turkmath.org

We prove a general dilation theorem for positive semidefinite kernels that are invariant under actions of $*$-semigroups and take values continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes). This theorem points out two necessary and sufficient conditions for the existence of $*$-representations of the given $*$-semigroup on VH-spaces or, equivalently, on reproducing kernel VH-spaces and it unifies many known dilation theorems, including the recent dilation theorems in Hilbert modules over $C^*$-algebras or locally $C^*$-algebras.