On an analogue of a property of singular <i>M</i> -matrices for the Lyapunov and Stein operators (Q6661779)

A well-known result for a singular irreducible \(M\)-matrix \(A\) is that the only nonnegative vector in the range of \(A\) is the zero vector. This paper extends the concept of trivial range monotonicity, a property of singular irreducible \(M\)-matrices, to Lyapunov and Stein operators acting on real symmetric matrices. The authors establish trivial range monotonicity for specific classes of matrices such as skew-symmetric and involutory matrices with \(A^2=I\), while providing counterexamples to demonstrate cases where this property does not hold. The results are derived using spectral conditions, group inverses, and a novel concept of generalized \(k\)-potency for operators. The findings are summarized in a table and future research directions are proposed, including the challenge of defining irreducibility for these operators.