Abstract
Inthispaper,weintroducetheconceptofcompletewanderingoperatorsfor as ystemU of unitary operators acting on a Hilbert space, which can be viewed as an abstract mathematical model for g-orthonormal bases of Hilbert spaces and operator- valued wavelets for L 2 (R). The idea comes from Dai and Larson's work (Mem Am Math Soc 134:640 1998), where the wandering vectors for a unitary system are intro- duced as an abstract model for orthogonal wavelets. The topological and algebraical properties of the set W(U) of all complete wandering operators for a unitary system U are studied. In particular, properties of the local commutant of U are established. A parametrization formula for W(U) and some interesting algebraic properties of com- pletewanderingoperatorsforaunitarysystemareobtained.Thespecialcaseofgreatest interest is the wavelet system {UD,T }. We pay certain attention on studying this more structured unitary system and some structural theorems are established. Lots of prop- erties of the wandering vectors for a unitary system are extended to the more general case, i.e. the wandering operators for a unitary system. However, operator-valued case is more complicated. We also give some examples to illustrate our results. Our works show that wavelet theory and frame theory are deeply connected with operator theory.
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