Operator Characterizations, Rigidity and Constructions of (Ω, μ)-Frames. (17th February 2018) - Guo, Xunxiang
ABSTRACT: In this paper, first, we give some operator characterizations of (Ω, μ )-frames. We obtain that normalized tight (Ω, μ )-frames are precisely the (Ω, μ )-frames which are unitary equivalent to normalized tight (Ω, μ )-frames for some closed subspace ℳ of L 2 (Ω, μ ) and (Ω, μ )-frames are precisely the (Ω, μ )-frames which are similar to normalized tight (Ω, μ )-frames for some closed subspace ℳ of L 2 (Ω, μ ). We also characterize the alternate dual (Ω, μ )-frames through an operator equation. Then we establish some rigidity in the pairs of dual (super) (Ω, μ )-frames related with disjointness. Finally, we consider the constructions of (Ω, μ )-frames, including the constructions of new (Ω, μ )-frames or new pair of dual (Ω, μ )-frames from known ones and the constructions of the canonical dual of a (Ω, μ )-frame under certain conditions, which generalize the corresponding results on discrete frames.