Pre‐Hilbert spaces with anomalous splitting and orthogonally‐closed subspace structures

Abstract

Key words Pre-Hilbert space (=inner product space), orthogonally-closed subspace, quasi-splitting subspace, splitting subspace, riesz interpolation property MSC (2010) 46C15, 81P10Four classes of closed subspaces of an inner product space S that can naturally replace the lattice of projections in a Hilbert space are: the complete/cocomplete subspaces C(S), the splitting subspaces E(S), the quasi-splitting subspaces E q (S) and the orthogonally-closed subspaces F(S). It is well-known that in general the algebraic structure of these families differ remarkably and they coalesce if and only if S is a Hilbert space. It is also known that when S is a hyperplane in its completion S i.e. d( S/S) = 1 then C(S) = E(S) and E q (S) = F(S). On the other extreme, when S = c 00 i.e. d( S/S) = 2 ℵ 0 then C(S) = E(S) and E q (S) = F(S). Motivated by thisand in contrast to it, we show that in general the codimension of S in S bears very little relation to the properties of these families. In particular, we show that the equalities E(S) = C(S) and E q (S) = F(S) can hold for inner product spaces with arbitrary codimension in S. At the end we also contribute to the study of the algebraic structure of E q (S) by testing it for the Riesz interpolation property. We show that E q (S) may fail to enjoy the Riesz interpolation property in both extreme situations when S is "very small" i.e. d( 2 /S) = 2 ℵ 0 and when S is 'very big' (i.e. d( 2 /S) = 2).