T ∗T kx∥2 ≤ ∥T k+2x∥2 + ∥T kx∥2,

for all x ∈ H.

In this paper we give basic properties of k−quasi class Q(N) and k−quasi class Q∗(N)
operators. We discuss some inclusion relations, the structural and spectral properties and
also we obtained a matrix representation of these new classes of operators.

2. Inclusion Relations and Basic Properties

First, we state a proposition which gives necessary and sufficient conditions for an
operator T to be of k−quasi class Q(N).

Proposition 1. An operator T ∈ L(H) is of k−quasi class Q(N) if and only if

T ∗k(T ∗2T 2 −NT ∗T + I)T k ≥ O,

for a fixed real number N ≥ 1 and k a natural number.

Proof. Since T is of k−quasi class Q(N), for a fixed real number N ≥ 1 and k a natural
number then

N∥T k+1x∥2 ≤ ∥T k+2x∥2 + ∥T kx∥2,
for all x ∈ H. Then,

(T k+2x|T k+2x)−N(T k+1x|T k+1x) + (T kx|T kx) ≥ 0

⇔ (T ∗(k+2)T k+2x|x)−N(T ∗(k+1)T k+1x|x) + (T ∗kT kx|x) ≥ 0

⇔ ((T ∗(k+2)T k+2 −NT ∗(k+1)T k+1 + T ∗kT k)x|x) ≥ 0

⇔ T ∗k(T ∗2T 2 −NT ∗T + I)T k ≥ 0.

From the definition of the k−quasi class Q(N), we see that this new class of operators
could compare to several classes of operators.



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1839

Proposition 2. The following assertions hold.

(i) class Q(N) ⊆ k−quasi class Q(N).

(ii) k−quasi class Q = k−quasi class Q(2).

(iii) k−quasi−paranormal ⊆ k−quasi class Q(N) for N ∈ [1, 2].

(iv) k−quasi−class A ⊆ k−quasi class Q(N) for N ∈ [1, 2].

(v) k−quasi class Q(N) ⊆ k−quasi class Q(N − 1) for N ≥ 2.

Proof.

(i) From the definition of the class Q(N) operator

T ∗2T 2 −NT ∗T + I ≥ O,

and the Proposition 1 we see that every operator of the class Q(N) is also an operator
of the k−quasi class Q(N). Thus, we have the following implication:

class Q(N) ⊆ k−quasi class Q(N).

(ii) It is clear from definitions.

(iii) From the definition of k−quasi−paranormal we have:

∥T k+1x∥2 ≤ ∥T k+2x∥∥T kx∥

≤ 1

2
(∥T k+2x∥2 + ∥T kx∥2)

≤ 1

N
(∥T k+2x∥2 + ∥T kx∥2),

for a real number N ∈ [1, 2]. This proves the result.

(iv) Since T belongs to k−quasi−class A, we have

T ∗k|T 2|T k ≥ T ∗k|T |2T k.

Let x ∈ H. Then

2∥T k+1x∥2 =
2⟨T ∗(k+1)T k+1x, x⟩ =

2⟨T ∗k|T |2T kx, x⟩ ≤
2⟨T ∗k|T 2|T kx, x⟩ ≤



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1840

2∥|T 2|T kx∥ · ∥T kx∥ =

2∥T k+2x∥ · ∥T k∥ ≤
∥T k+2x∥2 + ∥T kx∥2

Therefore

∥T k+1x∥2 ≤ 1

2
(∥T k+2x∥2 + ∥T kx∥2) ≤ 1

N
(∥T k+2x∥2 + ∥T kx∥2),

for a real number N ∈ [1, 2]. This proves the result.

(v) Since T is of k−quasi class Q(N), for a fixed real number N ≥ 2 and k a natural
number then

N∥T k+1x∥2 ≤ ∥T k+2x∥2 + ∥T kx∥2,

for all x ∈ H. Then,

(N − 1)∥T k+1x∥2 ≤ N∥T k+1x∥2 ≤ ∥T k+2x∥2 + ∥T kx∥2.

