Article
KYUNGPOOK Math. J. 2019; 59(3): 415431
Published online September 23, 2019
Copyright © Kyungpook Mathematical Journal.
Spectral Properties of kquasiclass A(s,t) Operators
Salah Mecheri, Naim Latif Braha?
Taibah University, College of Science Department of Mathematics, P. O. Box 20003 Al Madinah Al Munawarah, Saudi Arabia
email : mecherisalah@hotmail.com
Research Institute Ilirias, Rruga Janina, No2, ferizaj, 70000, Kosovo Department of Mathematics and Computer Sciences, University of Prishtina, Avenue Mother Teresa, No4, Prishtine, 10000, Kosova
email : nbraha@yahoo.com
Received: March 13, 2016; Revised: October 11, 2018; Accepted: October 16, 2018
In this paper we introduce a new class of operators which will be called the class of
Let
Let
Aluthge and Wang [3] introduced
An operator
An operator
In order to generalize the class
Definition 1.1
An operator
We have
If
In [38], H. Weyl proved that Weyl’s theorem holds for hermitian operators. Weyl’s theorem has been extended from hermitian operators to hyponormal operators [14], algebraically hyponormal operators [21],
More generally, M. Berkani investigated the generalized Weyl’s theorem which extends Weyl’s theorem, and proved that the generalized Weyl’s theorem holds for hyponormal operators ([6, 7, 8]). In a recent paper [25] the author showed that the generalized Weyl’s theorem holds for (
M. Berkani also investigated
Proposition 1.1

If R (T^{n} )is closed and T _{[n]}is Fredholm ,then R (T^{m} )is closed and T _{[m]}is Fredholm for every m ≥n. Moreover , indT _{[m]} = indT _{[n]}(= indT ). 
An operator T is BFredholm (BWeyl) if and only if there exist Tinvariant subspaces ℳ and such that where T ℳis Fredholm (Weyl) and is nilpotent.
The BWeyl spectrum
We define
We define
An operator
SVEP is possessed by many important classes of operators such as hyponormal operators and decomposable operators. The interested reader is referred to [26, 27, 28, 29, 30] for more details. A closed subspace of
In this paper, we show that an algebraically
Proposition 2.1
Let
Put
Lemma 2.1
([39])
Lemma 2.2
Suppose
By Hansen’s inequality, we have
Let
Lemma 2.3
Suppose
Let
Corollary 2.1
Since
An operator
Lemma 2.4
Assume that

Case 1: Suppose
T^{k} has dense range. It follows thatT is classA (s ,t ). Since a classA (s ,t ) operator is normaloid [20, 39], every quasinilpotent classA (s ,t ) operator is the zero operator. HenceT is nilpotent. 
Case 2: Suppose
T^{k} does not have dense range. Then by Lemma 2.1,T can be represented on$\mathscr{H}=\overline{R({T}^{k})}\oplus \text{ker}{T}^{*k}$ as$$T=\left(\begin{array}{cc}{T}_{1}& {T}_{2}\\ 0& {T}_{3}\end{array}\right).$$ SoT _{1} is inA (s ,t ),${T}_{3}^{k}=0$ andσ (T ) =σ (T _{1}) ∪ {0}. SinceT is quasinilpotent,σ (T ) = {0}. Butσ (T ) =σ (T _{1}) ∪ {0}. SinceT _{1} is inA (s ,t ),T _{1} = 0, soT is nilpotent.
Lemma 2.5
Assume
In the following theorem we will prove that an algebraically
Theorem 2.1
If
Corollary 2.2
If a Banach space operator
Theorem 2.2

If T ^{*}is an algebraically kquasiclass A (s ,t )operator ,then generalized aWeyl’s theorem holds for T. 
If T is an algebraically kquasiclass A (s ,t )operator ,then generalized aWeyl’s theorem holds for T ^{*}.
(i) It is well known that
Corollary 2.3

generalized Weyl’s theorem holds for T ^{*}. 
generalized Weyl’s theorem holds for T. 
Weyl’s theorem holds for T. 
Weyl’s theorem holds for T ^{*}.
If
Remark 2.1

Recall [5] that if
T is polaroid, then T satisfies generalized Weyl’s theorem (resp. generalized aWeyl’s) theorem if and only ifT satisfies Weyl’s theorem (resp. aWeyl’s theorem). Hence ifT is an algebraicallyk quasiclassA (s ,t ) operator, the above equivalences hold. 
Let
f (z ) be an analytic function onσ (T ). IfT is polaroid, thenf (T ) is polaroid too [5].
If
T ^{*} is algebraicallyk quasiparanormal, thenf (T ) satisfies generalized aWeyl’s theorem. Indeed, sinceT ^{*} is polaroid, the result holds by [5, Theorem 3.12] 
If
T is algebraicallyk quasiclassA (s ,t ), thenf (T ^{*}) satisfies generalized aWeyl’s theorem. Indeed, sinceT is polaroid, the result holds by [5, Theorem 3.12].

