 
Summary: DRAFT
Left invertibility and invertible solutions of the
Lyapunov equation
Hans Zwart
Department of Applied Mathematics,
Faculty of Electrical Engineering, Mathematics and Computer Science,
University of Twente,
P.O. Box 217, 7500 AE Enschede,
The Netherlands.
h.j.zwart@math.utwente.nl
August 25, 2011
Abstract
If the solution of a Lyapunov or Riccati equation corresponding to
a finitedimensional system is invertible, then this leads to additional
properties of this system. For infinitedimensional systems these re
sults do not need to hold. We show that it is necessary to put some
additional conditions on the spectrum of the system operator A. For
instance, if this operator has compact resolvent, then the results known
for finitedimensional systems extend to infinitedimensional ones. Us
ing this result, we obtain more insight in Lyapunov and Riccati equa
