Sets of Positive Operators with Suprema
William N. Anderson, Thomas D. Morley, George E. Trapp · SIAM Journal on Matrix Analysis and Applications · 1990
Let K be an n-by-n self adjoint matrix and let $K \otimes X$ be the Kronecker product of the matrix K and the linear operator X. Thus if $X:H \to H$, where H is a Hilbert space, then $K \otimes X:H^n \to H^n $. Given a positive operator A, operators of the form $A + K \otimes X$, where X is a positive operator, are studied. The signs of the eigenvalues of K and the rank of K play crucial roles in characterizing suprema of the following set: $\{ X\geqq 0|A +K \otimes X\geqq 0 \}$. It is shown that $\{ X\geqq 0 | A + K \otimes X\geqq 0 \}$. has a supremum for all $A\geqq 0$ if and only if K has exactly one negative eigenvalue. For the cases rank $K = 1$ and rank $K = 2$, the existence of the supremum is already known under the names “shorted operator” and “cascade limit,” respectively. The suprema in the case that rank $K > 2$ are new nonlinear operations.