Linear programming and operator means

AbstractThe shorted operator, the geometric mean, and the cascade limit are all examples of operations that are of the form sup{X¦C + K ⊗ X ≥ 0}, where K ⊗ X denotes the Kronecker product of the matrix K with the matrix X, K is a given n by n self-adjoint matrix, and C is a given positive semidefinite matrix. The supremum is taken with respect to the partial order generated by the positive semidefinite matrices. In all of the above examples the matrix K has exactly one negative eigenvalue. We show by linear programming techniques that if K has this property, and Xmax = sup{X¦C + K ⊗ X ≥ 0}, then (Xmaxc, c) = inf tr(AY), subject to: −∑i,j = 1nkijYij ≥ cc∗, Y = {Yij)i,j = 1n ≥ 0} In the case of the geometric mean A#B of two positive semidefinite matrices, this implies the new result that (A#Bc, c) = inf{tr(AY11 + BY22¦Y12 + Y21 ≥ cc∗, Y ≥ 0}