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 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 DʹAngelo (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.