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In this paper we obtain a multivariable commutator lifting inequality, which extends to several variables a recent result of Foias, Frazho, and Kaashoek. This inequality yields a multivariable lifting theorem that generalizes the noncommutative commutant lifting theorem. This is then used to solve new operator-valued interpolation problems of Schur?Caratheodory, Nevanlinna?Pick, and Sarason type on Fock spaces. Some consequences to norm constrained analytic interpolation in the unit ball of Cn are also considered.

Commutator lifting inequalities and interpolation