Left and right generalized Drazin invertible operators and Local spectral theory

In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property $(C)$, and property $(\beta)$, to its generalized Drazin inverse.


Introduction
The generalized Drazin inverse for operators arises naturally in the context of isolated spectral points and becomes a theoretical and practical tool in algebra and analysis (Markov chains, singular differential and difference equations, iterative methods...). The Drazin inverse was originally defined in 1958 for semigroups ([7]). When L(X) is the Banach algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space X, then S ∈ L(X) is the Drazin inverse of T ∈ L(X) if ST = T S ST S = S and T ST = T + U where U is a nilpotent operator. (1.1) The concept of Drazin invertible operators has been generalized by Koliha ([18]) by replacing the nilpotent operator U in (1.1) by a quasinilpotent operator. In this case, S is called a generalized Drazin inverse of T . Note that this extension was anticipated by Harte in [14]. Recently, in [24], the authors introduced the left and the right generalized Drazin invertible operators. These two classes of operators are a continuation and refinement of the research treatment of the Drazin inverse in Banach space operators theory. It proved that an operator T ∈ L(X) is left (resp. right) generalized Drazin inverse if and only if T = T 1 ⊕ T 2 where T 1 is bounded below (resp. surjective) and T 2 is quasinilpotent operator. Furthermore, these operators are characterized via the isolated points of the approximate point spectrum (resp. surjective spectrum) [24, Theorem 3.8; Theorem 3.10].
The main objective of this paper is to continue studying these operators via the local spectral theory. In Section 2, we give some preliminary results which our investigation will be need. In Section 3, we present many new and interesting characterizations of the left (resp. the right) generalized Drazin invertible operators in terms of the generalized Kato decomposition and the single-valued extension property. We also show that an operator admits a generalized Kato decomposition and has the SVEP at 0 is precisely left generalized Drazin invertible and conversely. Similarly, an operator T is right generalized Drazin invertible if and only if T admits a generalized Kato decomposition and its adjoint T * has the SVEP at 0. In particular, we prove that the left generalized Drazin spectrum and right generalized Drazin spectrum of a bounded operator are invariant under commuting finite rank perturbations. In section 4, we study the relationships between the local spectral properties of an operator and the local spectral properties of its generalized Drazin inverse, if this exists. In particular, a reciprocal relationship analogous to spectrum of invertible operator and its inverse, is established between the nonzero points of the local spectrum of a generalized Drazin invertible operator having SVEP and the nonzero points of the local spectrum of its generalized Drazin inverse. We also show that many local spectral properties, as SVEP, Dunford property (C), property (β), property (Q) and decomposability, are transferred from a generalized Drazin invertible operator to its generalized Drazin inverse. This section extends the results of [2] from the case of Drazin invertible operators to case of the generalized Drazin invertible operators. Finally, by a counterexample we show that these local spectral properties are not transferred in the case of the left (resp. the right) generalized Drazin invertible operators.

Preliminaries
Let L(X) be the Banach algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space X. For T ∈ L(X) write N(T ), R(T ), σ(T ) and ρ(T ) respectively, the null space, the range, the spectrum and the resolvent set of T . The nullity and the deficiency of T are defined respectively by α(T ) = dimN(T ) and β(T ) = dimX/R(T ). Here I denote the identity operator in X. By isoσ(T ) and accσ(T ) we define the set of all isolated and accumulation spectral points of T .
If M is a subspace of X then T M denote the restriction of T in M. Assume that M and N are two closed subspaces of X such that X = M ⊕N (that is H = M +N and M ∩N = 0). We say that T is completely reduced by the pair (M, N), denoted An operator is said to be bounded below if it is injective with closed range.
Recall that (see, e.g. [15]) the ascent a(T ) of an operator T ∈ L(X) is defined as the smallest nonnegative integer p such that N(T p ) = N(T p+1 ). If no such an integer exists, we set a(T ) = ∞. Analogously, the smallest nonnegative integer q such that R(T q ) = R(T q+1 ) is called the descent of T and denoted by d(T ). We set d(T ) = ∞ if for each q, R(T q+1 ) is a proper subspace of R(T q ). It is well known that if the ascent and the descent of an operator are finite, then they are equal.
