A real Banach space is called a
Functional Hilbertian Sum (FHS) if it is isometric to the direct sum of Hilbert spaces
of dimension at least two via a one-unconditional basis. Various isometric
permanence properties of Functional Hilbertian Sums are proved. Many of these
results are the real analogues of (and also imply) known theorems concerning
complex Banach spaces with one-unconditional (or “hyperorthogonal”) bases. For
example, it is proved that a space is FHS if and only if it equals the closed linear
span of the ranges of its rank-two skew-Hermitian operators. The complex
analogue due to Kalton and Wood is as follows: a complex Banach space has a
one-unconditional basis provided it equals the closed linear span of the ranges of its
rank-one skew-Hermitian operators. The isometries and skew-Hermitian
operators on FHS spaces are completely determined and FHS spaces are
isometrically classified. Skew-Hermitian operators on general real spaces with a
one-unconditional basis are also completely determined, using FHS spaces in an
essential manner. Various complementation results are established, insuring
that under certain circumstances, one-complemented subspaces of spaces
with one-unconditional bases are FHS spaces. One of these yields the real
analogue of (and also implies) the theorem of Kalton and Wood that the
family of complex Banach spaces with one-unconditional bases is closed
under contractive projections. In the course of this investigation, several
isometric invariants for real Banach spaces are introduced. Many of these
are natural analogues of known invariants for complex spaces and include
orthogonal projections, well-embedded spaces, Hilbert components and B. Lie
algebras.
