The spectral radius and Liapunov's theorem

AbstractA familiar theorem of Liapunov pertaining to stability of complex matrices is proved anew and extended to bounded linear operators on a Hilbert space. The extension depends on an identity of Taussky which connects equations of the form x − axb = c with those of the form ux + xv + w = 0. Another ingredient in our method is the notion of abscissa of stability, s(u), which corresponds under Taussky's transformation to the spectral radius r(a). When these ideas are combined it is found that a sharpened and generalized form of Liapunov's theorem follows from elementary properties of geometric series