turkmath.org

A VH-space (Vector Hilbert Space in the sense of Loynes) is a complex complete locally convex space with a topologically ordered $*$-space valued inner product. Examples of VH-spaces include the chain of locally Hilbert $C^*$-modules, Hilbert $C^*$-modules and Hilbert Spaces.

In this talk, after a brief discussion of VH-Spaces with examples and basic properties, we state a general dilation theorem for positive semidefinite kernels valued in adjointable operators on a barrelled VH-space. We prove that, under barrelledness assumption, a necessary and sufficient condition for the existence of a natural VH-space dilation, or equivalently, a reproducing kernel VH-space representation of the kernel, is satisfied automatically.