Equivalence up to a rank one perturbation (Q1587530)

The author studies the problem of unitary equivalence up to a rank one operator for unitary operators with simple spectra in Hilbert space. He gives a complete solution in the case of operators with pure singular spectrum which fills the whole unit circle. The operators \(U_1\) and \(U_2\) are said to be completely non-equivalent if there are no non-trivial invariant subspaces \(H_1\) and \(H_2\) for \(U_1\) and \(U_2\), respectively, such that the restriction of \(U_1\) on \(H_1\) is unitary equivalent to the restriction of \(U_2\) on \(H_2\). The authors shows that for completely non-equivalent singular cyclic unitary operators \(U\) and \(V\) such that \(\sigma(U)=\sigma(V)=T\) there exist a unitary operator \(W\) and a rank one operator \(K\) such that \(WUW^*=V+K\). Some function theoretic consequences of this result are presented. The argument is based on the Krein-Lifshits spectral shift for a unitary pair.