Commuting Outer Inverses

The group, Drazin and Koliha-Drazin inverses are particular classes of commuting outer inverses. In this note, we use the inverse along an element to study some spectral conditions related to these inverses in the case of bounded linear operators on a Banach space.


Introduction
Several of the useful properties of the group, Drazin and Koliha-Drazin inverses can be related to their spectral characterizations.Some of these can be traced to the property of being commuting outer inverses.
Let be the set of bounded linear operators on a Banach space   B X X , and let

 
A B X  .We denote the range by and the null space of A by   A  .Let M and be closed subspaces of N X .The outer inverse with prescribed range M and null space , denoted is the unique operator which satisfies: There is some advantage in prescribing the null space and range of an outer inverse by means of a third operator.In doing so, we will use the notion of invertibility along an element introduced by X. Mary ([1]).We say A is invertible along T if there exists

A
In this case, the inverse along is unique and we write .
we have that BA and AB are projections such that and . Thus, we are effectively prescribing the range of the projection and the null space of the projection BA AB .One of the useful properties of a generalized inverse is that, although the operator is not invertible, there is a subspace for which the reduction of the operator to that subspace is indeed invertible: In this case, such is unique and we write , then AT and TA are group invertible and , , , , , In the following section, we study an operator A such that A is invertible along T with AT TA  .Then, in Section 3 we study some projections related to the outer inverse with prescribed range and null space.Finally, in Section 4 we specialize to spectral projections, covering results from Dajić and Koliha ([3]).

Invertibility along a Commuting Operator Proposition 4. Let
. Proof.From Proposition 2 we have: If A is invertible along T , then we have the following matrix form ( [2]): A is invertible and  is a complement of , that is, .

T
When A and T commute, we can say a little more: Theorem 5. Let A be invertible along T and AT TA  .Then there exist an invertible operator 1 A on and an operator Then by Proposition 4 and hence we can consider the following matrix decomposition of A : : , ,0,0, .

Projections
Commuting outer inverses are naturally linked to projections.
, then there exists a bounded projection Thus, there exists a bounded projection . Hence, A is invertible along .P For a sort of converse, we give a necessary condition in Theorem 9.
Example 8.An operator A and a projection such that   be the (rotation) matrix defined by . To see that it is also 1-1, let   be the projection defined by 1 0 : .0 0 Then, it is easy to check that We have   2 0 .0 0  be a projection and suppose A is invertible and is the complement of , that is, is a complement for , we can take .Then, we have , , 0, 0, .

  
However, the reduction can not be invertible, and by Theorem 1 A is not invertible along .P Theorem 12. Let   P B X  be a projection and from Proposition 4 and the definition of Proposition 4 and the definition of . We use Theorem 1.The reduction we get that it is also 1-1.To see that . For the other inclusion, let is closed and complemented.Hence A is invertible along .P

Spectral Projections
are invariant under A , we have: and the matrix form of A , there exists a projection such that , . Thus, without loss of generality, we can suppose

PA AP 
A is invertible along a projection P .A very important class of projections which commutes with A is a class of spectral projections, which we now discuss.
is said to be a spectral set of A if and are both closed in .For a spectral set of is a point of the resolvent set or an isolated point of the spectrum   A  , then the operator A is called quasipolar.Let A be quasipolar and let 0 be the spectral projection associated with the spectral set .For the general case is a spectral set such that 0    , Dajić and Koliha have defined a generalized inverse and studied its properties ( [3]).

 
A B X  and be a spectral set for

2 M
 be the set of two by two matrices with real entries.Let

A.
If 0   then A is invertible along  .From the theorem above, A is invertible along