Multivalued linear projections

Abstract

A multivalued linear projection operator P defined on linear space X is
a multivalued linear operator which is idempotent and has invariant domain.
We show that a multivalued projection can
be characterized in terms of a pair of subspaces and then establish that the
class of multivalued linear projections is closed under taking adjoints and
closures.

We apply the characterizations of the adjoint and completion
of projection together with the closed graph and closed range theorems to give
criteria for the continuity of a projection defined ona  normed linear space.
A new proof of the theorem on closed sums of closed subspaces in a Banach space
(cf. Mennicken and Sagraloof [9, 10]) follows as a simple corollary.
We then show that the topological decomposition of a space may be expressed in
terms of multivalued projections. The paper is concluded with an application
to
multivalued semi-Fredholm relations with generalized inverses.


Mathematics Subject
Classification (2000): 47A06, 47A53



Quaestiones Mathematicae 25 (2002), 503-512