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

By Daniel Alpay, D. S. Kalyuzhnyı̆-Verbovetzkiı̆. 2004

69 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/04bcd308cfac35d5e036ed44aab64ab3a0008351

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Citations in the corpus

Foundations of Free Noncommutative Function Theory (2012, background): “In particular, transfer functions of conservative noncommutative multidimensional systems are characterized as formal power series whose values on a certain class of d-tuples of operators are contractive (matrix evaluations actually suffice — see [9]).”
Pick Interpolation for free holomorphic functions (2013, background): “Nc-functions have been studied for a variety of reasons: by Taylor [29], in the context of the functional calculus for non-commuting operators; Voiculescu [30, 31], in the context of free probability; Popescu [23, 24, 25, 26], in the context of extending classical function theory to d-tuples of boun”
Free holomorphic functions on the unit ball of B(H)n (2006, background): “We mention that, for the noncommutative polydisc, a representation theorem of the same flavor was obtained in [22] and [1].”
Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting (2009, methodology): “For example, Alpay and Kalyuzhnyĭ-Verbovetzkiĭ [2] use noncommutative Givone–Roesser realizations (see [5,27,28] for the original commutative version) in the setting of matrix-J-unitary noncommutative rational factorizations.”
Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, background): “Motivation comes from a result of Alpay–Kaliuzhnyi-Verbovetskyi [8] which says that that one need only plug in T = (T1, .”
Global Holomorphic Functions in Several Noncommuting Variables (2015, background): “This has been done in a variety of ways: by Taylor [24], in the context of the functional calculus for non-commuting operators; Voiculescu [25,26], in the context of free probability; Popescu [19–22], in the context of extending classical function theory to d-tuples of bounded operators; Ball, Groen”
The implicit function theorem and free algebraic sets (2014, unknown): “Nc-functions have been studied for a variety of reasons: by Anderson [4] as a generalization of the Weyl calculus; by Taylor [20], in the context of the functional calculus for non-commuting operators; Popescu [14, 15, 16, 17], in the context of extending classical function theory to d-tuples of bou”