Some Remarks about the Perron Condition for C 0 -semigroups

We prove the following. Let T = fT(t)g t0 be a bounded C 0-semigroup on a Banach space X and A its innnitesimal generator. Then <(A) < 0 if and only if sup t>0 jj t Z 0 e iii T()ddjj < 1; 8 2 R; 8x 2 X: In particular we obtain that a strongly continuous and exponentially bounded family of bounded linear operators U on X is uniformly exponentially stable if and only if the spectrum of the innnitesimal generator of the evolution semigroup associated to U; is lying in C ? := fz 2 C : <(z) < 0g: 1.INTRODUCTION In 1930 Oscar Perron found an interesting connection between the property of uniform exponential stability of diierential systems and a certain condition (called the Perron condition). For the reader's convenience we will