Commutant lifting and interpolation on the unit ball

We solve the commutant lifting and interpolation problems in the setting of the Hardy space and Schur functions on the open unit ball of $\mathbb{C}^n$. Our solutions also signify the role of inner functions on the unit ball, objects whose existence was once in doubt and to which Aleksandrov, Rudin, and others made fundamental contributions. Some of our results, particularly those related to inner functions, are new even in the classical, one-variable case.

💡 Research Summary

This paper presents a comprehensive solution to two fundamental problems in multivariable operator theory and complex analysis: the commutant lifting theorem and the Nevanlinna-Pick interpolation problem, set in the context of the Hardy space H^2(B_n) and Schur functions S(B_n) on the open unit ball of C^n.

The core achievement is a multivariable commutant lifting theorem (Theorem 1.4) for the Hardy space over the unit ball. For a quotient module Q of H^2(B_n) and a contractive module map X on Q, the theorem establishes a set of equivalent conditions for X to be liftable to a multiplication operator T_φ by a Schur function φ ∈ S(B_n). These conditions include: (1) the existence of a perturbation φ such that ψ - φ ∈ Q⊥, where ψ = X(P_Q 1); (2) the contractivity of a certain linear functional X_Q defined on a carefully constructed subspace M_Q of L^1(S_n); (3) a quantitative distance condition in the L^1 norm; and (4) the existence of a sequence of inner functions {u_i} that converges to ψ in the weak* topology on M_Q or Q_conj. The equivalence between liftability and the perturbation condition (ψ - φ ∈ Q⊥) provides a particularly geometric and useful characterization.

The authors then apply this lifting theorem to solve the Nevanlinna-Pick interpolation problem on the unit ball (Theorem 1.5). Given m distinct points Z = {z_i} in B_n and target values W = {w_i} in the unit disc D, they construct an explicit function ψ_{Z,W} from the Szegő kernel. The existence of an interpolating Schur function φ (with φ(z_i) = w_i) is shown to be equivalent to: the contractivity of a related linear functional J_{Z,W}; a specific distance inequality in L^1(S_n); and, most notably, the existence of a sequence of inner functions {u_i} such that lim u_i(z_j) = ψ_{Z,W}(z_j) for all j. This last condition, which involves approximation by inner functions, is new even in the classical one-variable setting.

A major theme of the paper is the central role played by inner functions on the unit ball. Once thought not to exist for n > 1, the pathological yet abundant inner functions constructed by Aleksandrov and Rudin are shown to be not mere curiosities but essential tools for characterizing the solutions to these central problems. The paper effectively bridges the gap between the abstract existence theory of inner functions and concrete problems in operator theory.

Methodologically, the work introduces and analyzes a key subspace M_Q = Q_conj +