Diagonalizing Operators with Reflection Symmetry

Let U be an operator in a Hilbert space H0, and let K⊂H0 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→H(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 ker(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 formf(x)↦f(cx), c>1, and the spectrum of S(Uc) is then computed in terms of the given number c.