Associated with an operator T ∈ L(X) there are two (not necessarily closed) linear subspaces of X invariant by T , played an important role in the development of the generalized Drazin inverse for T , the quasinilpotent part H 0 (T ) of T : and the analytical core K(T ) of T : there exist a sequence (x n ) in X and a constant δ > 0 such that T x 1 = x, T x n+1 = x n and x n ≤ δ n x for all n ∈ N}. It is well-known that necessary and sufficient condition for T ∈ L(X) to be generalized Drazin invertible is that 0 / ∈accσ(T ). Equivalently, K(T ) and H 0 (T ) are both closed, X = H 0 (T ) ⊕ K(T ), the restriction of T to H 0 (T ) is a quasinilpotent operator, and the restriction of T to K(T ) is invertible, Recently, by the use of this two subspaces, in [24], the authors defined and studied a new class of operators called left and right generalized Drazin invertible operators as a generalization of left and right Drazin invertible operators.  According to the Definitions 2.1 and 2.2, we also have Invertible operator =⇒ Generalized Drazin invertible operator =⇒ Right (resp. Left) generelazed Drazin invertible operator.
In the sequel the terms left (resp. right) generalized Drazin invertible operator is used for the nontrivial case of the bounded below (resp. surjective) operators.
The left Drazin spectrum, the right Drazin spectrum, the Drazin spectrum, the generalized Drazin spectrum, the left generalized Drazin spectrum and the right generalized Drazin spectrum of T are, respectively, defined by It is well known that these spectra are compact sets in the complex plane, and we have, σ su (T ) := {λ ∈ C : λI − T is not surjective}, are respectively the approximate point spectrum and the surjective spectrum of T .
An operator T ∈ L(X), T is said to be semi-regular if R(T ) is closed and N(T n ) ⊆ R(T ), for all n ∈ N. An important class of operators which involves the concept of semi-regularity is the class of operators admits a generalized Kato decomposition. If we assume in the definition above that T N is nilpotent, then there exists d ∈ N for which (T N ) d = 0. In this case T is said to be of Kato type operator of degree d. Examples of operators admits a generalized Kato decomposition, are Kato type operators, semi-regular operators, semi-Fredholm operators, quasi-Fredholm operators and generalized Drazin invertible operators, some other examples may be found in [20].
For operator T admits a generalized Kato decomposition we have the following properties of K(T ) and H 0 (T ).
Let M be a subspace of X and let X * be the dual space of X. As it is usual, is a GKD of its adjoint T * . Furthermore, if T is of a Kato type operator then T * is also of a Kato type.
For every operator T ∈ L(X), let us define the semi-regular spectrum, the Kato spectrum and the generalized Kato spectrum as follows: Kato decomposition} Recall that all the three sets defined above are always compact subsets of the complex plane, (see [1], [10]) and ordered by : Furthermore, the generalized Kato spectrum of an operator differs from the semiregular spectrum on at most countably many isolated points, more precisely the sets σ se (T ) \ σ gk (T ), σ se (T ) \ σ k (T ) and σ k (T ) \ σ gk (T ) are at most countable (see [1] and [10]).
Note that σ gk (T ) (resp. σ k (T )) is not necessarily non-empty. For example, a quasinilpotent (resp. nilpotent) operator T has empty generalized Kato spectrum (resp. Kato spectrum). Furthermore, the comparison between this spectra and the spectra defined by the Drazin inverses gives Definition 2.6. Let T ∈ L(X). The operator T is said to have the single-valued extension property at λ 0 ∈ C, abbreviated T has the SVEP at λ 0 , if for every neighborhood U of λ 0 the only analytic function f : U → X which satisfies the equation The operator T is said to have the SVEP if T has the SVEP at every λ ∈ C.
