An estimate of the Green matrix and the continuity of the dichotomy parameter (Q915930)

A matrix A is called exponentially dichotomic if its dichotomy parameter k(A) [see \textit{A. Ya. Bulgakov}, Sib. Mat. Zh. 21, No.3, 32-41 (1980; Zbl 0444.34050)] is finite. For example, every Hurwith matrix is exponentially dichotomic. In the above paper, the author showed the continuity of k on all Hurwitz matrices. More generally, it is shown here that, if \(k(A)<\infty\), then also \(k(A+B)<\infty\) for \(\| B\|\) sufficiently small, and \(| k(A+B)-k(A)| \leq C(\| B\| /\| A\|),\) where C does not depend on \(\| B\|\) or the rank of the matrix.