Hence, T is an operator of k−quasi class Q(N − 1). It follows that,

k−quasi class Q(N) ⊆ k−quasi class Q(N − 1) ⊆ . . . ⊆ k−quasi class
Q(3) ⊆ k−quasi class Q(2).

Similarly, we state a proposition giving necessary and sufficient conditions for an op-
erator T to be of k−quasi class Q∗(N) and also the proposition where we compare this
class of operator with other existing classes of operators.

Since the techniqes of the proofs of the resultes for both classes are almost the same
we omit the proofs of the results of k−quasi class Q∗(N).

Proposition 3. An operator T ∈ L(H) is of k−quasi class Q∗(N), if and only if

T ∗k(T ∗2T 2 −NTT ∗ + I)T k ≥ O,

for a fixed real number N ≥ 1 and k a natural number.

Proposition 4. The following assertions hold.

(i) class Q∗(N) ⊆ k−quasi class Q∗(N).

(ii) k−quasi class Q∗ = k−quasi class Q∗(2).

(iii) k−quasi− ∗ −paranormal ⊆ k−quasi class Q∗(N)for N ∈ [1, 2].

(iv) k−quasi− ∗ −class A ⊆ k−quasi class Q∗(N) for N ∈ [1, 2].

(v) k−quasi class Q∗(N) ⊆ k−quasi class Q∗(N − 1) for N ≥ 2.



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1841

In the following we state a proposition which gives conditions for an operator T of the
k−quasi class Q(N) to be an operator of the class Q(N).

Proposition 5. Let T ∈ L(H) be an operator of the k−quasi class Q(N). If T has dense
range, then T is an operator of the class Q(N).

Proof. Since T k has dense range then, 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 an operator of the
k−quasi class Q(N), then〈

(T ∗k(T ∗2T 2 −NT ∗T + I)T k)xn, xn

〉
≥ 0,

⟨(T ∗2T 2 −NT ∗T + I)T kxn, T
kxn⟩ ≥ 0, for all n ∈ N.

By the continuity of the inner product, we have

⟨(T ∗2T 2 −NT ∗T + I)y, y⟩ ≥ 0

Therefore T is an operator of the class Q(N).

Similarly, we state a proposition which gives conditions for an operator T of the
k−quasi class Q∗(N) to be an operator of the class Q∗(N).

Proposition 6. Let T ∈ L(H) be an operator of the k−quasi class Q∗(N). If T k has
dense range, then T is an operator of the class Q∗(N).

In the following we give a proposition which gives conditions for an operator T of the
k−quasi class Q(N) to be an k−quasi−paranormal operator.

Proposition 7. If T is a k−quasi class Q(N) operator and T 2 is an isometry, then T is
k−quasi−paranormal operator for N ≥ 2.

Proof. Let T be a k−quasi class Q(N) operator. Then

N∥T k+1x∥2 ≤ ∥T k+2x∥2 + ∥T kx∥2 = (∥T k+2x∥ − ∥T kx∥)2 + 2∥T k+2x∥∥T kx∥.

Suppose that T 2 is isometry, then

∥T 2x∥ = ∥x∥ ⇒ ∥T 4x∥ = ∥T 2x∥ ⇒ · · · ⇒ ∥T k+2x∥ = ∥T kx∥

for all x ∈ H. From this we have

∥T k+1x∥2 ≤ 2

N
∥T k+2x∥∥T kx∥ ≤ ∥T k+2x∥∥T kx∥,

for N ≥ 2. This proves the result.

Similarly, we give a proposition which gives conditions for an operator T of the k−quasi
class Q∗(N) to be an k−quasi− ∗ −paranormal operator.



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1842

Proposition 8. If T is a k−quasi class Q∗(N) operator and T 2 is an isometry, then T
is k−quasi− ∗ −paranormal operator for N ≥ 2.

Now we will prove some properties of these new classes of operators.

Proposition 9. Let T be an operator of k−quasi class Q(N).

(i) If T commutes with an isometric operator S, then TS is an operator of the k−quasi
class Q(N).

(ii) If S is unitarily equivalent to operator T, then S is an operator of the k−quasi class
Q(N).

Proof. Similarly as Proposition 2.4. in [7].

