Zeros of SISO Infinite-Dimensional Systems

We give a definition of the zeros of an infinite-dimensional system with bounded control and observation operators B and C respectively. The zeros are defined in terms of the spectrum of an operator on an invariant subspace. These zeros are shown to be exactly the invariant zeros of the system. For the case of SISO systems, where also the range of B is not in the kernel of C, we show that this subspace exists and it is the entire kernel of C. We calculate the operator K such that the spectrum of A + BK on ker(C) is the system zeros, and show that A + BK generates a Co-semigroup on ker(C). If the range of B is not in the kernel of C, a number of situations may occur, depending on the nature of B and C.