EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6739
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Invariant Subspace Problem for Norm Attaining
Operators

Aissa Nasli Bakir1,2, Ayyoub Fellag Ariouat1, Abdelkader Benali1,
Ibrahim Alraddadi3,∗, Saad M. Almuaddi4,5,∗

1 Department of Mathematics, Faculty of Exact Sciences and Informatics, Laboratory of
Mathematics and Application LMA, Hassiba Benbouali University of Chlef, Algeria
2 National Higher School of Cybersecurity, Sidi Abdellah, Algeria
3 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah,
Saudi Arabia
4 Basic & Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University,
P.O. Box 1982, Dammam 31441, Saudi Arabia
5 Mathematics Department, College of Science, Imam Abdulrahman Bin Faisal University,
Dammam 31441, Saudi Arabia

Abstract. Our aim is to characterize norm attaining and absolutely norm attaining quasi-∗-
paranormal operators and class Ωn operators defined on a separable Hilbert space. We define
invariant non trivial subspaces for the considered operators and we give a matrix representation
under certain condition. Compactness, reducing subspaces and the normality of such operators
are also established.

2020 Mathematics Subject Classifications: 47A30, 47B47, 47B20

Key Words and Phrases: Quasi-∗-paranormal operators, invariant subspaces, norm attaining
operators, absolutely norm attaining operators

1. Preliminaries and notations

Let H denote an infinite separable complex Hilbert space, and let B(H) be the Banach
algebra of all bounded linear operators on H. An operator T ∈ B(H) is said to be norm
attaining, if there exists a unit vector u ∈ H satisfying ∥Tu∥ = ∥T∥, [1], and T is said to be
absolutely norm attaining, briefly, AN -operator, if its restriction on any closed subspace
of H is norm attaining, [2]. Obviously, AN -operators are norm attaining. Many authors

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6739

Email addresses: a.nasli@univ-chlef.dz (A. Nasli Bakir),
a.fellagariouat@univ-chlef.dz (A. Fellag Ariouat), benali4848@gmail.com (A. Benali),
ialraddadi@iu.edu.sa (I. Alraddadi), smuaddi@iau.edu.sa (S. M. Almuaddi)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. Nasli Bakir et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6739 2 of 10

studied the structure of some classes of norm attaining non normal operators, see [3, 4]
and [5].

Authors in [5, 6] analyzed the properties of norm attaining and absolutely norm at-
taining operators, and provide a representation for the class of ∗-paranormal operators
under an orthogonal decomposition of H. Authors in [4] showed that compact (resp. iso-
metric) operators are AN -operators since their restrictions on closed invariant subspaces
remain compact (resp. isometric). Moreover, if T is AN -operator, then T ∗ may not be
one, see [4, 7] where it’s shown that an isometry U : ℓ2 → ℓ2 onto a subspace with infinite
codimension is AN -operator, whereas its adjoint U∗ is not. However, G. Ramesh in [5]
presented an additional condition for which T ∗ remains an AN -operator.

An operator T ∈ B(H) is said to be non-negative and we shall write T ≥ 0, if ⟨Tu, u⟩ ≥
0 for all u ∈ H and A is said to be normal if T ∗T = TT ∗, isometric if T ∗T = I, where I is
the identity operator on H. If T is isometric and onto, then T is said to be unitary. The
operator T ∈ B(H) is said to be ∗-paranormal if T ∗2T 2 − 2λTT ∗ + λ2 ≥ 0 for each λ > 0,
that is, ∥T ∗u∥2 ≤ ∥T 2u∥∥u∥ for all u ∈ H [8]. Also, T is said to be quasi-∗-paranormal
if T ∗(T ∗2T 2 − 2λTT ∗ + λ2)T ≥ 0 for all λ > 0, [9]. Clearly, a ∗-paranormal operator
is quasi-∗-paranormal while the converse is in general false, see [10]. It is known that
quasi-∗-paranormal operators are normaloid, that is, r(T ) = ∥T∥, where r(T ) = sup{|λ| :
λ ∈ σ(T )} is the spectral radius of T. Ample properties of these classes can be found in
[11] and [12]. The operator T in B(H) is said to be quasi-normal of order n for certain
integer n, and we shall write T ∈ Ωn, if TT

⋆nTn = T ⋆nTn+1. For more information on the
class Ωn, we refer the reader to [13, 14].