Since
Theorem 2.3

generalized Weyl’s theorem holds for ${d}_{A,B}^{*}$ . 
generalized Weyl’s theorem holds for d _{A,B}. 
Weyl’s theorem holds for d _{A,B}.
Theorem 2.4

If A ,B ^{*}are kquasiclass A (s ,t )operators ,then generalized aWeyl’s theorem holds for d _{A,B}. 
If A ,B ^{*}are kquasiclass A (s ,t )operators ,then generalized aWeyl’s theorem holds for ${d}_{A,B}^{*}$ .
Theorem 2.5

Weyl’s theorem holds for d _{A,B}if and only if generalized Weyl’s theorem holds for d _{A,B}. 
aWeyl’s theorem holds for d _{A,B}if and only if generalized aWeyl’s theorem holds for d2 _{A,B}.
Corollary 2.4

If A ,B ^{*}are kquasiclass A (s ,t )operators ,then Weyl’s theorem ,aWeyl’s theorem ,generalized Weyl’s theorem and generalized aWeyl’s theorem hold for d _{A,B}and are equivalent. 
If A ,B ^{*}are kquasiclass A (s ,t )operators ,then Weyl’s theorem ,aWeyl’s theorem ,generalized Weyl’s theorem and generalized aWeyl’s theorem hold for ${d}_{A,B}^{*}$ and are equivalent.
Let
Theorem 2.6

If A ,B ^{*}are kquasiclass A (s ,t )operators ,then f (d _{A,B})satisfies Weyl’s theorem ,aWeyl’s theorem ,generalized Weyl’s theorem and generalized aWeyl’s theorem. 
If A ,B ^{*}are kquasiclass A (s ,t )operators ,then $f\left({d}_{A,B}^{*}\right)$ satisfies Weyl’s theorem ,aWeyl’s theorem ,generalized Weyl’s theorem and generalized aWeyl’s theorem.
Corollary 2.16 may be extended as follows.
Theorem 2.7

If A ,B ^{*}are kquasiclass A (s ,t )operators ,then Weyl’s theorem ,aWeyl’s theorem ,generalized Weyl’s theorem and generalized aWeyl’s theorem hold for f (d _{A,B})and are equivalent. 
If A ,B ^{*}are kquasiclass A (s ,t )operators ,then Weyl’s theorem ,aWeyl’s theorem ,generalized Weyl’s theorem and generalized aWeyl’s theorem hold for $f\left({d}_{A,B}^{*}\right)$ and are equivalent.
Remark 2.2

Since a quasiclass
A operator isk quasiclassA (s ,t ), hence all Weyl’s theorems (generalized or not) hold for algebraically quasiclassA operators and are equivalent by Theorem 2.8, Corollary 2.9 and Remark 2.10. This subsumes and extends [1, Theorem 2.4 and Theorem 3.3]. 
Since a class
A (s ,t ) operator isk quasiclassA (s ,t ), hence Theorem 2.8, Corollary 2.9 and Remark 2.10 generalize the results on Weyl type theorems for classA (s ,t ) operators proved in [11, 41]. 
Our results on Weyl type theorems for
d _{A,B} generalize a recent result on generalized Weyl’s theorem ford _{A,B} whenA andB ^{*} are classA operators [24]. Also, since a quasiclassA operators is ak quasiclassA (s ,t ), hence all Weyl’s theorems (generalized or not) hold for algebraically quasiclassA and algebraicallyw hyponormal operators and are equivalent by Theorem 2.15, Theorem 2.17 and Theorem 2.18. This subsumes and extends [12, 16, 17]
Hyperinvariant Subspaces
The purpose of this section is to make a beginning on the hyperinvariant subspace problem for another class of operators closely related to the normal operators namely, the class of
Theorem 3.1
Assume that
Since a
Corollary 3.1
It is remarkable that simply knowing when solutions to
Lemma 3.1
([34, Rosemblum Theorem])
It is known that an invariant subspace for an operator
Theorem 3.2
It is clear that if
Thus ℳ is a nontrivial hyperinvariant subspace for
Corollary 3.2
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