Commutant lifting theorem for the Bergman space (Q2501062)

The author of the paper under review studies analogs of the Sarason commutant lifting theorem for the Bergman space and weighted Bergman spaces. Let \(S\) be multiplication by \(z\) on the Bergman space. It is shown that if \(T\) commutes with the restriction of \(S^*\) to an \(n\)-dimensional invariant subspace, then it can be lifted to an operator \(R\) commuting with \(S^*\). However, unlike the case of the Hardy class, the norm of \(R\) can jump. The author estimates the best possible constant \(C_n\) for which \[ \inf_R\| R\| \leq C_n\| T\| . \]