Spectral Multiplicity Theory

In a past life, I was a mathematician. I worked in the area of function theoretic operator theory, and in particular looked at question involving spectral multiplicity theory. I studied unitary operators on L²(R) of the form T = F M_g F^-1M_u, where F is the Fourier transform, M_f denotes multiplication by f, g is in H^infinity(R), and |u| = 1 almost everywhere. I obtained a complete description of the spectral multiplicity theory for these operators, and this allowed a description of T in terms of shift operators and (M_u)_ac, the well understood absolutely continuous part of M_u acting on the reducing subspace H_ac of L²(R). Specifically, if the dimension of the orthogonal complement (in H²) of gH², which is a shift invariant subspace, is infinite, then T is a bilateral shift of infinite multiplicity. If the subspace has finite dimension n, then T is unitarily equivalent to the direct sum of U_n and an operator A, where A is the restriction of the the n-fold direct sum of (M_u)_ac to a reducing subspace of the n-fold direct sum of H_ac (containing at least one summand H_ac), and U_n is a bilateral shift of multiplicity n. If in addition n=1, then T is unitarily equivalent to the orthogonal direct sum of U₁ and (M_u)_ac.