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Shift reducing subspaces and irreducible-invariant subspaces generated by wandering vectors and applications

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  • Kubrusly, Carlos S.
  • Levan, Nhan

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.

Suggested Citation

  • Kubrusly, Carlos S. & Levan, Nhan, 2004. "Shift reducing subspaces and irreducible-invariant subspaces generated by wandering vectors and applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 65(6), pages 607-627.
    • Handle: RePEc:eee:matcom:v:65:y:2004:i:6:p:607-627
    DOI: 10.1016/j.matcom.2004.02.010
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    References listed on IDEAS

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    1. Antoniou, I. & Gustafson, K., 1999. "Wavelets and stochastic processes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 49(1), pages 81-104.
    2. Levan, N. & Kubrusly, C.S., 2003. "A wavelet “time-shift-detail” decomposition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 63(2), pages 73-78.
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