LEFT AND RIGHT GENERALIZED DRAZIN INVERTIBILITY OF AN UPPER TRIANGULAR OPERATOR MATRICES WITH APPLICATION TO BOUNDARY VALUE PROBLEMS

∗Corresponding author: mohammed.benharrat@gmail.com Abstract. When A ∈ B(H) and B ∈ B(K) are given, we denote by MC the operator on the Hilbert space H ⊕ K of the form MC =  A C 0 B  . In this paper we investigate the quasi-nilpotent part and the analytical core for the upper triangular operator matrix MC in terms of those of A and B. We give some necessary and sufficient conditions for MC to be left or right generalized Drazin invertible operator for some C ∈ B(K,H). As an application, we study the existence and uniqueness of the solution for abstract boundary value problems described by upper triangular operator matrices with right generalized Drazin invertible component.


Introduction and preliminaries
Let B(H) be the Banach algebra of all bounded linear operators acting on an infinite-dimensional complex Hilbert space H. Associated with an operator T there are two (not necessarily closed) linear subspaces of H invariant by T , played an important role in the development of the generalized Drazin inverse for T ∈ B(H), the quasi-nilpotent part H 0 (T ) of T : and the analytical core K(T ) of T : K(T ) = {x ∈ H : there exist a sequence (x n ) in H 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}.
See also [1].It is well-known that if K(T ) and H 0 (T ) are both closed, H = H 0 (T ) ⊕ K(T ), the restriction of T to H 0 (T ) is a quasi-nilpotent operator, and the restriction of T to K(T ) is invertible, provided that T is generalized Drazin invertible, (c.f.[18]).Recently, by the use of this two subspaces, in [24], the authors We also proved that T ∈ B(H) is a right generalized Drazin invertible if and only if 0 is an isolated point on the surjective spectrum σ su (T ) of T and by duality T ∈ B(H) is a left generalized Drazin invertible if and only if 0 is an isolated point on the approximate spectrum σ ap (T ).So we are mainly interested in the case where the point 0 belongs to the spectrum σ(T ) of T or in its various distinguished parts.Here, we are interested in the analogous question for an upper triangular operator of the form (1.1) defined on the separable Hilbert space H ⊕ K. Recall that the problem of the relationship between the spectrum, various distinguished parts of the spectrum of an upper triangular and Drazin invertibility and its diagonal has been considered by a number of authors in the recent past, we can see [4,5,13,21,22,[28][29][30] and the references therein for recent reviews on this topic.A related, and seemingly more demanding, problem is the following.Let H be a Hilbert space, T is a bounded linear operator on H, and M is a T -invariant closed subspace of H, then T takes the form which motivated the interest in 2 × 2 upper-triangular operator matrices.
In this paper we use the results of [24] to give a necessary and sufficient conditions for M C to be left (resp.right) generalized Drazin invertible which generalizes the notion of generalized Drazin invertible operators to the matrix case.We characterize the quasi-nilpotent part and the analytical core of the operator M C in term of the pair (A, B) of bounded operators.We apply our results to study the existence and uniqueness of solutions of boundary value problems described by an upper triangular operator matrices (2 × 2) acting in Hilbert spaces with a complex spectral parameter λ : , where U L is right generalized Drazin invertible, Γ is a boundary operator, F and Φ are given.
For T ∈ B(H) 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 Here I denotes the identity operator in H.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 H then T M denote the restriction of T in M .Assume that M and N are two subspaces of H such that H = M ⊕ N (that is H = M + N and M ∩ N = 0).We say that T is completely reduced by the pair (M, N ), denoted as (M, An operator is said to be bounded below if it is injective with closed range. Recall that (see, e.g.[14]) the ascent a(T ) of an operator T ∈ B(H) 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.Furthermore, if a(T ) = p < ∞ then An operator T ∈ B(H) is said to be Drazin invertible, if there exists an operator S ∈ B(H) such that ST = T S ST S = S and T ST = T + U where U is a nilpotent operator. (1. 2) The concept of Drazin invertible operators has been generalized by Koliha ( [18]) by replacing the nilpotent operator U in (1.2) by a quasi-nilpotent operator one.In this case, S is called a generalized Drazin inverse of T , denoted by T D .Examples of generalized Drazin invertible operators are the operators of the following classes: • Invertible operators, right invertible operators and left invertible operators.

LD(H) ∩ RD(H).
According to the Definitions 1.It is well known that these spectra are compact sets in the complex plane, and we have, and where are respectively the approximate point spectrum and the surjective spectrum of T .
The basic existence results of generalized Drazin inverses and its relation to the quasi-nilpotent part and the analytical core are summarized in the following theorems.

