Analytic mappings between noncommutative pencil balls

By J. William Helton, Igor Klep, Scott McCullough. 2009

62 citations recorded by the source corpus.

Abstract In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems (Helton et al. (2009) [10] , de Oliviera et al. (2009) [8] ). In the earlier paper (Helton et al. (2009) [9] ) we characterized NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. In this paper we turn to a more general dimension-free ball B L , called a “pencil ball”, associated with a homogeneous linear pencil L ( x ) : = A 1 x 1 + ⋯ + A g x g , A j ∈ C d ′ × d . For X = col ( X 1 , … , X g ) ∈ ( C n × n ) g , define L ( X ) : = ∑ A j ⊗ X j and let B L : = ( { X ∈ ( C n × n ) g : ‖ L ( X ) ‖ 1 } ) n ∈ N . We study the generalization of NC ball maps to these pencil balls B L , and call them “pencil ball maps”. We show that every B L has a minimal dimensional (in a certain sense) defining pencil L ˜ . Up to normalization, a pencil ball map is the direct sum of L ˜ with an NC analytic map of the pencil ball into the ball. That is, pencil ball maps are simple, in contrast to the classical result of D'Angelo (1993) [7, Chapter 5] showing there is a great variety of such analytic maps from C g to C m when g ≪ m . To prove our main theorem, this paper uses the results of our previous paper (Helton et al. (2009) [9] ) plus entirely different techniques, namely, those of completely contractive maps. What we do here is a small piece of the bigger puzzle of understanding how Linear Matrix Inequalities (LMIs) behave with respect to noncommutative change of variables.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6496c5bcbe2c1453537676a3b24a90933a641313

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Citations in the corpus

Foundations of Free Noncommutative Function Theory (2012, methodology): “We mention also the work of Hadwin [46] and Hadwin–Kaonga–Mathes [47], of Popescu [85, 86, 88, 89], of Helton–Klep–McCullough [55, 51, 52, 53], and of Muhly–Solel [69, 71].”
Global Holomorphic Functions in Several Noncommuting Variables (2015, unknown): “of bounded operators; Ball, Groenewald and Malakorn [10], in the context of extending realization formulas from functions of commuting operators to functions of non-commuting operators; Alpay and Kalyuzhnyi-Verbovetzkii [6] in the context of realization formulas for rational functions that are J-uni”
Noncommutative reproducing kernel Hilbert spaces (2016, background): “ependently in the work of Voiculescu and collaborators (see [49, 50]) in the context of the needs of a free-probability theory, as well as in the work of Helton-Klep-McCullough and collaborators (see [25, 26, 27, 29, 11]) on free noncommutative Linear-Matrix-Inequality relaxations and connections wi”
The implicit function theorem and free algebraic sets (2014, unknown): “alization formulas for rational functions that are J-unitary on the boundary of the domain; Helton [7] in proving positive matrix-valued functions are sums of squares; and Helton, Klep and McCullough [8, 9] and Helton and McCullough [10] in the context of developing a descriptive theory of the domai”
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”
Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, background): “Helton, Klep, and McCullough in [46] introduced such nc domains DL in the finite-dimensional setting, and studied various aspects of the associated nc function theory; following these authors, we shall refer to any such set DL as a pencil ball.”