Positive semi-definite 2 × 2 block matrices and norm inequalities

Let M=[AXX⁎B]∈C2n×2n be positive semi-definite 2×2 block matrix, where A,B,X∈Cn×n. A characterization for the matrix M with A+B=kI to be positive partial transpose is given in terms of its spectral norm. Using this result, a counter-example is constructed for the conjecture that ‖M‖≤‖A+B‖ when X is normal for all unitarily invariant norms.