Trivially, an operator T has the SVEP at every point of the resolvent set ρ(T ). Moreover, from the identity theorem for analytic functions it easily follows that T has the SVEP at every point of the boundary ∂σ(T ) of the spectrum. Hence, we have the implications: (1) Every operator T has the SVEP at an isolated point of the spectrum.
∈ accσ su (T ), then T * has the SVEP at λ In particular, it has been showed that if λI − T admits a generalized Kato decomposition, then implications (2) and (3) may be reversed. For more properties of the SVEP, we can see [22].

Left and right generalized Drazin invertible operators and the SVEP
Now we give a characterization of the left (resp. the right) generalized Drazin invertible operators in terms of generalized Kato decomposition and the singlevalued extension property.  (i) T has the SVEP at 0, (ii) T M has the SVEP at 0, Similarly, by [1, Theorem 3.15] we have: The following result expresses a characterization of the isolated points of σ ap (T ) in terms of generalized Kato decomposition and the SVEP. Proof. Suppose that 0 is an isolated point in σ ap (T ), then T has the SVEP at 0 and by [13,Proposition 9.], H 0 (T ) and K(T ) are closed subspaces of X with K(T ) = X, H 0 (T ) = {0} and K(T ) ∩ H 0 (T ) = {0}. If K(T ) ⊕ H 0 (T ) = X, then 0 is also isolated point in σ(T ) and clearly T admits a GKD(K(T ), H 0 (T )). Now, assume that K(T ) ⊕ H 0 (T ) X, and denote by X 0 = K(T ) ⊕ H 0 (T ). Observe that X 0 is a Banach space and (K(T ), H 0 (T )) is a GKD of T on X 0 . So by In the other hand we have H 0 (T ) ⊥ +K(T ) ⊥ = X * and the adjoint of the inclusion map i : X 0 → X is a map from X * onto X * 0 with kernel X ⊥ 0 . This implies that T * admits a GKD over  The basic existence results of generalized Drazin inverses and their relation to the SVEP, the quasinilpotent part and the analytical core are summarized in the following theorems. We know that the properties to be right generalized Drazin invertible or to be left generalized Drazin invertible are dual to each other, (see [24, Proposition 3.9]), then we have, Theorem 3.9. Let T ∈ L(X). The following assertions are equivalent: admits a GKD(M, N) and T * has the SVEP at 0, (v) T admits a GKD(M, N) and satisfied one of the equivalent conditions of the Theorem 3.4, (vi) there exists a bounded projection P on X such that T P = P T , T + P is surjective, T P is quasinilpotent and N(P ) = K(T ).

Proof
where P is the bounded projection given in .
Denote by S(T ) = {λ ∈ C : T does not have the SVEP at λ}.
• If X is a Hilbert space and T is a self-adjoint operator, then σ lgD (T ) = σ rgD (T ) = σ gD (T ).
Similarity, for operators of Kato type we have, Corollary 3.13. Let T ∈ L(X).
In the following, we show that both σ lgD (T ) and σ rgD (T ) are stable under additive commuting finite rank operators. As a consequence of Proposition 3.14 we have   Proof. Proposition 3.14 implies that σ lgD (T −1 ) = σ lgD (S −1 ) , and by Proposition 3.16 we have σ lgD (T ) = σ lgD (S).  In particular if R = U we get

Generalized Drazin inverse and local spectral theory
We know that if T ∈ L(X) is not invertible then T is generalized Drazin invertible if and only if X = K(T ) ⊕ H 0 (T ) and, with respect tho this decomposition, Note that the generalized Drazin inverse T D of T , if it exists, is uniquely determined and represented, with respect of the same decomposition, as the direct sum is the inverse of T 1 , Furthermore, the nonzero part of the spectrum of T D is given by the reciprocals of the nonzero points of the spectrum of T , i.e., Since the spectral mapping theorem holds for the approximate spectrum and the surjective spectrum, we have and An interesting question given in [2] is that there is a reciprocal relationship between the nonzero part of the local spectrum of a Drazin invertible operator and the nonzero part of the local spectrum of its Drazin inverse. In the sequel we study this question in the case of the generalized Drazin invertible operators.
