Foguel-type operators similar to contractions

Pisier’s celebrated counterexample to Halmos’s similarity-to-contractions problem was based on $2\times 2$ upper triangular block operator matrices involving three classical operators: forward and backward shifts on the diagonal and Hankel operators in the off-diagonal entry. Together with another classical object, namely Toeplitz operators, one can formulate another $2^3-1=7$ types of $2\times 2$ upper triangular block operator matrices, which we refer to as Foguel-type operators. In this paper, we give a complete characterization of all the seven Foguel-type operators being similar to contractions.