Closed Complemented Subspaces of Banach Spaces and Existence of Bounded Quasi-linear Generalized Inverses

ABSTRACT

In this article, an equivalent condition for the existence of a bounded quasi-linear (BQL) generalized inverse of a closed linear operator with respect to projector between two Banach spaces is given. Using the BQL generalized inverse, we give a necessary and sufficient condition for a closed linear subspace in a Banach space to be complemented. Finally, an application of the main results to the Saddle–Node bifurcation theorem from multiple eigenvalues in nonlinear analysis is given.

KEYWORDS:

• Banach space
• bounded quasi-linear (BQL) generalized inverse with respect to projectors
• complemented subspace
• necessary and sufficient condition

MATHEMATICS SUBJECT CLASSIFICATION:

• 47A05
• 46B20