Noncommutative ball maps

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. In this paper we characterize 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". We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, "NC ball maps" are very simple, in contrast to the classical result of D'Angelo on such analytic maps over C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables.Comment: 46 page

A Halmos Doctrine and Shifts on Hilbert Space

“…I will omit a formal statement, but I bring it up here because the key step in [46] for understanding an automorphism of H ∞ (E) was to express it in terms of Schur-class functions. There are points of contact between this result and a recent preprint of Bill Helton, Igor Klep, Scott McCullough and Nick Slinglend [29] that I mentioned above in the context of matrix inequalities.…”

“…It is impossible to do justice to this subject here or even to cite all the relevant literature. However, I want to call special attention to [29] in which connections with the theory of fully matricial sets and functions are explicitly made. Also, although he does not express himself in terms of these objects, Popescu's work in [58,60] and elsewhere leads naturally to fully matricial domains and functions.…”

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.…”

Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

“…Voiculescu [Voi04,Voi10] or the forthcoming paper of Kaliuzhnyi-Verbovetskyi and Vinnikov for an introduction. The rigidity of nc bianalytic maps is investigated by Popescu [Pop10]; see also [HKMS09,HKM11a,HKM11b]. For other properties of nc analytic functions, a very interesting body of work, e.g.…”