Multipliers on Spaces of Functions on a Locally Compact Abelian Group with Values in a Hilbert Space

2000 Mathematics Subject Classification: Primary 43A22, 43A25.

We prove a representation theorem for bounded operators commuting with translations on L2ω(G,H), where G is a locally compact abelian group, H is a Hilbert space and ω is a weight on G. Moreover, in the particular case when G = R, we characterize completely the spectrum of the shift operator S1,ω on Lω2(R,H).