Semi-bounded unitary representations of infinite-dimensional Lie groups

In this note we introduce the concept of a semi-bounded unitary representations of an infinite-dimensional Lie group $G$. Semi-boundedness is defined in terms of the corresponding momentum set in the dual ${\text{g}}^{\mathrm{\prime }}$ of the Lie algebra $\text{g}$ of $G$. After dealing with some functional analytic issues concerning certain weak-$\ast$-locally compact subsets of dual spaces, called semi-equicontinuous, we characterize unitary representations which are bounded in the sense that their momentum set is equicontinuous, we characterize semi-bounded representations of locally convex spaces in terms of spectral measures, and we also describe a method to compute momentum sets of unitary representations of reproducing kernel Hilbert spaces of holomorphic functions.