Abstract. A characteristic function is a special operator-valued analytic function defined on the open unit ball of $\mathbb {C}^n$ associated with an $n$-tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of $n$-tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason's problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions.

DOI: 10.14232/actasm-020-303-x

AMS Subject Classification (1991): 47A45, 47A20, 47A48, 47A56

Keyword(s): characteristic functions, analytic model, nilpotent operators, operator-valued polynomials, Gleason's problem, factorizations

received 22.10.2020, revised 28.6.2021, accepted 3.7.2021. (Registered under 53/2020.)