Proposition 10. Let T be an operator of k−quasi class Q∗(N).

(i) If T commutes with an unitary operator S, then TS is an operator of the k−quasi
class Q∗(N).

(ii) If S is unitarily equivalent to operator T, then S is an operator of the k−quasi class
Q∗(N).

Proposition 11. Let T ∈ L(H). If ∥T∥ ≤ 1√
N

, then T is an operator of k−quasi class

Q(N).

Proof. From ∥T∥ ≤ 1√
N

, we have

∥T∥2 ≤ 1

N
.

Then,

∥Tx∥2 ≤ 1

N
∥x∥, for allx ∈ H

(Tx, Tx)− 1

N
(x, x) ≤ 0, for allx ∈ H

((I −NT ∗T )x, x) ≥ 0, for allx ∈ H

I −NT ∗T ≥ 0

T ∗2T 2 −NT ∗T + I ≥ 0

T ∗k(T ∗2T 2 −NT ∗T + I)T k ≥ 0

so T is an operator of k−quasi class Q(N).



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1843

Proposition 12. Let T ∈ L(H). If ∥T ∗∥ ≤ 1√
N

, then T is an operator of k−quasi class

Q∗(N).

Proposition 13. Let M be a closed invariant subspace of H for the operator T of k−quasi
class Q(N). Then, the restriction T|Tk(M) is of class Q(N).

Proof. Let u ∈ M. Then

T ∗k(T ∗2T 2 −NT ∗T + I)T k ≥ 0

(T ∗k(T ∗2T 2 −NT ∗T + I)T ku, u) ≥ 0

((T ∗2T 2 −NT ∗T + I)T ku, T ku) ≥ 0

((T ∗2T 2 −NT ∗T + I)y, y) ≥ 0, y = T ku ∈ M.

It follows that the restriction T|Tk(M) is of class Q(N).

Proposition 14. Let M be a closed invariant subspace of H for the operator T of k−quasi
class Q∗(N). Then, the restriction T|Tk(M) is of class Q∗(N).

Proposition 15. Let T ∈ L(H) be an invertible operator and S be an operator such that
S commutes with operator T ∗T. Then, S is of k−quasi class Q(N) if and only if TST−1

is of k−quasi class Q(N).

Proof. Let S be an operator of k−quasi class Q(N). Then

S∗k(S∗2S2 −NS∗S + I)Sk ≥ 0.

From this we have
TS∗k(S∗2S2 −NS∗S + I)SkT ∗ ≥ 0.

Now, we prove that operator TT ∗ commutes with operator TS∗k(S∗2S2−NS∗S+I)SkT ∗.
Since operator S commutes with operator T ∗T , operator S∗ also commutes with operator
T ∗T. From this we have

TS∗k(S∗2S2 −NS∗S + I)SkT ∗[TT ∗]

= TS∗k(S∗2S2 −NS∗S + I)Sk[T ∗T ]T ∗

= T [T ∗T ]S∗k(S∗2S2 −NS∗S + I)SkT ∗

= [TT ∗]TS∗k(S∗2S2 −NS∗S + I)SkT ∗

Thus operator TT ∗ commutes with operator TS∗k(S∗2S2−NS∗S+I)SkT ∗. Then, operator
[TT ∗]−1 also commutes with operator TS∗k(S∗2S2 −NS∗S + I)SkT ∗. Since the operator
[TT ∗]−1 and TS∗k(S∗2S2 −NS∗S + I)SkT ∗ are positive, then

TS∗k(S∗2S2 −NS∗S + I)SkT ∗[TT ∗]−1 ≥ 0.



