Stability of linear infinite-dimensional systems revisited

The paper is devoted to a study of stability questions for linear infinite-dimensional discrete-time and continuous-time systems. The concepts of power stability and l ^p Instability for a linear discrete-time system x _k+1 = Ax _k (where x _k ε X, X is a Banach space, A is linear and bounded) are introduced and studied. Relationships between these concepts and the inequality r(A) < 1, where r(A) denotes the spectral radius of A, are also given. The discrete-time results are used for a simple derivation of some well-known properties of exponentially stable and L^p-stable linear continuous-time systems described by xdot](t) = Ax(t) (A generates here a strongly continuous semigroup of linear and bounded operators on X). Some remarks on norms related to stable systems are also included.