Minimum eigenvalue inequalities for Z-transformations on proper and symmetric cones

A linear transformation L defined on a finite dimensional real Hilbert space is said to be a Z-transformation on a proper cone K ifx∈K,y∈K*,and〈x,y〉=0⇒〈L(x),y〉⩽0,where K∗ is the dual of K. Examples of such transformations include Z-matrices on R+n, Lyapunov and Stein transformations on the semidefinite cone. For a Z-transformation L,τ(L):=min(λ):λ∈σ(L)}is an eigenvalue of L with a corresponding eigenvector in K. In this article, when K is a product cone, we relate the Z-property/positive stable property/minimum real eigenvalue of L with those of a subtransformation of L and its Schur complement.