For an operator T ∈ B(H), the range of T , the null space and the modulus of T will be
denoted by R(T ), N(T ) and |T | =

√
T ∗T respectively. If T ∈ B(H), then T = U |T | is the

polar decomposition of T, where U is a partial isometry, that is, U
∣∣
N(A)⊥

is an isometry,

R(U) = R(|T |).According to [15], U is a partial isometry if and only if UU∗U = U. The sets
σ(T ), σp(T ) denote respectively, the spectrum and the set of eigenvalues of T. For a self-
adjoint operator T ∈ B(H), that is, T ∗ = T, the discrete spectrum of T is the set σd(T ) =
{λ ∈ σp(T ) : λ is isolated and has a finite multiplicity}. The set σess(T ) = σ(T ) \ σd(T ) is
said to be the essential spectrum of T, [16]. If dimH < +∞, then σess(T ) = ∅.

The positive real number m(T ) = inf{∥Tu∥ : u ∈ H and ∥u∥ = 1} is said to be the
minimum modulus of T ∈ B(H), and the quantity me(T ) = inf{λ : λ ∈ σ(|T |)} is said to
be the essential minimum modulus of T. For more details, reader is referred to [4, 17] and
[16].

In [18], authors gave a characterization of norm attaining and absolutely norm attaining
∗-paranormal operators and defined a closed non trivial invariant subspace for this class of
operators. In this article, we generalize these results for a large class of quasi-∗-paranormal
operators. We show several spectral properties. We also provide invariant subspaces for
both of classes of quasi-∗-paranormal operators and class Ωn operators, and we show that
the given subspaces become reducing under certain conditions. Other properties related
to the compactness, the normality and the matrix representation are also established.



A. Nasli Bakir et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6739 3 of 10

2. Norm attaining quasi-∗-paranormal operators

Definition 1. [19, 20] An operator T ∈ B(H) is said to be quasi-∗-paranormal if

T ∗(T ∗2T 2 − 2λTT ∗ + λ2)T ≥ 0

for all λ > 0.
The given definition implies that

∥T ∗Tu∥2 ≤ ∥T 3u∥∥Tu∥

for all u ∈ H.

Example 1. [11] Let µ = (µn)n≥1 be a positive real sequence. Define the weighted shift
Sµ on the Hilbert space ℓ2 by

Sµen = µnen+1, n ≥ 1

where (en)n≥1 is the standard basis of ℓ2. Then, Sµ is quasi-∗-paranormal if and only if
the inequality µ2

n ≤ µn+1µn+2 holds for any n, n ≥ 1.

[20] The restriction of a quasi-∗-paranormal operator on a closed invariant subspace
is also quasi-∗-paranormal. [10, Lemma 3.4] For any quasi-∗-paranormal operator T ∈
B(H), and each non-zero complex scalar λ, we’ve N(T − λI) ⊂ N(T − λI)∗.

Remark 1. Lemma 2 is in general not true for λ = 0. A counter-example can be found
in [9].

Definition 2. [4] An operator T ∈ B(H) is said to be norm attaining ( or achieving the
norm) if there exists a unit vector u ∈ H for which ∥Tu∥ = ∥T∥.

Example 2. [21] Let θ = (θn)n≥1 be a real strictly increasing sequence. The operator Tθ

defined on the usual Hilbert space ℓ2 by

Tθx = (θnxn)n≥1, x = (xn)n≥1 ∈ ℓ2

is not norm attaining.

Example 3. [18] The usual Hilbert space ℓ2 is equipped with its standard orthonormal
basis (en)n≥1, and S is the unilateral left shift on ℓ2 defined by

Sen = en−1, n ≥ 2 and Se1 = 0

Then, S is norm attaining since ∥Se2∥ = ∥S∥ = 1.



A. Nasli Bakir et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6739 4 of 10

Recall that a closed subspace M ⊂ H is said to be invariant for an operator T ∈ B(H),
if Tu ∈ M for each u ∈ M, and M is said to be reducing for T if M is invariant for both
T and T ∗.

