Unitary Invariants in Multivariable Operator Theory

The problems considered in this paper come as a natural continuation of our program to develop a free analogue of Sz.-Nagy–Foiaş theory, for row contractions. An n-tuple (T1, . . . , Tn) of operators acting on a Hilbert space is called row contraction if T1T ∗ 1 + · · ·+ TnT ∗ n ≤ I. In this study, the role of the unilateral shift is played by the left creation operators on the full Fock space with n generators, F (Hn), and the Hardy algebra H ∞(D) is replaced by the noncommutative analytic Toeplitz algebra F∞ n . The algebra F∞ n and its norm closed version, the noncommutative disc algebra An, were introduced by the author [47] in connection with a multivariable noncommutative von Neumann inequality. F∞ n is the algebra of left multipliers of F (Hn) and can be identified with the weakly closed (or w ∗-closed) algebra generated by the left creation operators S1, . . . , Sn acting on F (Hn), and the identity. The noncommutative disc algebra An is the norm closed algebra generated by S1, . . . , Sn, and the identity. When n = 1, F∞ 1 can be identified with H ∞(D), the algebra of bounded analytic functions on the open unit disc. The algebra F∞ n can be viewed as a multivariable noncommutative analogue of H∞(D). There are many analogies with the invariant subspaces of the unilateral shift on H(D), inner-outer factorizations, analytic operators, Toeplitz operators, H∞(D)–functional calculus, bounded (resp. spectral) interpolation, etc. The noncommutative analytic Toeplitz algebra F∞ n has been studied in several papers [45], [46], [48], [49], [50], [52], [1], and recently in [15], [16], [17], [3], [54], [18], [40], and [56].