A general Hua-type matrix equality and its applications

Let Mm×n be the set of all complex matrices of size m × n with Mn = Mn×n. For A ∈ Mm×n, we denote the conjugate transpose of A by A ∗ and call A strictly contractive if I − A∗A is positive definite, where I is the identity matrix of appropriate size. If A ∈ Mn is Hermitian positive (semi)definite, then we write A(≥) > 0. Also, we identify A > (≥)B with A−B > (≥)0, called the Löwner partial order. The starting point of this paper is the following Hua matrix equality, which arises in studying the theory of functions of several variables.