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1844

Since operator S commutes with operator T ∗T, we get

(TST−1)∗k = (TST−1)∗(TST−1)∗ . . . (TST−1)∗

= T ∗−1S∗T ∗ T ∗−1S∗T ∗ . . . T ∗−1S∗T ∗ = T ∗−1S∗kT ∗

(TST−1)k = TSkT−1

(TST−1)∗(TST−1) = T ∗−1S∗T ∗TST−1 = TS∗ST−1

(TST−1)∗2(TST−1)2 = T ∗−1S∗2T ∗TS2T−1 = TS∗2S2T−1

To prove that operator TST−1 = M is an operator of k−quasi class Q(N), we substitute
last equations in the above expression and obtain

M∗k(M∗2M2 −NM∗M + I)Mk

T ∗−1S∗kT ∗[TS∗2S2T−1 −NTS∗ST−1 + I]TSkT−1

= TS∗k(S∗2S2 −NS∗S + I)SkT−1

Now we prove that the last expression is positive. Since

TS∗k(S∗2S2 −NS∗S + I)SkT ∗[TT ∗]−1 ≥ 0

TS∗k(S∗2S2 −NS∗S + I)SkT ∗T ∗−1T−1 ≥ 0

TS∗k(S∗2S2 −NS∗S + I)SkT−1 ≥ 0

Hence, operator TST−1 is an operator of k−quasi class Q(N).
Conversely, let TST−1 = M be an operator of k−quasi class Q(N). Then

M∗k(M∗2M2 −NM∗M + I)Mk ≥ 0.

Similarly, we have that

TS∗k(S∗2S2 −NS∗S + I)SkT−1 ≥ 0

T ∗TS∗k(S∗2S2 −NS∗S + I)SkT−1T ≥ 0

[T ∗T ]S∗k(S∗2S2 −NS∗S + I)Sk ≥ 0.

Operator T ∗T commutes with operator S and hence with operator [T ∗T ]S∗k(S∗2S2 −
NS∗S+I)Sk. Therefore, operator [T ∗T ]−1 also commutes with operator [T ∗T ]S∗k(S∗2S2−
NS∗S + I)Sk. Since these operators are positive, we have

[T ∗T ]−1[T ∗T ]S∗k(S∗2S2 −NS∗S + I)Sk ≥ 0.

Therefore,
S∗k(S∗2S2 −NS∗S + I)Sk ≥ 0.

What does it mean that operator S is of k−quasi class Q(N).



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1845

Proposition 16. Let T ∈ L(H) be an invertible operator and S be an operator such that
S commutes with operator T ∗T. Then, S is of k−quasi class Q∗(N) if and only if TST−1

is of k−quasi class Q∗(N).

The next proposition give necessary and sufficient conditions for a weighted shift oper-
ator T with decreasing weighted sequence (αn) to be an operator of this class of operators.

Proposition 17. A weighted shift operator T with decreasing weighted sequence (αn) is
an operator of k− quasi class Q(N) if and only if

|αn+k|2|αn+k+1|2 −N |αn+k|2 + 1 ≥ 0.

for every n and k natural numbers.

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

∗(en) = |an−1|en−1. Thus, we have

(T ∗T )(en) = |an|2en
(T ∗2T 2)(en) = |an|2|an+1|2en
T k(en) = |an||an+1| . . . |an+k−1en+k

(T ∗T )(en+k) = |an+k|2en+k

(T ∗2T 2)(en+k) = |an+k|2|an+k+1|2en+k

T ∗k(en+k) = |an+k−1||an+k−2| . . . |an|en

Now, since T is an operator of k− quasi class Q(N), we have

T ∗k(T ∗2T 2 −NT ∗T + I)T k ≥ 0.

From

T ∗k(T ∗2T 2 −NT ∗T + I)T k(en)

= |an||an+1| . . . |an+k−1|T ∗k(T ∗2T 2 −NT ∗T + I)(en+k)

= |an||an+1| . . . |an+k−1|(|an+k|2|an+k+1|2 −N |an+k|2 + 1)T ∗k(en+k)

= |an|2|an+1|2 . . . |an+k−1|2(|an+k|2|an+k+1|2 −N |an+k|2 + 1)(en) ≥ o.

we have inequality
|an+k|2|an+k+1|2 −N |an+k|2 + 1 ≥ 0.

Example 1. Consider the operator T : l2 → l2 defined by

T (x) = (0, α1x1, α2x2, . . .)

where αn =
N

2N
for a natural number N ≥ 1.



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1846

This operator T is of k−quasi class Q(N) and quasi nilpotent but it is not quasi-
hyponormal.

