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Quasi-splitting subspaces in a pre-Hilbert space

✍ Scribed by David Buhagiar; Emmanuel Chetcuti

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
126 KB
Volume
280
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let S be a pre‐Hilbert space. Two classes of closed subspaces of S that can naturally replace the lattice of projections in a Hilbert space are E (S) and F (S), the classes of splitting subspaces and orthogonally closed subspaces of S respectively. It is well‐known that in general the algebraic structure of E (S) differs considerably from that of F (S) and the two coalesce if and only if S is a Hilbert space. In the present note we introduce the class E~q~ (S) of quasi‐splitting subspaces of S. First it is shown that E~q~ (S) falls between E (S) and F (S). It is also shown that, in contrast to the other two classes, E~q~ (S) can sometimes be a complete lattice (without S being complete) and yet, in other examples E~q~ (S) is not a lattice. At the end, the algebraic structure of E~q~ (S) is used to characterize Hilbert spaces. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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