A new characterization of generalized Browder's theorem and a Cline's formula for generalized Drazin-meromorphic inverses
aa r X i v : . [ m a t h . SP ] M a y A new characterization of generalized Browder’s theorem anda Cline’s formula for generalized Drazin-meromorphic inverses
Anuradha Gupta and Ankit Kumar Department of Mathematics, Delhi College of Arts and Commerce,University of Delhi, Netaji Nagar, New Delhi-110023, India.E-mail:[email protected] Department of Mathematics, University of Delhi, New Delhi-110007, India.E-mail [email protected]
Abstract
In this paper, we give a new characterization of generalized Browder’s theorem by consideringequality between the generalized Drazin-meromorphic Weyl spectrum and the generalized Drazin-meromorphic spectrum. Also, we generalize Cline’s formula to the case of generalized Drazin-meromorphic invertibility under the assumption that A k B k A k = A k +1 for some positive integer k . Mathematics Subject Classification:
Keywords:
SVEP, generalized Drazin-meromorphic invertible, meromorphic operators, operatorequation.
Throughout this paper, let N and C denote the set of natural numbers and complex numbers,respectively. Let B ( X ) denote the Banach algebra of all bounded linear operators acting on a complexBanach space X . For T ∈ B ( X ), we denote the spectrum of T , null space of T , range of T andadjoint of T by σ ( T ), ker( T ), R ( T ) and T ∗ , respectively. For a subset A of C the set of accumulationpoints of A is denoted by acc( A ). Let α ( T ) = dim ker( T ) and β ( T ) = codim R ( T ) be the nullity of T and deficiency of T , respectively. An operator T ∈ B ( X ) is called a lower semi-Fredholm operatorif β ( T ) < ∞ . An operator T ∈ B ( X ) is called an upper semi-Fredholm operator if α ( T ) < ∞ and R ( T ) is closed . The class of all lower semi-Fredholm operators (upper semi-Fredholm operators,respectively) is denoted by φ + ( X ) ( φ − ( X ), respectively). An operator T is called semi-Fredholm if itis upper or lower semi-Fredholm. For a semi-Fredholm operator T ∈ B ( X ), the index of T is definedby ind ( T ):= α ( T ) − β ( T ). The class of all Fredholm operators is defined by φ ( X ) := φ + ( X ) ∩ φ − ( X ).The class of all lower semi-Weyl operators (upper semi-Weyl operators, respectively) is defined by W − ( X ) = { T ∈ φ − ( X ) : ind ( T ) ≥ } ( W + ( X ) = { T ∈ φ + ( X ) : ind ( T ) ≤ } , respectively). Anoperator T ∈ B ( X ) is called Weyl if T ∈ φ ( X ) and ind ( T ) = 0. The lower semi-Fredholm , lower emi-Fredholm , Fredholm , lower semi-Weyl , upper semi-Weyl and Weyl spectra are defined by σ lf ( T ) := { λ ∈ C : λI − T is not lower semi-Fredholm } ,σ uf ( T ) := { λ ∈ C : λI − T is not upper semi-Fredholm } ,σ f ( T ) := { λ ∈ C : λI − T is not Fredholm } ,σ lw ( T ) := { λ ∈ C : λI − T is not lower semi-Weyl } ,σ uw ( T ) := { λ ∈ C : λI − T is not upper semi-Weyl } ,σ w ( T ) := { λ ∈ C : λI − T is not Weyl } , respectively . A bounded linear operator T is said to be bounded below if it is injective and R ( T ) is closed. For T ∈ B ( X ) the ascent denoted by p ( T ) is the smallest non negative integer p such that ker T p =ker T p +1 . If no such integer exists we set p ( T ) = ∞ . For T ∈ B ( X ) the descent denoted by q ( T )is the smallest non negative integer q such that R ( T q ) = R ( T q +1 ). If no such integer exists we set q ( T ) = ∞ . By [1, Theorem 1.20] if both p ( T ) and q ( T ) are finite then p ( T ) = q ( T ). An operator T ∈ B ( X ) is called left Drazin invertible if p ( T ) < ∞ and R ( T p +1 ) is closed. An operator T ∈ B ( X )is called right Drazin invertible if q ( T ) < ∞ and R ( T q ) is closed. Moreover, T is called Drazininvertible if p ( T ) = q ( T ) < ∞ . An operator T ∈ B ( X ) is called upper semi-Browder if it is anupper semi-Fredholm and p ( T ) < ∞ . An operator T ∈ B ( X ) is called lower semi-Browder if it is anlower semi-Fredholm and q ( T ) < ∞ . We say that an operator T ∈ B ( X ) is Browder if it is uppersemi-Browder and lower semi-Browder. The lower semi-Browder , upper semi-Browder and Browderspectra are defined by σ lb ( T ) : = { λ ∈ C : λI − T is not lower semi-Browder } ,σ ub ( T ) : = { λ ∈ C : λI − T is not upper semi-Browder } ,σ b ( T ) : = { λ ∈ C : λI − T is not Browder } , respectively . Clearly, every Browder operator is Drazin invertible.An operator T ∈ B ( X ) is said to possess the single-valued extension property (SVEP) at λ ∈ C if for every neighbourhood V of λ the only