Generalized Projections and Equivalent Representations of James Orthogonal Decompositions in Banach Spaces

Abstract

We present a short observe of all the generalized projection operators in Banach spaces. These operators have been created and studied in our works [1, 2, 6, 10]. We pay a special heed to the new properties of the projection operator P which is afterwards used to construct the equivalent decompositions of arbitrary elements of Banach spaces onto their subspaces and cones. These decompositions are complete analogies of the well known classical orthogonal decompositions in Hilbert spaces (see (4.8) and (5.4) and compare with (4.6) and (5.3), respectively). Generalized projections and orthogonal decompositions in Banach spaces have lately found numerous applications in different areas of the functional analysis.