A class of operators and similarity to contractions (Q1345489)

Let \(H^2\) be the Hardy space on the disk in the complex plane. Let \(S\) be the shift operator on \(H^2\), \(Sf(z)= zf(z)\). For a bounded linear operator \(Q\) on \(H^2\) a corresponding operator \(R\) on \(H^2 \oplus H^2\) is defined by the matrix of operators \(R= \left[ \begin{smallmatrix} S^* &Q\\ 0 &S \end{smallmatrix} \right]\). Let \(\Gamma_\varphi f= P(\varphi \widetilde {f})\), where \(\widetilde {f} (z)= f(\widetilde {z})\) for \(|z|=1\) and \(P\) is the orthogonal projection of \(L^2\) onto \(H^2\). (A precise definition is given in the paper.) When \(Q= \Gamma_\varphi\), it was shown by Peller that if \(\varphi'\in \text{BMOA}\) then \(R\) is polynomially bounded, and Bourgain showed that in fact \(R\) is similar to a contraction. The author shows that a stronger conclusion may be drawn, \(R\) is similar to \(S^* \oplus S\), and that a converse result also holds.