Positive Operators and an Inertia Theorem

In recent years there has been interest in a theorem on positive definite matrices known as Lyapunov's theorem. Several authors have proved generalizations of this theorem, (WIELANDT [29J, TAUSSKY [24J, [25J, [26J , OSTROWSKISCHNEIDER [20J, GIVENS [10J, CARLSON-SCHNEIDER [3J, CARLSON [4J) . Lyapunov's theorem and its generalizations have become known as inertia theorems. In this note we shall use a generalization of the theorem of Perron-Frobenius on matrices with non-negative elements (KREIN-RuTMAN [14J) to prove a new inertia theorem. Our theorem is closely related to known results on M-matrices (OSTROWSKI [19J, SCHNEIDER [20J, FAN [6J, FAN-HoUSEHOLDER [7J, FIEDLERPTAK [8J) . The relation between M-matrix theorems and inertia theorems does not seem to have been observed before.