analytic function f : V → X which satisfies the equation( λI − T ) f ( λ ) = 0 is the function f = 0. If an operator T has SVEP at every λ ∈ C , then T is saidto have SVEP. Morever, the set of all points λ ∈ C such that T does not have SVEP at λ is an openset contained in interior of σ ( T ). Therefore, if T has SVEP at each point of an open punctured disc D \ { λ } centered at λ , T also has SVEP at λ . p ( λI − T ) < ∞ ⇒ T has SVEP at λ and q ( λI − T ) < ∞ ⇒ T ∗ has SVEP at λ. An operator T ∈ B ( X ) is called Riesz if λI − T is Browder for all λ ∈ C \ { } . An operator T ∈ B ( X )is called meromorphic if λI − T is Drazin invertible for all λ ∈ C \ { } . Clearly, every Riesz operatoris meromorphic. A subspace M of X is said to be T - invariant if T ( M ) ⊂ M . For a T -invariantsubspace M of X we define T M : M → M by T M ( x ) = T ( x ) , x ∈ M . We say T is completely reducedby the pair ( M, N ) (denoted by (
M, N ) ∈ Red ( T )) if M and N are two closed T -invariant subspacesof X such that X = M ⊕ N .An operator T ∈ B ( X ) is called semi-regular if R ( T ) is closed and ker( T ) ⊂ R ( T n ) for every n ∈ N . An operator T ∈ B ( X ) is called nilpotent if T n = 0 for some n ∈ N and called quasi-nilpotentif || T n || n →
0, i.e λI − T is invertible for all λ ∈ C \ { } .2or T ∈ B ( X ) and a non negative integer n , define T [ n ] to be the restriction of T to T n ( X ).If for some non negative integer n the range space T n ( X ) is closed and T [ n ] is Fredholm (a lowersemi B-Fredholm, an upper semi B-Fredholm, a lower semi B-Browder, an upper semi B-Browder,B-Browder, respectively) then T is said to be B-Fredholm (a lower semi B-Fredholm, an upper semiB-Fredholm, a lower semi B-Browder, an upper semi B-Browder, B-Browder, respectively). For asemi B-Fredholm operator T , (see [6]), the index of T is defined as index of T [ n ] . The lower semiB-Fredholm , upper semi B-Fredholm and B-Fredholm , lower semi B-Browder , upper semi B-Browder and B-Browder spectra are defined by σ lsbf ( T ) := { λ ∈ C : λI − T is not lower semi B-Fredholm } ,σ usbf ( T ) := { λ ∈ C : λI − T is not upper semi B-Fredholm } ,σ bf ( T ) := { λ ∈ C : λI − T is not B-Fredholm } ,σ lsbb ( T ) := { λ ∈ C : λI − T is not lower semi B-Browder } ,σ usbb ( T ) := { λ ∈ C : λI − T is not upper semi B-Browder } ,σ bb ( T ) := { λ ∈ C : λI − T is not B-Browder } , respectively.By [1, Theorem 3.47] an operator T ∈ B ( X ) is upper semi B-Browder (lower semi B-Browder,B-Browder, respectively) if and only if T is left Drazin invertible (right Drazin invertible, Drazininvertible, respectively).An operator T ∈ B ( X ) is called a lower semi B-Weyl (an upper semi B-Weyl, respectively) ifit is an lower semi B-Fredholm (an upper semi B-Fredholm, respectively) having ind ( T ) ≤ T ) ≥
0, respectively). An operator T ∈ B ( X ) is called B-Weyl if it is B-Fredholm and ind ( T ) = 0.The lower semi B-Weyl , upper semi B-Weyl and B-Weyl spectra are defined by σ lsbw ( T ) := { λ ∈ C : λI − T is not lower semi B-Weyl } ,σ usbw ( T ) := { λ ∈ C : λI − T is not upper semi B-Weyl } ,σ bw ( T ) := { λ ∈ C : λI − T is not B-Weyl } , respectively.It is known that (see [6, Theorem 2.7]) T ∈ B ( X ) is B-Fredholm (B-Weyl, respectively) if there exists( M, N ) ∈ Red ( T ) such that T M is Fredholm (Weyl, respectively) and T N is nilpotent. Recently,(see [15, 17]) have generalized the class of B-Fredholm and B-Weyl operators and introduced theconcept of pseudo B-Fredholm and pseudo B-Weyl operators. An operator T ∈ B ( X ) is said to bepseudo B-Fredholm (pseudo B-Weyl, respectively) if there exists ( M, N ) ∈ Red ( T ) such that T M isFredholm (Weyl, respectively) and T N is quasi-nilpotent. The pseudo B-Fredholm and pseudo B-Weylspectra are defined by σ pBf ( T ) := { λ ∈ C : λI − T is not pseudo B-Fredholm } ,σ pBw ( T ) := { λ ∈ C : λI − T is not pseudo B-Weyl } , respectively.An operator T is said to admit a generalized kato decomposition ( GKD ), if there exists a pair(
M, N ) ∈ Red ( T ) such that T M is semi-regular and T N is quasi-nilpotent. In the above definition ifwe assume T N to be nilpotent, then T is said to be of Kato Type. (See [14]) An operator is said toadmit a Kato-Riesz decomposition ( GKRD ), if there exists a pair (
M, N ) ∈ Red ( T ) such that T M issemi-regular and T N is Riesz.Recently, ˇZivkovi´c-Zlatanovi´c and Duggal [16] introduced the notion of generalized Kato-meromorphicdecomposition. An operator T ∈ B ( X ) is said to admit a generalized Kato-meromorphic decompo-sition ( GKM D ), if there exists a pair (
