Some functions preserving positive semidefiniteness of 2 × 2 block matrices

Let n∈Nn∈N and [Ajk]j,k=1,2[Ajk]j,k=1,2 be a Hermitian 2n×2n2n×2n matrix partitioned into four quadratic matrices AjkAjk of order n  . Marcus and Watkins proved in 1971 that [trAjk2] is positive semidefinite whenever [Ajk][Ajk] is positive semidefinite. Let A   be an n×nn×n matrix. If f   is a CC-valued function on CC, denote by f(A)f(A) the value of the primary matrix function associated with f on the matrix A. If F   is a symmetric CC-valued function on CnCn, let F(A)F(A) be the value of F on the eigenvalues of A. Generalizing Marcus and Watkins' result we describe those functions f and F  , for which [F(f(Ajk))][F(f(Ajk))] is positive semidefinite whenever [Ajk][Ajk] is positive semidefinite. We extend an example by Choudhury, which gives a negative result for 3×33×3 block matrices.