Real Algebraic and Analytic Geometry
e-mail: , ,
Submission: 2010, January 5.
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 the earlier paper (see 261) 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_m x_m, where A_j are complex g'-by-g 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 "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 D'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.
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.
Mathematics Subject Classification (2000): 47A56, 46L07, 32H99, 32A99, 46L89.
Keywords and Phrases: noncommutative analytic function, complete isometry, ball map, linear matrix inequality.
Full text, 30p.: dvi 211k, ps.gz 424k, pdf 503k.