Multivalued linear projections

Authors: Cross, R.W.; Wilcox, D.L.

Source: Quaestiones Mathematicae, Volume 25, Number 4, 1 December 2002 , pp. 503-512(10)

Publisher: NISC Pty Ltd

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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 characterised 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 characterisations of the adjoint and completion of a projection together with the closed graph and closed range theorems to give criteria for the continuity of a projection defined on a normed linear space. A new proof of the theorem on closed sums of closed subspaces in a Banach space (cf. Mennicken and Sagraloff [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 generalised inverses.

Keywords: MULTIVALUED LINEAR PROJECTIONS; SUBSPACE PAIRS; MULTIVALUED LINEAR OPERATORS; LINEAR RELATIONS; NORMED LINEAR SPACES; TOPOLOGICAL COMPLEMENTATION; SEMI-FREDHOLM RELATIONS

Document Type: Research article

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