Some Extremal Problems for Positive Definite Matrices and Operators

Let C be a real-valued function defined on the set 9& of all positive definite complex hermitian or real symmetric matrices according as F = C (the complex field) or F = R (the real field). Suppose A, B E 9&. We study the optimization problems of (1) finding max C(X) subject to A X, B X positive semidefinite, (2) finding minG(X) subject to X A, X B positive semidefinite. For a general class of functions G, we construct the optimal solutions, and give conditions under which the solutions obtained are unique. The particular case of the determinant function, which motivated this work, is studied in detail. We then extend the results to the infinitedimensional case using the theory of symmetrically normed ideals. Similar optimization problems with more constraints are also briefly discussed.