To prove that operator T is of k−quasi class Q(N) and quasi nilpotent it is similarly
as Example 2.5 in [6]. This operator is not quasi-hyponormal from the fact that:

αn ̸≤ αn+1

(see Proposition 3.4 in [8]).

Similarly, the next proposition give necessary and sufficient conditions for a weighted
shift operator T with decreasing weighted sequence (αn) to be an operator of this class of
operators.

Proposition 18. A weighted shift operator T with decreasing weighted sequence (αn) is
an operator of k− quasi class Q∗(N) if and only if

|αn+k|2|αn+k+1|2 −N |αn+k−1|2 + 1 ≥ 0

for every n.

In following propositions we give the inclusion of approximate point spectrum of these
classes of operators.

Proposition 19. Let T ∈ L(H) be a regular k−quasi class Q(N) operator. Then the
approximate point spectrum of operator T lies in the disc

σa(T ) ⊆ {λ ∈ C :

√
N

∥T−k−1∥ ·
√
∥T k+1∥2 + ∥T k−1∥2

≤ |λ| ≤ ∥T∥}.

Proof. Let T be a regular k−quasi class Q(N) operator. For every unit vector x in
Hilbert space H, we have:

∥x∥2 = ∥(T k+1)−1 · (T k+1)x∥2

≤∥(T k+1)−1∥2 · ∥T k+1x∥2

≤∥(T k+1)−1∥2 · 1

N
· (∥T k+2x∥2 + ∥T kx∥2)

≤ 1

N
· ∥(T k+1)−1∥2 · (∥T k+1∥2 · ∥Tx∥2 + ∥T k−1∥2 · ∥Tx∥2).

So,

N ≤ ∥Tx∥2 · ∥(T k+1)−1∥2 · (∥T k+1∥2 + ∥T k−1∥2),



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1847

where we have

∥Tx∥ ≥
√
N

∥T−k−1∥ ·
√
∥T k+1∥2 + ∥T k−1∥2

.

Now, assume that λ ∈ σa(T ), then there exists a sequence (xn), such as ∥xn∥ = 1 and
∥(T − λI)xn∥ → 0, n → ∞.

From the last inequation we have:

∥Txn − λxn∥ ≥ ∥Txn∥ − |λ| · ∥xn∥ ≥
√
N

∥T−k−1∥ ·
√
∥T k+1∥2 + ∥T k−1∥2

− |λ|.

Now, when n → ∞ we have

|λ| ≥
√
N

∥T−k−1∥ ·
√
∥T k+1∥2 + ∥T k−1∥2

.

So, we have

σa(T ) ⊆ {λ ∈ C :

√
N

∥T−k−1∥ ·
√
∥T k+1∥2 + ∥T k−1∥2

≤ |λ| ≤ ∥T∥}.

Therefore the proof is completed.

Proposition 20. Let T ∈ L(H) be a regular k−quasi class Q∗(N) operator. Then the
approximate point spectrum of operator T lies in the disc

σa(T ) ⊆ {λ ∈ C :

√
N

∥(T ∗T k)−1∥ ·
√

∥T k+1∥2 + ∥T k−1∥2
≤ λ ≤ ∥T∥}.

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

of operator T given by T̃ = |T |
1
2U |T |

1
2 was introduced by Aluthge (see [1]). The adjoint of

Aluthge transformation, the ∗−Aluthge transformation is defined by Yamazaki (see [11])

as T̃ (∗) def
= (T̃ ∗)∗ = |T ∗|

1
2U |T ∗|

1
2 .

The following propositions give the equivalence between Aluthge transformation and
∗−Aluthge transformation of these new classes of operators.

Proposition 21. Let T ∈ L(H). Then, T̃ is an operator of quasi class Q(N) if and only
if T̃ (∗) is an operator of quasi class Q(N).

Proof. Similarly as Theorem 2.14 in [7].

Proposition 22. Let T ∈ L(H). Then, T̃ is an operator of quasi class Q∗(N) if and only
if T̃ (∗) is an operator of quasi class Q∗(N).



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1848

3. A Matrix Representation

In this section we give some results for the matrix representation of these classes of
operators.