As an extension of results given in [18] and [5], where are provided invariant non
trivial subspaces for both of norm achieving ∗-paranormal and norm achieving paranormal
operators respectively, we shall define in the following, a non trivial subspace for norm
attaining quasi-∗-paranormal operators as a positive answer to the problem of invariant
subspaces for operators on Hilbert spaces that asks if any bounded linear operator acting
on a Hilbert space admits at least, a non trivial invariant subspace.

Theorem 1. Let T ∈ B(H) be a quasi-∗-paranormal operator that achieves the norm.
Then, the subspace M = {u ∈ H : ∥Tu∥ = ∥T∥∥u∥} is invariant for T.

Proof. Since T is norm attaining, M = N(T ∗T −∥T∥2I) is a non-zero closed subspace
of H. Next, as T is quasi-∗-paranormal,

∥T∥2∥u∥2 = ∥Tu∥2 = ⟨T ∗Tu, u⟩ ≤ ∥T ∗Tu∥∥u∥ ≤
√

∥T 3u∥∥Tu∥∥u∥
≤

√
∥T 2∥∥Tu∥2∥u∥

≤
√

∥T∥2∥Tu∥2∥u∥
≤ ∥T∥∥Tu∥∥u∥
≤ ∥T∥2∥u∥2

for each u ∈ M. Hence,
∥T∥2∥u∥2 = ∥T ∗Tu∥∥u∥ = ∥Tu∥2

i.e.,

∥T∥∥Tu∥ = ∥T∥2∥u∥ = ∥T ∗Tu∥, u ∈ M (1)

Using equality (1), the fact that ∥T ∗T∥ = ∥T∥2 and by Cauchy-Schwarz’s inequality,
we get for each vector u in M,

∥T 2u∥2 = ⟨T ∗T 2u, Tu⟩ ≤ ∥T ∗T∥∥Tu∥2 = ∥T∥2∥Tu∥2 = ∥T ∗Tu∥2

That is,

∥T 2u∥ ≤ ∥T ∗Tu∥ (2)

for all u ∈ M. On another hand,

∥T ∗Tu∥2 ≤ ∥T 3u∥∥Tu∥ ≤ ∥T 2u∥∥T∥∥Tu∥ = ∥T 2u∥∥T ∗Tu∥

Hence, for all u ∈ M,

∥T ∗Tu∥ ≤ ∥T 2u∥ (3)



A. Nasli Bakir et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6739 5 of 10

By (2) and (3),
∥T ∗Tu∥ = ∥T∥∥Tu∥ = ∥T 2u∥

for each u ∈ M. This shows that M is invariant for T.

By a similar way as in [5, Lemma 3.1], we can easily prove the following result For an
operator T ∈ B(H), M = N(∥T∥2I−TT ∗) = N(|T ∗|−∥T∥I). Furthermore, if T ∗ is norm
attaining, then M ̸= {0}. [7] Let T ∈ B(H). The following statements are equivalent

A. T achieves the norm.

b. T ∗ achieves the norm.

c. |T | achieves the norm.

d. |T ∗| achieves the norm.

e. ∥T∥ is an eigenvalue of T.

f. ∥T∥ is an eigenvalue of T ∗.

Corollary 1. Let T ∈ B(H) be a quasi-∗-paranormal operator achieving the norm. Then
∥T∥ is an eigenvalue of |T |.

Proof. The operator |T ∗|−∥T∥I is not one-to-one acoording to Corollary 1 and Lemma
2. Then, the result holds by Lemma 2.

Corollary 2. If both of T and T ∗ are quasi-∗-paranormal in B(H), then the subspace
M = N(∥T∥2I − TT ∗) reduces T.

Proof. By Corollary 1, M is invariant for T, and M ⊂ N(∥T∥2I − T ∗T ) according to
the proof of Theorem 1. Since T ∗ is also quasi-∗-paranormal, N(∥T∥2I −T ∗T ) ⊂ M is an
invariant subspace for T ∗. Thus, M = N(∥T∥2I − T ∗T ) is a reducing subspace for T.

Theorem 2. Let T ∈ B(H) be a norm attaining quasi-∗-paranormal operator. If M =
N(T ∗T − ∥T∥2I) is finite dimensional, then

a. M is a reducing subspace for T.

b. The restriction T
∣∣
M⊥ of T on M is also quasi-∗-paranormal.

