Paper detail

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. In an earlier paper 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 &#34;NC ball maps&#34;. In this paper we turn to a more general dimension-free ball B_L, called a &#34;pencil ball&#34;, associated with a homogeneous linear pencil L(x):= A_1 x_1 + ... + A_m x_m, where A_j are complex matrices. For an m-tuple X of square matrices of the same size, define L(X):=\sum A_j \otimes X_j and let B_L denote the set of all such tuples X satisfying ||L(X)||<1. We study the generalization of NC ball maps to these pencil balls B_L, and call them &#34;pencil ball maps&#34;. We show that every B_L has a minimal dimensional (in a certain sense) defining pencil L&#39;. Up to normalization, a pencil ball map is the direct sum of L&#39; 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&#39;Angelo on such analytic maps in C^m. To prove our main theorem, this paper uses the results of our previous paper mentioned above plus entirely different techniques, namely, those of completely contractive maps.