2015/07/27 by Tepper L. Gill, Gill, Tepper L., Marzett Golden +1
Mathematics · #46C99 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 47B37 #Secondary 46B10 #math.FA #math.OA #msc:46B10 #msc:46C99 #msc:47B37
paper · pdf · doi:10.48550/arxiv.1507.08611
arXiv admin note: text overlap with arXiv:1010.4922
arxiv created 2015/07/27 · arxiv updated 2015/07/31
The purpose of this note is to show that, if \mcB is a uniformly convex Banach, then the dual space \mcB' has a "Hilbert space representation" (defined in the paper), that makes \mcB much closer to a Hilbert space then previously suspected. As an application, we prove that, if \mcB also has a Schauder basis (S-basis), then for each A ∈ \C[\mcB] (the closed and densely defined linear operators), there exists a closed densely defined linear operator A^* ∈ \C[\mcB] that has all the expected properties of an adjoint. Thus for example, the bounded linear operators, L[\mcB], is a ^*algebra. This result allows us to give a natural definition to the Schatten class of operators on a uniformly convex Banach space with a S-basis. In particular, every theorem that is true for the Schatten class on a Hilbert space, is also true on such a space. The main tool we use is a special version of a result due to Kuelbs \citeK, which shows that every uniformly convex Banach space with a S-basis can be densely and continuously embedded into a Hilbert space which is unique up to a change of basis.