Diagonalizing operators with reflection symmetry

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Let U be an operator in a Hilbert space H₀, and let K₀ be a closed and invariant subspace. Suppose there is a period-2 unitary operator J in H₀ such that JUJ=U*, and PJP ≥ 0, where P denotes the projection of H₀ onto K. We show that there is then a Hilbert space H(K), a contractive operator W:K(K), and a selfadjoint operator S=S(U) in H(K) such that W*W=PJP, W has dense range, and SW=WUP. Moreover, given (K,J) with the stated properties, the system (H(K),W,S) is unique up to unitary equivalence, and subject to the three conditions in the conclusion. We also provide an operator-theoretic model of this structure where U|ₖ is a pure shift of infinite multiplicity, and where we show that (W)=0. For that case, we describe the spectrum of the selfadjoint operator S(U) in terms of structural properties of U. In the model, U will be realized as a unitary scaling operator of the form f(x) f(cx), c>1, and the spectrum of S(Uc) is then computed in terms of the given number c.