Proposition 23. Let T ∈ L(H) be the operator defined as

T =

(
A B
0 0

)
,

B is any operator. If A is operator of class Q(N) then T is an operator of k−quasi
class Q(N).

Proof. Let be D = A∗2A2 −NA∗A+ I. A simple calculation shows that:

T ∗ =

(
A∗ 0
B∗ 0

)
,

T ∗(k+2) =

(
A∗(k+2) 0

B∗A∗(k+1) 0

)
,

T (k+2) =

(
A(k+2) A(k+1)B

0 0

)
,

T ∗(k+2)T (k+2) =

(
A∗(k+2)A(k+2) A∗(k+2)A(k+1)B

B∗A∗(k+1)A(k+2) B∗A∗(k+1)A(k+1)B

)
.

T ∗k(T ∗2T 2 −NT ∗T + I)T k

=T ∗(k+2)T (k+2) −NT ∗(k+1)T (k+1) + T ∗kT k

=

(
A∗kDAk A∗kDA(k−1)B

B∗A∗(k−1)DAk B∗A∗(k−1)DA(k−1)B

)
Let u = x⊕ y ∈ H ⊕H. Then,

⟨(T ∗(k+2)T (k+2) −NT ∗(k+1)T (k+1) + T ∗kT k)u, u⟩
= ⟨A∗kDAkx, x⟩+ ⟨A∗kDA(k−1)By, x⟩
+ ⟨B∗A∗(k−1)DAkx, y⟩+ ⟨B∗A∗(k−1)DA(k−1)By, y⟩
= ⟨DAkx,Akx⟩+ ⟨DA(k−1)By,Akx⟩
+ ⟨DAkx,A(k−1)By⟩+ ⟨DA(k−1)By,A(k−1)By⟩
= ⟨D(Akx+A(k−1)By), (Akx+A(k−1)By)⟩ ≥ 0

because A is operator of class Q(N) then, D = A∗2A2 −NA∗A+ I ≥ O, so T is operator
of k−quasi class Q(N).

Proposition 24. Suppose that T k does not have a dense range, then the following state-
ments are equivalent:



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1849

(i) Operator T is a k−quasi class Q(N) operator;

(ii) T =

(
A B
0 C

)
on H = T k(H)⊕kerT ∗k, where A is an operator of the class Q(N)

on T k(H), Ck = 0 and σ(T ) = σ(A) ∪ {0}, B is any operator.

Proof. (1) ⇒ (2) Suppose that T ∈ L(H) is an operator of k−quasi class Q(N). Since
that T k does not have dense range, we can represent T as the upper triangular matrix:

T =

(
A B
0 C

)
on H = T k(H)⊕ kerT ∗k.

Since T is an operator of k−quasi class Q(N), we have

T ∗k(T ∗2T 2 −NT ∗T + I)T k ≥ 0.

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

⟨(T ∗2T 2 −NT ∗T + I)x, x⟩ = ⟨(A∗2A2 −NA∗A+ I)x, x⟩ ≥ 0,

for all x ∈ T k(H).
Hence

A∗2A2 −NA∗A+ I ≥ 0.

This shows that A is an operator of the class Q(N) on T k(H).

Let P be the orthogonal projection of H onto T k(H). For any

x =

(
x1
x2

)
∈ H = T k(H)⊕ kerT ∗k.

Then
⟨Ckx2, x2⟩ = ⟨T k(I − P )x, (I − P )x⟩ = ⟨(I − P )x, T ∗k(I − P )x⟩ = 0.

Thus T ∗k = 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 [3, Corollary 7]. Since σ(A)∩σ(C) has no interior points,
then σ(T ) = σ(A) ∪ σ(C) = σ(A) ∪ {0} and Ck = 0.

