Algebraic Properties of Generalized Inverses by Dragana S. Cvetković Ilić & Yimin Wei

Author:Dragana S. Cvetković Ilić & Yimin Wei
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

.

Proof

Indeed, in this case the subspace defined in Theorem 3.11 by is equal to . Hence, for any , we have that which implies that the injectivity of operator is equivalent with . i.e.,

(3.33)

Also, operator has range if and only if , which is equivalent with . Now, by (3.33), we have that

(3.34)

Using (3.34), the fact that the intersection of two operator ranges is an operator range and Theorem 2.​3 [22], we conclude that is closed. Now, the proof follows by Theorem 3.11.

Obviously, from Theorem 3.17 we get the following corollary:

Corollary 3.4

Let be given projections and , . Then the invertibility of the linear combination is independent of the choice of the scalars , .

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