Inertia and controllability in infinite dimensions (Q1100148)

The inertia of an \(n\times n\) matrix A with complex entries is the integer triple \(In(A)=(\pi (A),v(A),\delta (A))\), where \(\pi\) (A) is the number of eigenvalues in the open right half plane, v(A) is the number of eigenvalues in the open left half plane and \(\delta\) (A) is the number of eigenvalues on the imaginary axis. There are various theorems which relate the inertia of a matrix to controllability properties and the solvability of various linear equations [see \textit{F. R. Gantmacher}, The theory of matrices, Vol. I and II, Chelsea Publ. Co. (New York 1959) (for a review of the Russian original (1953) see Zbl 0050.248); and \textit{C. T. Chen}, SIAM J. Appl. Math. 25, 158-161 (1973; Zbl 0273.15009)]. The present paper generalizes these concepts to bounded operators on a Hilbert space.