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MatrixJunitary noncommutative rational formal power series, in The State Space Method: Generalizations and Applications
 OT 161, Birkhäuser, BaselBostonBerlin
, 2006
"... Abstract. Formal power series in N noncommuting 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 ..."
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Abstract. Formal power series in N noncommuting 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 Ntuples of n×n matrices (with n = 1,2,...) or operators on an infinitedimensional 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 matrixvalued functions which are Junitary on the imaginary line or on the unit circle is extended to the setting of noncommutative rational formal power series. The property of Junitarity holds on Ntuples of n×n skewHermitian versus unitary matrices (n = 1,2,...), and a rational formal power series is called matrixJunitary 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
Interpolation in the noncommutative SchurAgler class
 J. OPERATOR THEORY
, 2007
"... The class of SchurAgler functions over a domain D ⊂ C d is defined as the class of holomorphic operatorvalued functions on D for which a certain von Neumann inequality is satisfied when a commuting tuple of operators satisfying a certain polynomial norm inequality is plugged in for the variables. ..."
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The class of SchurAgler functions over a domain D ⊂ C d is defined as the class of holomorphic operatorvalued functions on D for which a certain von Neumann inequality is satisfied when a commuting tuple of operators satisfying a certain polynomial norm inequality is plugged in for the variables. There now has been introduced a noncommutative version of the SchurAgler class which consists of formal power series in noncommuting indeterminates satisfying a noncommutative version of the von Neumann inequality when a tuple of operators (not necessarily commuting) coming from a noncommutative operator ball is plugged in for the formal indeterminates. The purpose of this paper is to extend the previously developed interpolation theory for the commutative SchurAgler class to this noncommutative setting.
Schurclass multipliers on the Fock space: De Branges–Rovnyak reproducing kernel spaces and transferfunction realizations
"... We introduce and study a Fockspace noncommutative analogue of reproducing kernel Hilbert spaces of de BrangesRovnyak type. Results include: use of the de BrangesRovnyak space H(KS) as the state space for the unique (up to unitary equivalence) observable, coisometric transferfunction realization ..."
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We introduce and study a Fockspace noncommutative analogue of reproducing kernel Hilbert spaces of de BrangesRovnyak type. Results include: use of the de BrangesRovnyak space H(KS) as the state space for the unique (up to unitary equivalence) observable, coisometric transferfunction realization of the Schurclass multiplier S, realizationtheoretic characterization of inner Schurclass multipliers, and a calculus for obtaining a realization for an inner multiplier with prescribed left zerostructure. In contrast with the parallel theory for the Arveson space on the unit ball B d ⊂ C d (which can be viewed as the symmetrized version of the Fock space used here), the results here are much more in line with the classical univariate case, with the extra ingredient of the existence of all results having both a “left ” and a “right” version.
CARATHÉODORY INTERPOLATION ON THE NONCOMMUTATIVE POLYDISK
, 2004
"... Abstract. The Carathéodory problem in the Nvariable noncommutative Herglotz–Agler class and the Carathéodory–Fejér problem in the Nvariable noncommutative Schur–Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory–Fejér) problem has a solution if and only if the noncomm ..."
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Abstract. The Carathéodory problem in the Nvariable noncommutative Herglotz–Agler class and the Carathéodory–Fejér problem in the Nvariable noncommutative Schur–Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory–Fejér) problem has a solution if and only if the noncommutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set Λ) takes operator values with positive semidefinite real part (resp., contractive operator values) on Ntuples of Λjointly nilpotent contractive n × n matrices, for all n ∈ N. 1.