Matrix-J-unitary Non-commutative Rational Formal Power Series

Abstract

Abstract. Formal power series in N non-commuting indeterminates can be considered as a counterpart of functions of one variable holomorphic at 0, and some of their properties are described in terms of coefficients. However, really fruitful analysis begins when one considers for them evaluations on N -tuples of n × n matrices (with n = 1, 2, . . .) or operators on an infinite-dimensional separable Hilbert space. Moreover, such evaluations appear in control, optimization and stabilization problems of modern system engineering.In this paper, a theory of realization and minimal factorization of rational matrix-valued functions which are J-unitary on the imaginary line or on the unit circle is extended to the setting of non-commutative rational formal power series. The property of J-unitarity holds on N -tuples of n × n skew-Hermitian versus unitary matrices (n = 1, 2, . . .), and a rational formal power series is called matrix-J-unitary in this case. The close relationship between minimal realizations and structured Hermitian solutions H of the Lyapunov or Stein equations is established. The results are specialized for the case of matrix-J-inner rational formal power series. In this case H > 0, however the proof of that is more elaborated than in the one-variable case and involves a new technique. For the rational matrix-inner case, i.e., when J = I, the theorem of Ball, Groenewald and Malakorn on unitary realization of a formal power series from the non-commutative Schur-Agler class admits an improvement: the existence of a minimal (thus, finite-dimensional) such unitary realization and its uniqueness up to a unitary similarity is proved. A version of the theory for matrix-selfadjoint rational formal power series is also presented. The concept of non-commutative formal reproducing kernel Pontryagin spaces is introduced, and in this framework the backward shift realization of a matrix-J-unitary rational formal power series in a finite-dimensional non-commutative de Branges-Rovnyak space is described. Mathematics Subject Classification (2000). Primary 47A48; Secondary 13F25, 46C20, 46E22, 93B20, 93D05.

Free holomorphic functions on the unit ball ofB(H)n, II

“…. , X n ) ∈ [B(H) n ] 1 . In particular, in the scalar case, we obtain a characterization and parametrization of all bounded free holomorphic functions on the unit ball [B(H) n ] 1 .…”

“…. , X n ) ∈ B(H) n : X 1 X * n + · · · + X n X * n 1/2 < γ , as the set of all power series α∈F + n a α Z α with radius of convergence γ , i.e., {a α } α∈F + n are complex numbers with lim sup k→∞ ( |α|=k |a α | 2 ) 1 where the convergence is in the operator norm topology. Due to the fact that a free holomorphic function is uniquely determined by its representation on an infinite-dimensional Hilbert space, we identify, throughout this paper, a free holomorphic function with its representation on a separable infinite-dimensional Hilbert space.…”

Aspects of non-commutative function theory

“…In addition to the motivations above, let us mention the work of Voiculescu [37], in the context of free probability; Popescu [26][27][28][29], in the context of extending classical function theory to d -tuples of bounded operators; Ball, Groenewald and Malakorn [8], in the context of extending realization formulas from functions of commuting operators to functions of non-commuting operators; Alpay and Kalyuzhnyi-Verbovetzkii [5] in the context of realization formulas for rational functions that are J -unitary on the boundary of the domain; and Helton, Klep and McCullough [13,14] and Helton and McCullough [18] in the context of developing a descriptive theory of the domains on which LMI and semi-definite programming apply; Muhly and Solel [23], in the context of tensorial function theory; Cimpric, Helton, McCullough and Nelson [10] in the context of non-commutative real Nullstellensätze; the second author and Timoney [22] and of Helton, Klep, McCullough and Slinglend, in [17] on non commutative automorphisms; and the work of Pascoe and TullyDoyle [25] on non-commutative operator monotonicity.…”

Carathéodory interpolation on the non-commutative polydisk

“…. , z N such that for every C ∈ D N (or equivalently, for every C ∈ D N matr , see [8]) the series w∈F N F w ⊗C w converges in the operator norm to the contractive operator F (C).…”