M, N ) ∈ Red ( T ) such that T M is semi-regular and T N ismeromorphic. For T ∈ B ( X ) the generalized Kato-meromorphic spectrum is defined by σ gKM ( T ) := { λ ∈ C : λI − T does not admit a GKMD } . T ∈ B ( X ) is said to be Drazin invertible if there exists S ∈ B ( X ) such that T S = ST , ST S = S and T ST − T is nilpotent. This definition is equivalent to the fact that there existof a pair ( M, N ) ∈ Red ( T ) such that T M is invertible and T N is nilpotent. Koliha [13] generalizedthis concept by replacing the third condition with T ST − T is quasi-nilpotent. An operator is saidto be generalized Drazin invertible if there exist a pair ( M, N ) ∈ Red ( T ) such that T M is invertibleand T N is quasi-nilpotent. The generalized Drazin spectrum is defined by σ gD ( T ) := { λ ∈ C : λI − T is not generalized Drazin invertible } . Recently, ˇZivkovi´c-Zlatanovi´c and Cvetkovi´c [14] introduced the concept of generalized Drazin-Rieszinvertible by replacing the third condition with
T ST − T is Riesz. They proved that the an operator T ∈ B ( X ) is generalized Drazin-Riesz invertible if and only if there exists a pair ( M, N ) ∈ Red ( T )such that T M is invertible and T N is Riesz. An operator T ∈ B ( X ) is called generalized Drazin-Riesz bounded below (surjective, respectively) if there exists a pair ( M, N ) ∈ Red ( T ) such that T M is bounded below (surjective, respectively) and T N is Riesz. The generalized Drazin-Riesz boundedbelow , generalized Drazin-Riesz surjective and generalized Drazin-Riesz invertible spectra are definedby σ gDR J ( T ) := { λ ∈ C : λI − T is not generalized Drazin-Riesz bounded below } ,σ gDR Q ( T ) := { λ ∈ C : λI − T is not generalized Drazin-Riesz surjective } ,σ gDR ( T ) := { λ ∈ C : λI − T is not generalized Drazin-Riesz invertible } , respectively.Also, they introduced the notion of operators which are direct sum of a Riesz and a Fredholm(lower (upper) semi-Fredholm, lower (upper) semi-Weyl, Weyl). An operator is called general-ized Drazin-Riesz Fredholm (generalized Drazin-Riesz lower (upper) semi-Fredholm, generalizedDrazin-Riesz lower (upper) semi-Weyl, generalized Drazin-Riesz Weyl, respectively) if there exists( M, N ) ∈ Red ( T ) such that T M is Fredholm (lower (upper) semi-Fredholm, lower (upper) semi-Weyl,Weyl, respectively) and T N is Riesz. The The generalized Drazin-Riesz lower (upper) semi-Fredholm , generalized Drazin-Riesz Fredholm , generalized Drazin-Riesz upper(lower) semi-Weyl and generalizedDrazin-Riesz Weyl spectra , are defined by σ gDRφ − ( T ) := { λ ∈ C : λI − T is not generalized Drazin-Riesz lower semi-Fredholm } ,σ gDRφ + ( T ) := { λ ∈ C : λI − T is not generalized Drazin-Riesz upper semi-Fredholm } ,σ gDRφ ( T ) := { λ ∈ C : λI − T is not generalized Drazin-Riesz Fredholm } ,σ gDRW − ( T ) := { λ ∈ C : λI − T is not generalized Drazin-Riesz lower semi-Weyl } ,σ gDRW + ( T ) := { λ ∈ C : λI − T is not generalized Drazin-Riesz upper semi-Weyl } ,σ gDRW ( T ) := { λ ∈ C : λI − T is not generalized Drazin-Riesz Weyl } , respectively.Also, ˇZivkovi´c-Zlatanovi´c and Duggal [16] introduced the notion of generalized Drazin-meromorphicinvertible by replacing the third condition with T ST − T is meromorphic. They proved that the anoperator T ∈ B ( X ) is generalized Drazin-meromorphic invertible if and only if there exists a pair( M, N ) ∈ Red ( T ) such that T M is invertible and T N is meromorphic. An operator T ∈ B ( X ) is saidto be generalized Drazin-meromorphic bounded below (surjective, respectively) if there exists a pair( M, N ) ∈ Red ( T ) such that T M is bounded below (surjective, respectively) and T N is meromorphic.The generalized Drazin-meromorphic bounded below , generalized Drazin-meromorphic surjective and generalized Drazin-meromorphic invertible spectra are defined by σ gDM J ( T ) := { λ ∈ C : λI − T is not generalized Drazin-meromorphic bounded below } σ gDM Q ( T ) := { λ ∈ C : λI − T is not generalized Drazin-meromorphic surjective } ,σ gDM ( T ) := { λ ∈ C : λI − T is not generalized Drazin-meromorphic invertible } , respectively.4lso, they introduced the notion of operators which are direct sum of a meromorphic and Fred-holm (lower (upper) semi-Fredholm, lower (upper) semi-Weyl, Weyl). An operator is called general-ized Drazin-meromorphic Fredholm (generalized Drazin-meromorphic lower (upper) semi-Fredholm,generalized Drazin-meromorphic lower (upper) semi-Weyl, generalized