Before this down, we shall give the relevant definitions concerning the local spectral theory. Given a bounded linear operator T ∈ L(X) , the local resolvent set ρ T (x) of T at a point x ∈ X is defined as the union of all open subsets U of C such that there exists an analytic function f : U −→ X satisfying The local spectrum σ T (x) of T at x is the set defined by σ T (x) := C \ ρ T (x). Obviously, σ T (x) ⊆ σ(T ).
The SVEP for T is equivalent to saying that σ T (x) = ∅ if and only if x = 0, see [22,Proposition 1.2.16]. Note that if T has SVEP then a spectral theorem holds for the local spectrum, i.e., if f is an analytic function defined on an open neighborhood U of σ(T ) then See also [26].
An important invariant subspace in local spectral theory is given by the local spectral subspace of T associated at a subset Ω ⊆ C, defined as Obviously, for every closed set Ω ⊆ C we have X T (Ω) = X T (Ω ∩ σ(T )).
For a closed subset Ω ⊆ C, the glocal subspace X T (Ω) is defined as the set of all x ∈ X for which there exists an analytic function f : Obviously, for Ω a closed set, X T (Ω) ⊂ X T (Ω), and we have equality when T satisfies the SVEP.  See the monograph [22] for a detailed study of these properties.
The next first result shows that the the SVEP is transmitted from T to its generalized Drazin inverse T D , In the following result, we show that the relation (4.1) holds also in the local sens.
We establish now that that also the property (C) is transferred to the generalized Drazin inverse.   Proof. Suppose that T ∈ L(X) is generalized Drazin invertible and 0 ∈ σ(T ). Then T = T 1 ⊕ T 2 with T 1 is invertible and T 2 is quasi-nilpotent. From [22, Proposition 2.1.6], we can identify H(U, X) with the direct sum H(U, K(T )) ⊕ H(U, H 0 (T )). T 1 has the property (β) and hence its inverse T −1 1 has the property (β). Now . Clearly T D U has closed range in H(U, X), so T D has the property (β). Conversely; if T D = T −1 1 ⊕ 0 has the property (β). Then as above T 1 has the property (C). Since the quasinilpotent operator T 2 has the property (β) and the fact that we conclude that T D U has closed range in H(U, X), so T D has the property (β). An operator T ∈ L(X) is said to have the decomposition property (δ) if the decomposition X = X T (U ) + X T (V ) holds for every open cover {U, V } of C. Note that T ∈ L(X) has property (δ) (respectively, property (β) ) if and only if T * has property (β) (respectively, property (δ)), see [22,Theorem 2.5.5]. If T ∈ L(X) has both property (β) and property (δ) then T is said to be decomposable. Corollary 4.9. Suppose that T is generalized Drazin invertible. If T has property (δ) then T D has property (δ), and analogously, if T is decomposable then T D is decomposable.
Proof. Clearly, from the definition of the generalized Drazin invertibility it follows that if T is generalized Drazin invertible then its adjoint T * is also generalized Drazin invertible, with Drazin inverse T D * . If T has property (δ) then T * has property (β) and hence, by Theorem 4.8, also T D * has property (β). By duality this implies that T D has property (δ). The second assertion is clear: if T is decomposable then T D has both properties (δ) and (β) and the same holds for T D , again by Theorem 4.8 and the first part of the proof. Hence T D is decomposable.
A natural question suggested by all the results of this section is whether the local spectral properties are transmitted from a left (resp. right) generalized Drazin invertible operator to its left (resp. right) generalized Drazin inverse. The next example shows that the answer to this question is negative.
Example 4.10. Let X = ℓ 2 be the Hilbert space of all square summable complex sequences x = (x n ) n = (x 1 , x 2 , . . .), indexed by the a nonnegative integers. We define the right shift operator R and the left shift operator L in ℓ 2 by • If T is left generalized Drazin invertible and T * K(T * ) has the SVEP (respectively, property (C), property (Q), property (β)), then T is generalized Drazin invertible.