Turkish Journal of Analysis and Number Theory, 2016, Vol. 4, No. 4, 87-91
Available online at http://pubs.sciepub.com/tjant/4/4/1
©Science and Education Publishing
DOI:10.12691/tjant-4-4-1
On −Quasi Class
∗
Operators
Valdete Rexhëbeqaj Hamiti
*
, Shqipe Lohaj, Qefsere Gjonbalaj
Faculty of Electrical and Computer Engineering, University of Prishtina, Prishtinë, Kosova
*Corresponding author: valdete_r@hotmail.com
Abstract Let be a bounded linear operator on a complex Hilbert space . In this paper we introduce a new class
of operators: − quasi class
∗
operators. An operator is said to be − quasi class
∗
if it satisfies
2 2 2
* 2
1
2
k k k
TT T x T x
+
≤ +
, for all ∈, where is a natural number. We prove the basic properties of
this class of operators.
Keywords: −quasi class
∗
, quasi class
∗
, −quasi −∗− paranormal operators, quasi −∗− paranormal
operators
Cite This Article: Valdete Rexhëbeqaj Hamiti, Shqipe Lohaj, and Qefsere Gjonbalaj, “On −Quasi Class
∗
Operators.” Turkish Journal of Analysis and Number Theory, vol. 4, no. 4 (2016): 87-91. doi: 10.12691/tjant-4-4-1.
1. Introduction
Throughout this paper, let be a complex Hilbert
space with inner product 〈∙,∙〉 . Let () denote the
∗
algebra of all bounded operators on . For ∈(), we
denote by ker the null space, by () the range of
and by (T) the spectrum of . The null operator and the
identity on will be denoted by 0 and , respectively. If
is an operator, then
∗
is its adjoint, and‖‖ = ‖
∗
‖. For
an operator ∈(), as usual ||=(
∗
)
1
2
.
We shall denote the set of all complex numbers by ℂ,
the set of all non-negative integers by ℕ and the complex
conjugate of a complex number by
̅
. The closure of a
set will be denoted by
�
. An operator ∈() is a
positive operator, ≥ 0, if (, ) ≥ 0 for all ∈. We
write () for the spectral radius. It is well known
that () ≤ ‖‖ . The operator is called normaloid
if ()= ‖‖. The operator is an isometry if ‖‖ =
‖‖ , for all ∈ . The operator is called unitary
operator if
∗
=
∗
= .
An operator ∈(), is said to be paranormal [4], if
‖‖
2
≤ ‖
2
‖ for any unit vector in . Further, is
said to be ∗− paranormal [1,9], if ‖
∗
‖
2
≤ ‖
2
‖ for
any unit vector in . An operator ∈(), is said to be
quasi−paranormal operator if ‖
2
‖
2
≤ ‖
3
‖‖‖, for
all ∈().
Mecheri [7] introduced a new class of operators called
− quasi paranormal operators. An operator is called
−quasi −paranormal if ‖
+1
‖
2
≤ ‖
+2
‖‖
‖,for
all ∈, where is a natural number. An operator is
called quasi −∗− paranormal [8,11], if ‖
∗
‖
2
≤
‖
3
‖‖‖, for all ∈.
An operator is called − quasi −∗− paranormal if
‖
∗
‖
2
≤ ‖
+2
‖‖
‖ for all ∈, where is a
natural number, [6].
Shen, Zuo and Yang [13] introduced a new class of
operator quasi−∗−class . An operator ∈() is said
to be a quasi−∗−class , if
∗
|
2
|≥
∗
|
∗
|
2
.
Mecheri [12] introduced −quasi−∗−class operator.
An operator ∈() is said to be a − quasi −∗
−class , if
∗
|
2
|
≥
∗
|
∗
|
2
.
Duggal, Kubrusly, Levan [3] introduced a new class of
operators, the class . An operator ∈() belongs to
class if
∗2
2
− 2
∗
+ ≥ 0, or equivalent ‖‖
2
≤
1
2
(‖
2
‖
2
+ ‖‖
2
), for all ∈.
Senthilkumar, Prasad [10] introduced a new class of
operators, the class
∗
. An operator ∈() belongs to
class
∗
if
∗2
2
− 2
∗
+ ≥ 0, or equivalent
‖
∗
‖
2
≤
1
2
(‖
2
‖
2
+ ‖‖
2
) for all ∈.
Senthilkumar, Naik and Kiruthika [2] introduced a new
class of operators, the quasi class
∗
. An operator
∈() is said to belong to the quasi class
∗
if
∗3
3
− 2(
∗
)
2
+
∗
≥ 0, or equivalent ‖
∗
‖
2
≤
1
2
(‖
3
‖
2
+ ‖‖
2
) for all ∈.
Now we introduce the class of − quasi class
∗
operators defined as follows:
Definition 1.1. An operator ∈() is said to be of the
−quasi class
∗
if
2 2 2
* 2
1
,
2
k k k
TT x T x T x
+
≤ +
for all ∈, where is a natural number.
Remark 1.2. For =1, a 1 −quasi class
∗
operators is a
quasi class
∗
operators.
2. Main Results
Proposition 2.1. An operator ∈() is of the −quasi
class
∗
, if and only if
Received June 21, 2016; Revised August 21, 2016; Accepted August 29, 2016