MATRIX CONVEXITY OF FUNCTIONS OF 2 VARIABLES

Let I, J be intervals such that 0 is-an-element-of I and J. Let M(m) be the algebra of all m x m complex matrices, and let M(m)(I) be the set of all hermitian members of M. whose spectrum is contained in I. Let pi(m) be the set of all hermitian projections in M(m). We give a complete characterization of the set of functions f:I X J --> R which satisfy the inequality (P X Q)f(PAP, QBQ)(P X Q) less-than-or-equal-to (P X Q)f(A, B)(P X Q) for all A is-an-element-of M(m)(I), B is-an-element-of M(n)(J), P is-an-element-of pi(m), Q is-an-element-of pi(n). We also generalize some results on matrix convexity of functions of one variable to the case of functions of two variables.