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The closed subspaces of the Hilbert space L2ðRÞ which are invariant under multiplication by HyðRÞ functions and the dilation operators f ðxÞ ! f ðsxÞ, 1 < s < y, are determined as the two parameter family of subspaces L2½a; b, 0ea, bey, which are reducing for multiplication operators, together with a four parameter family of nonreducing subspaces. The lattice and topological structure are determined and using operator algebra methods the corresponding family of orthogonal projections, with the weak operator topology, is identified as a compact connected 4-manifold

Translation and dilation invariant subspaces of L 2 (R)