Drazin-meromorphic Weyl)if there exists ( M, N ) ∈ Red ( T ) such that T M is Fredholm (lower (upper) semi-Fredholm, lower(upper) semi-Weyl, Weyl) and T N is Riesz. The generalized Drazin-meromorphic lower (upper) semi-Fredholm , generalized Drazin-meromorphic Fredholm , generalized Drazin-meromorphic lower (upper)semi-Weyl and generalized Drazin-meromorphic Weyl spectra are defined by σ gDMφ − ( T ) := { λ ∈ C : λI − T is not generalized Drazin-meromorphic lower semi-Fredholm } ,σ gDMφ + ( T ) := { λ ∈ C : λI − T is not generalized Drazin-meromorphic upper semi-Fredholm } ,σ gDMφ ( T ) := { λ ∈ C : λI − T is not generalized Drazin-meromorphic Fredholm } ,σ gDMW − ( T ) := { λ ∈ C : λI − T is not generalized Drazin-meromorphic lower semi-Weyl } ,σ gDMW + ( T ) := { λ ∈ C : λI − T is not generalized Drazin-meromorphic upper semi-Weyl } ,σ gDMW ( T ) := { λ ∈ C : λI − T is not generalized Drazin-meromorphic Weyl } , respectively.From [14, 16] we have σ gD ∗ φ ( T ) = σ gD ∗ φ + ( T ) ∪ σ gD ∗ φ − ( T ) ,σ gK ∗ ( T ) ⊂ σ gD ∗ φ + ( T ) ⊂ σ gD ∗ W + ( T ) ⊂ σ gD ∗J ( T ) ,σ gK ∗ ( T ) ⊂ σ gD ∗ φ − ( T ) ⊂ σ gD ∗ W − ( T ) ⊂ σ gD ∗Q ( T ) ,σ gK ∗ ( T ) ⊂ σ gD ∗ φ ( T ) ⊂ σ gD ∗ W ⊂ σ gD ∗ ( T ) , where ∗ stands for Riesz or meromorphic operators.Recall that an operator T satisfies Browder’s theorem if σ b ( T ) = σ w ( T ) and generalized Brow-der’s theorem if σ bb ( T ) = σ bw ( T ). Amouch et al. [7] and Karmouni and Tajmouati [12] gave anew characterization of Browder’s theorem using spectra arised from Fredholm theory and Drazininvertibilty. Motivated by them, we give a new characterization of operators satisfying generalizedBrowder’s theorem. We prove that an operator T satisfies generalized Browder’s theorem if andonly if σ gDMW ( T ) = σ gDM ( T ). In the last section, we generalize the Cline’s formula for the case ofgeneralized Drazin-meromorphic invertibility under the assumption that A k B k A k = A k +1 for somepositive integer k . The following result will be used in the sequel:
Theorem 2.1. [16, Theorem 2.1]
Let T ∈ B ( X ) , then T is generalized Drazin-meromorphic uppersemi-Weyl (lower semi-Weyl, upper semi-Fredholm, lower semi-Fredholm, Weyl, respectively) if andonly if T admits a GKM D and / ∈ acc σ usbw ( T ) (acc σ lsbw ( T ) , acc σ usbf ( T ) , acc σ lsbf ( T ) , acc σ bw ( T ) ,respectively). Proposition 2.2.
Let T ∈ B ( X ) , then σ gDM J ( T ) = σ gDMW + ( T ) if and only if T has SVEP at every λ / ∈ σ gDMW + ( T ) .Proof. Suppose that σ gDM J ( T ) = σ gDMW + ( T ). Let λ / ∈ σ gDMW + ( T ), then λI − T generalized Drazin-meromorphic bounded below. Therefore, by [16, Theorem 2.5] T has SVEP at λ . Conversely, supposethat T has SVEP at every λ / ∈ σ gDMW + ( T ). It suffices to show that σ gDM J ( T ) ⊂ σ gDMW + ( T ). Let5 / ∈ σ gDMW + ( T ) which implies that λI − T is generalized Drazin-meromorphic upper semi-Weyl.Therefore, by Theorem 2.1 λI − T admits a GKM D . Thus, there exists (
M, N ) ∈ Red ( λI − T )such that ( λI − T ) M is semi-regular and ( λI − T ) N is meromorphic. Since T has SVEP at every λ / ∈ σ gDMW + ( T ), ( λI − T ) has SVEP at 0. As SVEP at a point is inherited by the restrictions onclosed invariant subspaces, ( λI − T ) M has SVEP at 0. Therefore, by [1, Theorem 2.91] ( λI − T ) M is bounded below. Thus, by [16, Theorem 2.6] we have λI − T is generalized Drazin-meromorphicbounded below. Hence, λ / ∈ σ gDM J ( T ). Proposition 2.3.
Let T ∈ B ( X ) , then σ gDM Q ( T ) = σ gDMW − ( T ) if and only if T ∗ has SVEP atevery λ / ∈ σ gDMW − ( T ) .Proof. Suppose that σ gDM Q ( T ) = σ gDMW − ( T ). Let λ / ∈ σ gDMW − ( T ), then λI − T generalized Drazin-meromorphic surjective. Therefore, by [16, Theorem 2.6] T ∗ has SVEP at λ . Conversely, supposethat T ∗ has SVEP at every λ / ∈ σ gDMW − ( T ). It suffices to show that σ gDM Q ( T ) ⊂ σ gDMW − ( T ).Let λ / ∈ σ gDMW − ( T ) which implies that λI − T is generalized Drazin-meromorphic lower semi-Weyl.Then by Theorem 2.1 λI − T admits a GKM D and λ / ∈ acc σ lsbw ( T ). Since T ∗ has SVEP atevery λ / ∈ σ gDMW − ( T ) and σ gDMW − ( T ) ⊂ σ lw ( T ) then T ∗ has SVEP at every λ / ∈ σ lw ( T ) = σ uw ( T ∗ ).Therefore, by [1, Theorem 5.27] we have σ lw ( T ) = σ uw ( T ∗ ) = σ ub ( T ∗ ) = σ lb ( T ). Thus, by [1, Theorem5.38] we have σ lsbw ( T ) = σ lsbb ( T ). This implies that λ / ∈ acc σ lsbb ( T ). Therefore, by [16, Theorem 2.6] λI − T is generalized Drazin-meromorprhic surjective and it follows that λ / ∈ σ gDM Q ( T ). Corollary 2.4.