Proof. a. By Theorem 1, M is an invariant subspace for T. Since M is of finite
dimension, the isometry T ∗

∥T∥ is unitary on M. We can then write

T =

(
T
∣∣
M

S

0 R

)



A. Nasli Bakir et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6739 6 of 10

under the decomposition H = M ⊕ M⊥, where S ∈ B(M⊥,M) and R ∈ B(M⊥,M⊥).
Since T is quasi-∗-paranormal,

0 ≤ T ∗(T ∗2T 2 − 2λTT ∗ + λ2)T

=

(
(T

∣∣
M
)∗XT

∣∣
M

(T
∣∣
M
)∗(XS + Y R)

(S∗X +R∗Y ∗)T
∣∣
M

(S∗X +R∗Y ∗)S + (S∗Y +R∗Z)R

)
for all λ > 0, where

X = (T
∣∣
M
)∗2(T

∣∣
M
)2 − 2λ(T

∣∣
M
(T

∣∣
M
)∗ + SS∗ + λ2I

Y = (T
∣∣
M
)∗2(T

∣∣
M
S + SR)− 2λSR∗

Z = (T
∣∣
M
S + SR)∗(T

∣∣
M
S + SR) +R∗2R2 − 2λRR∗ + λ2I

By [22, Theorem 6], we get
(T

∣∣
M
)∗XT

∣∣
M

≥ 0

and
(S∗X +R∗Y ∗)S + (S∗Y +R∗Z)R ≥ 0

Hence,
(T

∣∣
M
)∗2

(
T
∣∣
M
)2 − 2λ(T

∣∣
M
(T

∣∣
M
)∗ + SS∗)+ λ2I ≥ 0

for all λ > 0. As the operator 1
∥T∥(T

∣∣
M
)∗ is unitary, we get for λ = 1 that SS∗ ≤ 0. Hence,

S = 0. This shows that

T =

(
T
∣∣
M

0

0 R

)
Thus, the subspace M reduces T.

b. The operator R∗ZR = R∗(R∗2R2 − 2λRR∗ + λ2I)R is non-negative for all λ > 0.
Thus, the restriction R = T

∣∣
M⊥ is also quasi-∗-paranormal.

Corollary 3. Let T ∈ B(H) be a compact quasi-∗-paranormal operator. Then, the sub-
space M = N(∥T∥2I − TT ∗) is reducing for T.

Proof. By the hypothesis, the operator TT ∗ is also compact. Then, M is a nonzero
finite dimensional subspace by Fredholm Alternative. The desired result follows then by
Theorem 2.

3. Absolutely norm attaining quasi-*-paranormal operators

In the sequel, we present certain structure results on the absolutely norm attaining
quasi-∗-paranormal operators as an extension of certain results given in [3] and [18].



A. Nasli Bakir et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6739 7 of 10

Definition 3. [4] An operator T ∈ B(H) is said to be absolutely norm attaining, briefly
AN -operator, if the restriction T

∣∣
V
is norm attaining for any closed subspace V ⊂ H, that

is, there exists a unit vector u ∈ V for which

∥T
∣∣
V
u∥ = ∥Tu∥ = ∥T

∣∣
V
∥

Example 4. [7] The operator S defined on the Hilbert space ℓ2 by

Se1 =
1
2e1 and Sen = en, (n ≥ 2)

is absolutely norm attaining.

Example 5. In [3], author showed that the operator A ∈ L (ℓ2 ⊕ ℓ2) defined by

A(x, y) =
(
(y1, x1,

x2
2
,
x3
3
, ...,

xn
n
, ....), (y2, y3, ...., yn, yn+1, ....)

)
for all x = (xk)k≥1, y = (yk)k≥1 ∈ ℓ2, is not absolutely norm attaining on ℓ2 ⊕ ℓ2.

