2017/01/01 by Serdar Ay, Ay, Serdar, Aurelian Gheondea +1
Mathematics · #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.1701.04740
We consider weakly positive semidefinite kernels valued in ordered *-spaces\nwith or without certain topological properties, and investigate their\nlinearisations (Kolmogorov decompositions) as well as their reproducing kernel\nspaces. The spaces of realisations are of VE (Vector Euclidean) or VH (Vector\nHilbert) type, more precisely, vector spaces that possess gramians (vector\nvalued inner products). The main results refer to the case when the kernels are\ninvariant under certain actions of *-semigroups and show under which\nconditions *-representations on VE-spaces, or VH-spaces in the topological\ncase, can be obtained. Finally we show that these results unify most of\ndilation type results for invariant positive semidefinite kernels with operator\nvalues as well as recent results on positive semidefinite maps on\n*-semigroups with values operators from a locally bounded topological vector\nspace to its conjugate Z-dual space, for Z an ordered *-space.\n