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 (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 BL, called a “pencil ball”, associated with a homogeneous linear pencilL(x):=A1x1+⋯+Agxg,Aj∈Cd′×d. For X=col(X1,…,Xg)∈(Cn×n)g, define L(X):=∑Aj⊗Xj and letBL:=({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 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 Cg to Cm 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.