Theorem 3. Let T ∈ B(H) be an absolutely norm attaining quasi-∗-paranormal operator.
If σess(|T |) = {∥T∥}, then

T =

(
∥T∥S B
0 C

)
under the orthogonal decomposition H = M ⊕M⊥, where

1. S ∈ B(M) is an isometry.

2. B∗S = 0.

Proof. 1. Let T = U |T | be the polar decomposition of T. Then, for all u ∈ M, we get

Tu = U |T |u = ∥T∥Uu

That is,
T
∣∣
M

= ∥T∥U
∣∣
M

= ∥T∥S
Since U

∣∣
M

is an isometry, the operator S so is.

2. The subspace M is invariant for T. Hence, on H = M ⊕M⊥,

T =

(
∥T∥S B
0 C

)
Since T is quasi-∗-paranormal, we get for all λ > 0,

T ∗(T ∗2T 2 − 2λTT ∗ + λ2)T =

(
X Y
Z W

)
≥ 0

where
X = ∥T∥2S∗(∥T∥4 + λ2 − 2λ∥T∥2SS∗ − 2λBB∗)S ≥ 0

and for some bounded linear operators Y,Z,W. Then, for λ = ∥T∥2, and since S is an
isometry, S∗BB∗S = (B∗S)∗(B∗S) ≤ 0. Thus, B∗S = 0 since B∗S is a positive operator.



A. Nasli Bakir et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6739 8 of 10

4. Norm attaining class Ωn operators

Definition 4. [13] An operator T ∈ B(H) is said to be quasi-normal of order n for some
integer n, or a class Ωn operator if TT ⋆nTn = T ⋆nTn+1.

Example 6. [13] Matrices S =

(
1 0
0 0

)
and B =

(
0 0
1 0

)
are quasi-normal operators

of order 2, i.e., B,S ∈ Ω2.

In the following, we provide invaraiant subspaces for operators belonging to class Ωn.

Theorem 4. Let T ∈ B(H) be quasi-normal operator of order 2 such that T 2 achieves
the norm. Then, the subspace V = {u ∈ H : ∥T 2u∥ = ∥T 2∥∥u∥} is invariant for T 2.

Proof. Since T ∈ Ω2, TT ⋆2T 2 = T ⋆2T 3. Then, (T ∗2T 2)2 = T ∗4T 4. Hence, for all
u ∈ H, ∥T ∗2T 2u∥ = ∥T 4u∥. By Cauchy-Schwarz’s inequality,

∥T 2u∥2 = ⟨T 2u, T 2u⟩ = ⟨T ∗2T 2u, u⟩ ≤ ∥T ∗2T 2u∥∥u∥ ≤ ∥T 4u∥∥u∥
≤ ∥T 2(T 2u)∥∥u∥
≤ ∥T 2∥∥T 2u∥∥u∥

That is for all u ∈ V ,
∥T 2∥2∥u∥ ≤ ∥T 4u∥ ≤ ∥T 2∥2∥u∥

since ∥T 2u∥ = ∥T 2∥∥u∥, u ∈ V. Thus,

∥T 4u∥ = ∥T 2(T 2u)∥ = ∥T 2∥2∥u∥ = ∥T 2∥∥T 2u∥

for each u ∈ V. This achieves the proof.

Corollary 4. Let T ∈ B(H) a class Ω2 operator. If T ∗2 achieves the norm, then the
subspace V∗ = {u ∈ H : ∥T ∗2u∥ = ∥T 2∥∥u∥} is invariant under T ∗2.

5. Conclusion

Structures of norm attaining quasi-∗-paranormal operators and class Ωn are established
in the present manuscript. It’s shown that elements of these classes of operators admit
at least an invariant non trivial subspace. Some results depending on compactness, and
finite dimension are given too. As perspective works, we ask if a such structure can be
provided for large classes of norm achieving k-quasi-∗-paranormal operators, and class
Ωn,k operators defined for certain integer k as T ∗k(TT ⋆nTn − T ⋆nTn+1)T k = 0.



A. Nasli Bakir et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6739 9 of 10

Author Contributions

A. Nasli Bakir: Conceptualization, methodology, writing—original draft, supervision.
A. Fellag Ariouat: Investigation, data analysis, writing—original draft. A. Benali: Formal
analysis, validation, writing—review and editing. I. Alraddadi: Software, visualization,
project administration, funding acquisition. S.M. Almuaddi: Resources,formal analysis,
data curation, writing—review and editing. All authors have read and agreed to the
published version of the manuscript

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