(2) ⇒ (1) Suppose that T =

(
A B
0 C

)
on H = T k(H) ⊕ kerT ∗k, where A is an

operator of the class Q(N) on T k(H), and Ck = O.
A simple calculation shows that:

T ∗ =

(
A∗ 0
B∗ C∗

)
,

T ∗T =

(
A∗A A∗B
B∗A B∗B + C∗C

)
,



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1850

T ∗k =

(
A∗k 0

(
∑k−1

j=0 A
jBCk−1−j)∗ 0

)
,

T k =

(
Ak (

∑k−1
j=0 A

jBCk−1−j)

0 0

)
,

Then, we have

T ∗k(T ∗2T 2 −NT ∗T + I)T k

=

(
A∗k 0

(
∑k−1

j=0 A
jBCk−1−j)∗ 0

)
×
(

D A∗2AB +A∗2BC
B∗A∗A2 + C∗B∗A2 − 2B∗A |AB +BC|2 + |C2|2 −N(B∗B + C∗C) + I

)
×
(
Ak

∑k−1
j=0 A

jBCk−1−j

0 0

)
=

(
A∗kDAk A∗kD

∑k−1
j=0 A

jBCk−1−j

(
∑k−1

j=0 A
jBCk−1−j)∗DAk M

)
,

where D = A∗2A2 − NA∗A + I,M = (
∑k−1

j=0 A
jBCk−1−j)∗D

∑k−1
j=0 A

jBCk−1−j . Let

v = x⊕ y be a vector in H = T k(H)⊕ kerT ∗k, where x ∈ T k(H) and y ∈ kerT ∗k. Then,

〈
T ∗k(T ∗2T 2 −NT ∗T + I)T kv, v

〉
=
〈
A∗kDAkx, x

〉
+

〈
A∗kD

k−1∑
j=0

AjBCk−1−jy, x

〉

+

〈
(

k−1∑
j=0

AjBCk−1−j)∗DAkx, y

〉

+

〈
(

k−1∑
j=0

AjBCk−1−j)∗D
k−1∑
j=0

AjBCk−1−jy, y

〉

=

〈
D(Akx+

k−1∑
j=0

AjBCk−1−jy), Akx+
k−1∑
j=0

AjBCk−1−jy

〉
.

Since A is an operator of the class Q(N), we have that D = A∗2A2 −NA∗A+ I ≥ 0.
Therefore, 〈

T ∗k(T ∗2T 2 −NT ∗T + I)T kv, v
〉
≥ 0



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1851

for all v ∈ H. Hence,
T ∗k(T ∗2T 2 −NT ∗T + I)T k ≥ 0

So we have that T is a k−quasi class Q(N) operator.

Proposition 25. If T k does not have a dense range, then the following statements are
equivalent:

(i) T =

(
A B
0 C

)
on H = T k(H) ⊕ kerT ∗k, where A∗2A2 − N(AA∗ + BB∗) + I ≥

0, Ck = O and σ(T ) = σ(A) ∪ {0}, B is any operator;

(ii) Operator T is a k−quasi class Q∗(N) operator.

Proof.

(1) ⇒ (2) Suppose that T =

(
A B
0 C

)
on H = T k(H) ⊕ kerT ∗k, where A∗2A2 −

N(AA∗ +BB∗) + I ≥ 0 and Ck = O.
A simple calculation shows that:

T ∗ =

(
A∗ 0
B∗ C∗

)
,

TT ∗ =

(
AA∗ +BB∗ BC∗

CB∗ CC∗

)
,

T ∗k =

(
A∗k 0

(
∑k−1

j=0 A
jBCk−1−j)∗ 0

)
,

T k =

(
Ak (

∑k−1
j=0 A

jBCk−1−j)

0 0

)
,

Then, we have

T ∗k(T ∗2T 2 −NTT ∗ + I)T k

=

(
A∗kDAk A∗kD

∑k−1
j=0 A

jBCk−1−j

(
∑k−1

j=0 A
jBCk−1−j)∗DAk M

)
,

where D = A∗2A2 −N(AA∗ + BB∗) + I,M = (
∑k−1

j=0 A
jBCk−1−j)∗D

∑k−1
j=0 A

jBCk−1−j .

