The infinite-time admissibility of observation operators and operator Lyapunov equations

Let Sigma(A, -, C) be an abstract dynamical system with A being the generat or of a C-0-semigroup on a Hilbert space H, C : D(A) --> Y a linear operato r, Y another Hilbert space. In this paper, some sufficient and necessary co nditions are obtained for the observation operator C to be infinite-time ad missible. For a control system Sigma(A, B, -), due to duality argument, som e sufficient and necessary conditions are also given for the control operat or B to be extended admissible. It is wellknown that observation operator C is admissible if and only if the operator Lyapunov equation associated wit h the system has a nonnegative solution. In this paper, all nonnegative sol utions to this equation are represented parametrically.