The Equivalence of L,StabiIity, the Resolvent Condition, and Strict H-Stability

The Kreiss matrix theorem asserts that a family of N X N matrices is L,-stable if and only if either a resolvent condition (R) or a Hennitian norm condition (H) is satisfied. We give a direct, considerahly shorter proof of the power-houndedness of an N X N matrix satisfying (R), sharpening former results by showing that powerhoundedness depends, at most, linearly on the dimension M. We also show that L,-stability is characterized by an H-condition employing a general H-numerical radius instead of the usual H-norm, thus generalizing a sufficient stability criterion, clue to Lax and Wendroff.