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An important tool in the theory are the following separation theorems.
Let
be a (non degenerate) duality of complex vector spaces. Thus
and
have weak topologies, and the matrix levels have the corresponding product
topology. For
and
,
is defined by the joint amplifications of the duality:
![$\displaystyle \langle v,w\rangle=[\langle v_{i,j},w_{k,l}\rangle]_{(i,k),(j,l)}$](img1102.png)
.
Note that the matrices, ordered by the cone of the positive semidefinite matrices, are
not totally ordered;
does not imply
.
Theorem:62Let
be a duality of complex vector spaces,
a closed set of
matrices over
and
for some
.
- a)
- [WW99, Thm. 1.6] If
is matrix convex, then there
are
and
such that for all
and

, but

.
- b)
- [EW97b, Thm. 5.4] If
is matrix convex and
,
then there is
such that for all
and

, but

.
- c)
- [Bet97, p. 57] If
is a matrix cone, then there is
sucht that for all
and

, but

.
- d)
- [EW97a, Thm. 4.1] If
is absolutely matrix convex,
then tere is
such that for all
and

, but

.
One can prove Ruan's theorem using the separation theorem for
absolutely matrix convex sets, applied to the unit ball of a matricially normed
space .
If
is a complex involutive vector space, one can find selfadjoint separating
functionals:
Theorem: 63Let
be a duality of complex involutive vector spaces,
a closed
set of selfadjoint matrices over
and
for some
.
- b)
- If
is matrix convex and
,
then there is a
such that for all
and

, but

.
Footnotes
- ...Theorem:62
-
From this theorem one can easily get the following sharper version of the parts a), b)
and d):
- a)
- If
is matrix convex, then there are
,
and
such that for all
and

, but

.
- b)
- If
is matrix convex and
, then there are
and
such that for all
and

, but

.
- d)
- If
is absolutely matrix convex, then there are
and
such that for all
and

, but

.
- ... 63
-
This theorem can be obtained from the above separation theorem, part a) and b). Note
that for selfadjoint
, the mapping
is selfadjoint.
Next: Bipolar theorems
Up: Matrix convexity
Previous: Examples
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Prof. Gerd Wittstock
2001-01-07