Let T ∈ B ( X ) , then σ gDM ( T ) = σ gDMW ( T ) if and only if T and T ∗ have SVEP atevery λ / ∈ σ gDMW ( T ) .Proof. Suppose that σ gDM ( T ) = σ gDMW ( T ) . Let λ / ∈ σ gDMW ( T ), then λI − T is generalized Drazin-meromorphic invertible. Therefore, by [16, Theorem 2.4] T and T ∗ have SVEP at λ . Conversely, let λ / ∈ σ gDMW ( T ) = σ gDMW + ( T ) ∪ σ gDMW − ( T ). Then by proofs of Theorem 2.2 and Theorem 2.3 wehave λ / ∈ σ gDM J ( T ) ∪ σ gDM Q ( T ) = σ gDM ( T ) . Theorem 2.5.
Let T ∈ B ( X ) , then following statements are equivalent:(i) σ gDM ( T ) = σ gDMW ( T ) ,(ii) T or T ∗ have SVEP at every λ / ∈ σ gDMW ( T ) .Proof. Suppose that T has SVEP at every λ / ∈ σ gDRW ( T ). It suffices to prove that σ gDM ( T ) ⊂ σ gDMW ( T ). Let λ / ∈ σ gDMW ( T ) then λI − T admits a GKM D and λ / ∈ acc σ bw ( T ). Since σ gDRW ( T ) ⊂ σ bw ( T ), T has SVEP at every λ / ∈ σ bw ( T ). Therefore, σ bw ( T ) = σ bb ( T ). Thus, λ / ∈ acc σ bb ( T ) whichimplies that λI − T is generalized Drazin-meromorphic invertible.Now suppose that T ∗ has SVEP at every λ / ∈ σ gDRW ( T ). Since σ bb ( T ) = σ bb ( T ∗ ) and σ bw ( T ) = σ bw ( T ∗ ) we have σ gDR ( T ) = σ gDRW ( T ). The converse is an immediate consequence of Corollary2.4.Recall that an operator T ∈ B ( X ) is said satisfy generalized a-Browder’s theorem if σ usbb ( T ) = σ usbw ( T ). An operator T ∈ B ( X ) satisfies a-Browder’s theorem if σ ub ( T ) = σ uw ( T ). By [4, Theorem2.2] we know that a-Browder’s theorem is equivalent to generalized a-Browder’s theorem. Theorem 2.6.
Let T ∈ B ( X ) , then the following holds:(i) generalized a-Browder’s theorem holds for T if and only if σ gDM J ( T ) = σ gDMW + ( T ) ,(ii) generalized a-Browder’s theorem holds for T ∗ if and only if σ gDM Q ( T ) = σ gDMW − ( T ) ,(iii)generalized Browder’s theorem holds for T if and only if σ gDM ( T ) = σ gDMW ( T ) . roof. (i) Suppose that generalized a-Browder’s theorem holds for T which implies that σ usbb ( T ) = σ usbw ( T ). It suffices to prove that σ gDM J ( T ) ⊂ σ gDMW + ( T ). Let λ / ∈ σ gDMW + ( T ), then λI − T is generalized Drazin-meromorphic upper semi-Weyl. By Theorem 2.1 it follows that λI − T ad-mits a GKM D and λ / ∈ acc σ usbw ( T ). This gives λ / ∈ acc σ usbb ( T ). Therefore, by [16, Theorem 2.5] λI − T is generalized Drazin-meromorphic bounded below which gives λ / ∈ σ gDM J ( T ). Conversely,suppose that σ gDM J ( T ) = σ gDMW + ( T ). Using Proposition 2.2 we deduce that T has SVEP at every λ / ∈ σ gDMW + ( T ). Since σ gDMW + ( T ) ⊂ σ uw ( T ), T has SVEP at every λ / ∈ σ uw ( T ). By [1, Theorem5.27] T satisfies a-Browder’s theorem. Therefore, generalized a-Browder’s theorem holds for T .(ii) Suppose that generalized a-Browder’s theorem holds for T ∗ which implies that σ lsbb ( T ) = σ lsbw ( T ).It suffices to prove that σ gDM Q ( T ) ⊂ σ gDMW − ( T ). Let λ / ∈ σ gDMW − ( T ), then λI − T is gen-eralized Drazin-meromorphic lower semi-Weyl. By Theorem 2.1 it follows that λI − T admitsa GKM D and λ / ∈ acc σ lsbw ( T ). This gives λ / ∈ acc σ lsbb ( T ). Therefore, by [16, Theorem 2.6] λI − T is generalized Drazin-meromorphic surjective which gives λ / ∈ σ gDM Q ( T ). Conversely, sup-pose that σ gDM Q ( T ) = σ gDMW − ( T ). Using Proposition 2.3 we deduce that T ∗ has SVEP at every λ / ∈ σ gDMW − ( T ). Since σ gDMW − ( T ) ⊂ σ lw ( T ), T ∗ has SVEP at every λ / ∈ σ lw ( T ) = σ uw ( T ∗ ). There-fore, generalized a-Browder’s theorem holds for T ∗ .(iii) Suppose that generalized Browder’s theorem holds for T which implies that σ bb ( T ) = σ bw ( T ). Itsuffices to prove that σ gDM ( T ) ⊂ σ gDMW ( T ). Let λ / ∈ σ gDMW ( T ), then λI − T is generalized Drazin-meromorphic Weyl. By Theorem 2.1 it follows that λI − T admits a GKM D and λ / ∈ acc σ bw ( T ).This gives λ / ∈ acc σ bb ( T ). Therefore, by [16, Theorem 2.4] λI − T is generalized Drazin-meromorphicinvertible which gives λ / ∈ σ gDM ( T ). Conversely, suppose that σ gDM ( T ) = σ gDMW ( T ). Using Propo-sition 2.4 we deduce that T and T ∗ have SVEP at every λ / ∈ σ gDMW ( T ). Since σ gDMW ( T ) ⊂ σ bw ( T ), T and T ∗ have SVEP at every λ / ∈ σ bw ( T ). Therefore, by [1, Theorem 5.14] generalized Browder’stheorem holds for T .Using Theorem 2.6, [2, Theorem 2.3], [4, Theorem 2.1], [5, Proposition 2.2] and [12, Theorem 2.6]we have the following theorem: Theorem 2.7.
