Abstract
We introduce the notions of elementary reducing subspaces and elementary irreducible-invariant subspaces-- generated from wandering vectors--of a shift operator of countably infinite multiplicity, defined on a separable Hilbert space H. Necessary and sufficient conditions for a set of shift wandering vectors to span a wandering subspace are obtained. These lead to characterizations of shift reducing subspaces and shift irreducible-invariant subspaces, as well as a new decomposition of H into orthogonal sum of elementary reducing subspaces. Applications of elementary reducing subspaces to wavelet expansion, and of elementary irreducible-invariant subspaces to wavelet multiresolution analysis (MRA) will be discussed.
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