2014/01/20 by Palle E. T. Jørgensen, Jorgensen, Palle, Steen Pedersen +3
Mathematics · #22E70 #31A15 #42C15 #46N20 #46N30 #46N50 #47L60 #58J65 #65R10 #81S25 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1401.4782
openalex publication_date 2014/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study two extension problems, and their interconnections: (i) extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups G; and (ii) (in case of Lie groups G) representations of the associated Lie algebras La(G), i.e., representations of La(G) by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space \mathscrHF (RKHS). Our analysis is non-trivial even if G=ℝn, and even if n=1. If G=ℝn, (ii), we are concerned with finding systems of strongly commuting selfadjoint operators \ Ti\ extending a system of commuting Hermitian operators with common dense domain in \mathscrHF. Specifically, we consider partially defined positive definite (p.d.) continuous functions F on a fixed group. From F we then build a reproducing kernel Hilbert space \mathscrHF, and the operator extension problem is concerned with operators acting in \mathscrHF, and with unitary representations of G acting on \mathscrHF. Our emphasis is on the interplay between the two problems, and on the harmonic analysis of our RKHSs \mathscrHF.