Let T ∈ B ( X ) , then the following statements are equivalent:(i) Browder’s theorem holds for T ,(ii) Browder’s theorem holds for T ∗ ,(iii) T has SVEP at every λ / ∈ σ w ( T ) ,(iv) T ∗ has SVEP at every λ / ∈ σ w ( T ) .(v) T has SVEP at every λ / ∈ σ bw ( T ) .(vi) generalized Browder’s theorem holds for T .(vii) T or T ∗ has SVEP at every λ / ∈ σ gDRW ( T ) . (viii) σ gDR ( T ) = σ gDRW ( T ) ,(ix) T or T ∗ has SVEP at every λ / ∈ σ gDMW ( T ) ,(x) σ gDM ( T ) = σ gDMW ( T ) ,(xi) σ gD ( T ) = σ pBW ( T ) . Using [4, Theorem 2.2] and [12, Theorem 2.7] a similar result for a-Browder’s theorem can bestated as follows:
Theorem 2.8.
Let T ∈ B ( X ) , then the following statements are equivalent:(i) a-Browder’s theorem holds for T ,(ii) generalized a-Browder’s theorem holds for T ,(iii) T has SVEP at every λ / ∈ σ gDRW + ( T ) ,(iv) σ gDR J ( T ) = σ gDRW + ( T ) , v) T has SVEP at every λ / ∈ σ gDMW + ( T ) ,(vi) σ gDM J ( T ) = σ gDMW + ( T ) . Lemma 2.9.
Let T ∈ B ( X ) , then(i) σ uf ( T ) = σ ub ( T ) ⇔ σ usbf ( T ) = σ usbb ( T ) ,(ii) σ lf ( T ) = σ lb ( T ) ⇔ σ lsbf ( T ) = σ lsbb ( T ) .Proof. (i) Let σ uf ( T ) = σ ub ( T ). It suffices to show that σ usbb ( T ) = σ usbf ( T ). Let λ / ∈ σ usbf ( T ). Then λ I − T is upper semi B-Fredholm. Therefore, by [1, Theorem 1.117] there exists an open disc D centered at λ such that λI − T is upper semi-Fredholm for all λ ∈ D \ { λ } . Since σ uf ( T ) = σ ub ( T ), λI − T is upper semi-Browder for all λ ∈ D \ { λ } . Therefore, p ( λI − T ) < ∞ for all λ ∈ D \ { λ } .Thus, T has SVEP at every λ ∈ D \ { λ } which gives T has SVEP at λ . Thus, by [3, Theorem2.5] it follows that λ / ∈ σ usbb ( T ). Conversely, let σ usbb ( T ) = σ usbf ( T ). It suffices to show that σ ub ( T ) ⊂ σ uf ( T ). Let λ / ∈ σ uf ( T ). Then λ / ∈ σ usbf ( T ) = σ usbb ( T ). Therefore, p ( λI − T ) < ∞ whichimplies that λ / ∈ σ ub ( T ). (ii) Using a similar argument as above we can get the desired result. Theorem 2.10.
Let T ∈ B ( X ) , then the following statements are equivalent:(i) σ usbf ( T ) = σ usbb ( T ) ,(ii) T has SVEP at every λ / ∈ σ usbf ( T ) ,(iii) T has SVEP at every λ / ∈ σ gDMφ + ( T ) ,(iv) σ gDM J ( T ) = σ gDMφ + ( T ) .Proof. (i) ⇔ (ii) Suppose that σ usbf ( T ) = σ usbb ( T ). Let λ / ∈ σ usbf ( T ), then λ / ∈ σ usbb ( T ) which gives p ( λI − T ) < ∞ . Therefore, T has SVEP at λ . Now suppose that T has SVEP at every λ / ∈ σ usbf ( T ).It suffices to prove that σ usbb ( T ) ⊂ σ usbf ( T ). Let λ / ∈ σ usbf ( T ), then λI − T is upper semi B-Fredholmoperator. Since T has SVEP at λ then by [3, Theorem 2.5] it follows that λ / ∈ σ usbb ( T ).(iii) ⇔ (iv) Suppose that T has SVEP at every λ / ∈ σ gDMφ + ( T ) which implies that λI − T is generalizedDrazin-meromorphic upper semi-Fredholm. It suffices to show that σ gDM J ( T ) ⊂ σ gDMφ + ( T ). Let λ / ∈ σ gDMφ + ( T ), then by Theorem 2.1 there exists ( M, N ) ∈ Red ( λI − T ) such that ( λI − T ) M issemi-regular and ( λI − T ) N is meromorphic. Since T has SVEP at λ , ( λI − T ) M has SVEP at 0.Therefore, by [1, Theorem 2.91] ( λI − T ) M is bounded below. Thus, λ / ∈ σ gDM J ( T ). Conversely,suppose that σ gDM J ( T ) = σ gDMφ + ( T ). Let λ / ∈ σ gDRφ + ( T ), then λI − T is generalized Drazin-meromorphic bounded below. Therefore, by [16, Theorem 2.5] it follows that T has SVEP at λ .(i) ⇔ (iv) Suppose that σ usbf ( T ) = σ usbb ( T ). It suffices to prove that σ gDM J ( T ) ⊂ σ gDMφ + ( T ). Let λ / ∈ σ gDMφ + ( T ), then λI − T is generalized Drazin-meromorphic upper semi-Fredholm. By Theorem2.1 it follows that λI − T admits a GKM D and λ / ∈ acc σ usbf ( T ). This gives λ / ∈ acc σ usbb ( T ).Therefore, by [16, Theorem 2.5] λI − T is generalized Drazin-meromorphic bounded below whichgives λ / ∈ σ gDM J ( T ). Conversely, suppose that σ gDM J ( T ) = σ gDMφ + ( T ). Then by (iv) ⇒ (iii) T has SVEP at every λ / ∈ σ gDMφ + ( T ). Since σ gDMφ + ( T ) ⊂ σ uf ( T ), T has SVEP at every λ / ∈ σ uf ( T ).Therefore, by [12, Theorem 2.8] we have σ uf = σ ub ( T ). Thus, by Lemma 2.9 σ usbf = σ usbb ( T ). Theorem 2.11.
