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On wavelet induced isomorphisms for reducing subspaces

In this paper, we adapt the notion of a wavelet induced isomorphism of [0, 1) associated with a wavelet set, introduced in [E. J. Ionascu, A new construction of wavelet sets, Real Anal. Exchange 28(2) (2002/03) 593–610], to the case of an ℋ-wavelet set, where ℋ is a reducing subspace [X. Dai and S. Lu, Wavelets in subspaces, Michigan Math. J. 43 (1996) 81–98]. We characterize all these wavelet induced isomorphisms similar to those given in Ionascu paper and provide specific examples of this theory in the case of symmetric ℋ-wavelet sets. Examples when ℋ is the classical Hardy space are also considered.