Commutant lifting and factorization of reproducing kernels

A general version of the commutant lifting theorem for operators between different spaces is proved. It includes as special cases the lifting theorems of Ball-Trent-Vinnikov and Volberg-Treil. A multivariable variant of the Volberg-Treil theorem is obtained as a corollary. A certain factorization property of reproducing kernels is shown to be a sufficient condition for the lifting. Another factorization property is shown to be a necessary condition.