Theorem 1.1 ( [18]
). Assume that T ∈ B(H).The following assertions are equivalent: (i) T is generalized Drazin invertible, (ii) 0 is an isolated point in the spectrum σ(T ) of T ; (v) there is a bounded projection P on H such that R(P ) = K(T ) and N (P ) = H 0 (T ). (vi ⊕ denotes the algebraic direct sum and T M denote the restriction of T to a subspace M of H. Theorem 1.2.Assume that T ∈ B(H).The following assertions are equivalent: (i) T is left generalized Drazin invertible; (ii) 0 is an isolated point in σ ap (T ); We know that the properties to be right generalized Drazin invertible or to be left generalized Drazin invertible are dual each other, (see [24, Proposition 3.9]), then we have, The following assertions are equivalent: (i) T is right generalized Drazin invertible; (ii) 0 is an isolated point in σ su (T ); The reduced minimum modulus γ(T ) of T is defined by The paper is organized as follows.In Section 2 we give a relationship between the quasi-nilpotent part and the analytical core of a pair (A, B) of operators and that of 2 × 2 block triangular matrices M C , we show that the quasi-nilpotent part of M C is a direct sum of the quasi-nilpotent part of A and B and that is the same for analytical core.In Section 3, we study the left (resp.right) generalized Drazin invertibility of M C using the isolated point in the approximate spectrum (resp.the surjective spectrum) of A and B.
Finally in section 4 we illustrate our approach by studying a boundary value problems described by an upper triangular operator matrices.
2. The quasi-nilpotent part and the analytical core of the operator M C In the following, we find the relationship between the quasi-nipotent part (resp.the analytical core) of the pair (A, B) of operators and that of M C defined in (1.1) and we give fundamental results concerning this operator.
Proof.Suppose that We have Then Therefore, Then lim n→∞ A n x 2 n = 0 and lim n→∞ B n y Proof.Let x ∈ K(A) and y ∈ K(B), by definition there exist two sequences (x n ) in H, (y n ) in K and a constants δ 1 > 0, δ 2 > 0 such that Ax 1 = x, Ax n+1 = x n and x n ≤ δ n 1 x and By 1 = y, By n+1 = y n and y n ≤ δ n 2 y for all n ∈ N. We have (2.1) where δ = max(δ 1 , δ 2 ).Hence Conversely, suppose that Then there exist a sequence We obtain from (2.1) that By n+1 = y n , Ax n+1 = x n , Ax 1 = x and By 1 = y.
Consequently, x n ≤ δ n x and y n ≤ δ n y , that is . This completes the proof.
As a consequence of Propositions 2.1 and 2.2 we have the following result.

Left and right generalized Drazin invertibility of M C
Hwang and Lee , [13], give a necessary and sufficient condition for M C to be bounded below for some C ∈ B(K, H) and they are characterized the intersection of the approximate point spectrum, the surjective spectrum and the spectrum of M C .
The next theorem is an extension of [13, Theorem 1], we will give some necessary and sufficient conditions for M C to be left generalized Drazin invertible operator for some C ∈ B(K, H).
Theorem 3.1.For a given pair (A, B) of bounded operators, the following statements are equivalent: (i) M C is left generalized Drazin invertible for some C ∈ B(K, H), (ii) A is left generalized Drazin invertible and there exists a constant δ such that To prove this theorem we need the following lemma.We next claim that if A is left generalized Drazin invertible and there exists a constant δ such that for every λ with 0 Drazin invertible for some C ∈ B(K, H).Since R(B − λI) is not closed and β(A − λI) = ∞, there exists an isomorphism J : K → R(B − λI).Define an operator C : K → H in the following way: By a similar proof we check easily that bounded below for 0 < |λ| < δ and by Theorem 1.2 M C is left generalized Drazin invertible.
For the converse, suppose in the contrary that By duality, we have: Theorem 3.2.For a given pair (A, B) of operators, the following statements are equivalent: B is right generalized Drazin invertible and there exists a constant δ such that As a direct application of Theorem 3.1, the following corollary can be derived to give a characterization of σ lgD (M C ) for all C ∈ B(K, H).The following is the dual statement of Corollary 3.1.
By combining Corollaries 3.1 and 3.2 we obtain; This result gives a generalization of [4, Theorem 2.1].

Application to a spectral boundary value matrix problem
This section is devoted to the study of boundary value problems described by an upper triangular operator matrices (2 × 2) acting in Hilbert spaces with a complex spectral parameter λ, , where F and Φ are given and U L is the matrix operator defined on H ⊕ K by with L : K → H a given linear operator.We first define the boundary value problem (P) by ordered pairs (U L , M C ) of an upper triangular operator matrix M C where U L is a right generalized Drazin invertible and we construct the adapted boundary operator Γ of U L .We prove the existence of an unique solution of (P) and we give an explicit expression for this solution.Before this down, we define the boundary operator for a right generalized Drazin invertible operator.
If S be a right generalized Drazin inverse of the operator A ∈ B(H), then Now, let E another complex Hilbert space, called boundary space.
Proof.We have that K(T ) ⊂ N (Γ 1 ), K(D) ⊂ N (Γ 2 ) and there exist Π 1 : E → H and Π 2 : Since T and D are right generalized Drzain invertibles, then so is Our purpose is to establish the existence and uniqueness of solutions for the boundary value problem (P).
In the theorem below, we give an explicit expression for the solution of the problem (P).

Definition 1 . 1 .
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.An operator T ∈ B(H) is said to be right generalized Drazin invertible if K(T ) is closed and complemented with a subspace N in H such that T (N ) ⊂ N ⊆ H 0 (T ).Definition 1.2.An operator T ∈ B(H) is said to be left generalized Drazin invertible if H 0 (T ) is closed and complemented with a subspace M in H such that T (M ) ⊂ M and T (M ) is closed.

Corollary 2 . 1 .
If R(C) ⊂ H 0 (A) and R(B) ⊂ N (C).Then M C is generalized Drazin invertible if and only if both A and B are generalized Drazin invertible.
Since R((M C −λI) * ) is closed and that is R((B − λI) * ) is closed.This contradicts our assumption.Therefore we must have

Definition 4 . 1 .
An operator Γ : H → E is said to be a boundary operator for a right generalized Drazin invertible operator A corresponding to its right generalized Drazin inverse S ∈ B(H) if, (i) K(A) ⊂ N (Γ); (ii) There exists an operator Π : E → H such that ΓΠ = I E and R(Π) = N (A) ∩ K(A).