Linear elastic systems with damping (1) ÿ

$+B\dot{y}+A\dot{y}=0$ in Hilbert spaces are analysed. In (1), A is a positive definite unbounded linear operator, B is a closed linear operator. Equivalent first-order linear systems to (1) are used to study the systems (1). In the present paper the author investigates the widely used linear elastic systems (1) with damping B related in various ways to

${A}^{\alpha}$ (1/2

$\le \alpha \le 1)$.

${A}^{\alpha}$ is the fractional power of the positive definite operator A for a real number 1/2

$\le \alpha \le 1$. Some preliminary results are describes in section 2 of the paper. The spetral property of the systems (1) associated with

$\alpha \in [1/2,1]$ is discussed and some new results are proved for the analytic property and exponential stability of the semigroup associated with the systems (1). The obtained results have many advantages in engineering applications.