Analytic Mappings between Noncommutative Pencil Balls

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 [HMPV, dOHMP]. In the earlier paper [HKMS] 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 BL, called a “pencil ball”, associated with a homogeneous linear pencil L(x) := A1x1 + · · ·+ Agxg, Aj ∈ C ′×d. For X = col(X1, . . . , Xg) ∈ (Cn×n)g, define L(X) := ∑ Aj ⊗Xj and let BL := ( {X ∈ (Cn×n)g : ‖L(X)‖ < 1} ) n∈N. We study the generalization of NC ball maps to these pencil balls BL, and call them “pencil ball maps”. We show that every BL has a minimal dimensional (in a certain sense) defining pencil L̃. Up to normalization, a pencil ball map is the direct sum of L̃ 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 on such analytic maps in C. To prove our main theorem, this paper uses the results of our previous paper [HKMS] 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.