Let v = x ⊕ y be a vector in H = T k(H) ⊕ kerT ∗k, where x ∈ T k(H) and y ∈ kerT ∗k.
Then,

〈
T ∗k(T ∗2T 2 −NTT ∗ + I)T kv, v

〉
=
〈
A∗kDAkx, x

〉



Sh. Lohaj / Eur. J. Pure Appl. Math, 15 (4) (2022), 1836-1853 1852

+

〈
A∗kD

k−1∑
j=0

AjBCk−1−jy, x

〉

+

〈
(

k−1∑
j=0

AjBCk−1−j)∗DAkx, y

〉

+

〈
(

k−1∑
j=0

AjBCk−1−j)∗D

k−1∑
j=0

AjBCk−1−jy, y

〉

=

〈
D(Akx+

k−1∑
j=0

AjBCk−1−jy), Akx+

k−1∑
j=0

AjBCk−1−jy

〉
.

Since D = A∗2A2 −N(AA∗ +BB∗) + I ≥ 0, we have〈
T ∗k(T ∗2T 2 −NTT ∗ + I)T kv, v

〉
≥ 0

for all v ∈ H. Hence,
T ∗k(T ∗2T 2 −NTT ∗ + I)T k ≥ 0

So we have that T is a k−quasi class Q∗(N) operator.
(2) ⇒ (1) Suppose that T ∈ L(H) is an operator of k−quasi class Q∗(N). Since that

T k does not have dense range, we can represent T as the upper triangular matrix:

T =

(
A B
0 C

)
on H = T k(H)⊕ kerT ∗k.

Since T is an operator of k−quasi class Q∗(N), we have

T ∗k(T ∗2T 2 −NTT ∗ + I)T k ≥ 0.

Let P be the projection onto T k(H). Than we have

P (T ∗2T 2 −NTT ∗ + I)P ≥ 0.

Therefore,
A∗2A2 −N(AA∗ +BB∗) + I ≥ 0.

On the other hand, for any x = (x1, x2) ∈ H, we have

⟨Ckx2, x2⟩ = ⟨T k(I − P )x, (I − P )x⟩ = ⟨(I − P )x, T ∗k(I − P )x⟩ = 0,

which implies Ck = 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 [3, Corollary 7] and σ(A) ∩ σ(C) has no interior points,
then σ(T ) = σ(A) ∪ σ(C) = σ(A) ∪ {0}.



REFERENCES 1853

Corollary 1. Let T ∈ L(H) be a k−quasi class Q∗(N) operator, the range of T k not to
be dense, then

T =

(
A B
O C

)
on H = T k(H)⊕ kerT ∗k.

where A is an operator of the class Q∗(N) on T k(H), Ck = O.

References

[1] A. Aluthge. On p-hyponormal operators for 0 < p < 1, integral equations operator
theory. Springer Link, 13:307–315, 1990.

[2] F. Gao and X. Fang. On k−quasi-class A operators. Journal of Inequalities and
Applications, pages 1–10, 2009.

[3] J. K. Han H. Y. Lee and W. Y. Lee. Invertible completions of 2x2 upper triangular
operator matrices. Proc. Amer. Math. Soc., 128:119–123, 2000.

[4] V.R. Hamiti. On k−quasi class Q operators. Bulletin of Mathematical Analysis and
Applications, 6:31–37, 2014.

[5] I. Hoxha and N. L. Braha. A note on k−quasi∗−paranormal operators. Journal of
Inequalities and Applications, 1(350), 2013.

[6] Sh. Lohaj and V. R. Hamiti. Some properties k−quasi class Q∗ operators. The
Australian Journal of Mathematical Analysis and Applications, 14:1–10, 2017.

[7] Sh. Lohaj and V. R. Hamiti. A note on class Q(N) operators. Missouri Journal of
Math. Sci., 2:185–196, 2018.

[8] N.L. Braha M. Lohaj F. Marevci Sh. Lohaj. Some properties of paranormal and
hyponormal operators. Bulletin of Mathematical Analysis and Applications, 1:23–35,
2009.

[9] Salah Mecheri. Bishop s property b and riesz idempotent for k−quasi-paranormal
operators. Banach J. Math. Anal., 6(1):147 – 154, 2012.

[10] Salah Mecheri. Isolated points of spectrum of k− quasi∗−class A operators. Studia
Mathematica, 1(208):87–96, 2012.

[11] T. Yamazaki. Parallelisms between aluthge transformation and powers of operators.
Acta Sci. Math. (Szeged), 67:809–820, 2001.