Let T ∈ B ( X ) , then the following statements are equivalent:(i) σ lsbf ( T ) = σ lsbb ( T ) ,(ii) T ∗ has SVEP at every λ / ∈ σ lsbf ( T ) ,(iii) T ∗ has SVEP at every λ / ∈ σ gDMφ − ( T ) ,(iv) σ gDM Q ( T ) = σ gDMφ − ( T ) .Proof. (i) ⇔ (ii) Suppose that σ lsbf ( T ) = σ lsbb ( T ). Let λ / ∈ σ lsbf ( T ), then λ / ∈ σ lsbb ( T ) which gives q ( λI − T ) < ∞ . Therefore, T ∗ has SVEP at λ . Now suppose that T ∗ has SVEP at every λ / ∈ σ lsbf ( T ).It suffices to prove that σ lsbb ( T ) ⊂ σ lsbf ( T ). Let λ / ∈ σ lsbf ( T ), then λI − T is lower semi B-Fredholm8perator. Since T ∗ has SVEP at λ then by [3, Theorem 2.5] we have λ / ∈ σ lsbb ( T ).(iii) ⇔ (iv) Suppose that T ∗ has SVEP at every λ / ∈ σ gDMφ − ( T ) which implies that λI − T is general-ized Drazin-meromorphic lower semi-Fredholm. It suffices to show that σ gDM Q ( T ) ⊂ σ gDMφ − ( T ). ByTheorem 2.1 it follows that λI − T admits a GKM D and λ / ∈ acc σ lsbf ( T ). Since σ gDMφ − ( T ) ⊂ σ lf ( T ), T ∗ has SVEP at every λ / ∈ σ lf ( T ). Therefore, by [12, Theorem 2.9] we have σ lf = σ lb ( T ). Thus, byLemma 2.9 b σ lsbf = σ lsbb ( T ) which implies that λ / ∈ acc σ lsbb ( T ). Hence, λ / ∈ σ gDM Q ( T ) . Conversely,suppose that σ gDM Q ( T ) = σ gDMφ − ( T ). Let λ / ∈ σ gDMφ − ( T ), then λI − T is generalized Drazin-meromorphic surjective. Therefore by [16, Theorem 2.6] it follows that T ∗ has SVEP at λ .(i) ⇔ (iv) Suppose that σ lsbf ( T ) = σ lsbb ( T ). It suffices to prove that σ gDM Q ( T ) ⊂ σ gDMφ − ( T ). Let λ / ∈ σ gDMφ − ( T ), then λI − T is generalized Drazin-meromorphic lower semi-Fredholm. By Theo-rem 2.1 it follows that λI − T admits a GKM D and λ / ∈ acc σ lsbf ( T ). This gives λ / ∈ acc σ lsbb ( T ).Therefore, by [16, Theorem 2.6] λI − T is generalized Drazin-meromorphic surjective which gives λ / ∈ σ gDM Q ( T ). Conversely, suppose that σ gDM Q ( T ) = σ gDMφ − ( T ). Then by (iv) ⇒ (iii) T ∗ hasSVEP at every λ / ∈ σ gDMφ − ( T ). Since σ gDMφ − ( T ) ⊂ σ lf ( T ) , T ∗ has SVEP at every λ / ∈ σ lf ( T ). Thisgives σ lsbf ( T ) = σ lsbb ( T ).Using [12, Corollary 2.10] and Theorems 2.10, 2.11 we have the following result: Corollary 2.12.
