ordering of self-adjoints
Since is a proper convex cone (http://planetmath.org/Cone5) (see this entry (http://planetmath.org/PositiveElement3)), we can define a partial order on the set , by setting
if and only if , i.e. is positive.
The proof that is partial order makes no use of the self-adjointness . In fact, itself is an ordered topological vector space under the relation .
However, it turns out that this ordering relation provides its most usefulness when restricted to self-adjoint elements. For example, some of the above would not hold if we did not restrict to .
|Title||ordering of self-adjoints|
|Date of creation||2013-03-22 17:30:37|
|Last modified on||2013-03-22 17:30:37|
|Last modified by||asteroid (17536)|