Let T ∈ B ( X ) , then the following statements are equivalent:(i) σ f ( T ) = σ b ( T ) ,(ii) T and T ∗ have SVEP at every λ / ∈ σ f ( T ) ,(iii) σ bf ( T ) = σ bb ( T ) ,(iv) T and T ∗ have SVEP at every λ / ∈ σ bf ( T ) ,(v) σ gD ( T ) = σ pbf ( T ) ,(vi) T and T ∗ have SVEP at every λ / ∈ σ pbf ( T ) ,(viii) σ gDR ( T ) = σ gDRφ ( T ) ,(viii) T and T ∗ have SVEP at every λ / ∈ σ gDRφ ( T ) ,(ix) σ gDM ( T ) = σ gDMφ ( T ) ,(x) T and T ∗ have SVEP at every λ / ∈ σ gDMφ ( T ) . Let R be a ring with identity. Drazin [9] introduced the concept of Drazin inverses in a ring. Anelement a ∈ R is said to be Drazin invertible if there exist an element b ∈ R and r ∈ N such that ab = ba, bab = b, a r +1 b = a r . If such b exists then it is unique and is called Drazin inverse of a and denoted by a D . For a, b ∈ R ,Cline [8] proved that if ab is Drazin invertible, then ba is Drazin invertible and ( ba ) D = b (( ab ) D ) a .Recently, Gupta and Kumar [10] generalized Cline’s formula for Drazin inverses in a ring with identityto the case when a k b k a k = a k +1 for some k ∈ N and obtained the following result: Theorem 3.1. ( [10, Theorem 2.20])
Let R be a ring with identity and suppose that a k b k a k = a k +1 for some k ∈ N . Then a is Drazin invertible if and only if b k a k is Drazin invertible. Moreover, ( b k a k ) D = b k ( a D ) a k and a D = a k ( b k a k ) D ) k +1 . Recently, Karmouni and Tajmouati [11] investigated for bounded linear operators
A, B, C satisfy-ing the operator equation
ABA = ACA and obtained that AC is generalized Drazin-Riesz invertibleif and only if BA is generalized Drazin-Riesz invertible. Also, they generalized Cline’s formula to9he case of generalized Drazin-Riesz invertibility. In this section, we establish Cline’s formula forthe generalized Drazin- Riesz invertibility for bounded linear operators A and B under the condi-tion A k B k A k = A k +1 . By [10, Theorem 2.1, Theorem 2.2, Proposition 2.4 and Lemma 2.1] and aresult [1, Corollary 3.99] we can deduce the following result: Proposition 3.2.
Let
A, B ∈ B ( X ) satisfies A k B k A k = A k +1 for some k ∈ N , then A is meromor-phic if and only if B k A k is meromorphic. Theorem 3.3.
Suppose that
A, B ∈ B ( X ) and A k B k A k = A k +1 for some k ∈ N . Then A isgeneralized Drazin-meromorphic invertible if and only if B k A k is generalized Drazin-meromorphicinvertible.Proof. Suppose that A is generalized Drazin-meromorphic invertible, then there exists T ∈ B ( X )such that T A = AT, T AT = T and AT A − A is meromorphic . Let S = B k T A k . Then( B k A k ) S = ( B k A k )( B k T A k ) = B k ( A k B k A k ) T = B k A k +1 T = B k A k T and S ( B k A k ) = ( B k T A k )( B k A k ) = B k T A k +1 = B k A k T. Therefore, S ( B k A k ) = ( B k A k ) S . Consider S ( B k A k ) S = B k T A k ( B k A k ) B k T A k = ( B k T A k )( B k A k T ) = B k T A k +1 T = B k T A k = S. Let Q = I − AT , then Q is a bounded projection commuting with A . Therefore, Q n = Q for all n ∈ N . We observe that( QA ) k B k ( QA ) k = Q k A k B k Q k A k = Q k A k +1 Q k = Q k +1 A k +1 = ( QA ) k +1 and B k A k − ( B k A k ) S = B k A k − ( B k A k ) B k T A K = B k A k − B k ( A k B k A k ) B k T A k = B k A k − B k A k +2 T = B k ( I − A T ) A k = B k ( I − AT ) A k = B k QA k = B k Q k A k = B k ( QA ) k . Since QA is meromorphic and( QA ) k B k ( QA ) k = ( QA ) k +1 , by Proposition 3.2 B k A k − ( B k A k ) S ismeromorphic.Conversely, suppose that B k A k is generalized Drazin-meromorphic invertible. Then there exists T ′ ∈ B ( X ) such that T ′ B k A k = B k A k T ′ , T ′ B k A k T ′ = T ′ and B k A k T ′ B k A k − B k A k is meromorphic . Let S ′ = A k T ′ k +1 . Then S ′ A = A k T ′ k +1 A = A k T ′ k +2 B k A k A = A k T ′ k +2 B k A k +1 = A k T ′ k +2 ( B k A k ) = A k T ′ k and AS ′ = A k +1 T ′ k +1 = A k T ′ k . AS ′ = ( A k T ′ k +1 A ) A k T ′ k +1 = ( A k T ′ k ) A k T ′ k +1 = A k v k +1 B k A k T ′ k +1 = A k T ′ k +1 ( B k A k ) k +1 = S k +1 = A k T ′ k +1 = S ′ . We claim that for all n ∈ N we have( A − A S ′ ) n = ( A n − A n +1 S ′ ) . We prove it by induction. Evidently, the result is true for n = 1. Assume it to be true for n = p .Consider ( A − A S ′ ) p +1 = ( A − A S ′ )( A − A S ′ ) p = ( A − A S ′ )( A p − A p +1 S ′ )= A P +1 − A P +2 S ′ − A P +2 S ′ + A P +3 S ′ = A p +1 − A p +2 S ′ . Also, B k ( A − A S ′ ) k = B k ( A k − A k +1 S ′ ) = B k A k − B k A k − A S ′ = B k A k − B k A k − A k T ′ k − = B k A k − B k A k − T ′ k − = B k A k − ( B k A k ) k T ′ k − = B k A k − ( B k A k ) S ′ . Now consider( A − A S ′ ) k B k ( A − A S ′ ) k = ( A k − A k +1 S ′ ) B k ( A k − A k +1 S ′ )= A k B k A k − A k +1 S ′ B k A k − A k B k A k B k A k S ′ + A k +1 ( B k A k ) S ′ = A k +1 − A k +2 S ′ = ( A − A S ′ ) k +1 . Since B k ( A − A S ′ ) k = B k A k − ( B k A k ) T ′ is meromorphic, by Proposition 3.2 it follows that A − A S ′ is meromorphic. Acknowledgement
The second author is supported by Department of Science and Technology, New Delhi, India (GrantNo. DST/INSPIRE Fellowship/[IF170390]).
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