The Problem Of 2×2 Block Operator Matrices Generating C₀ Semigroups

In this paper, the problem of C0 semigroups generated by 2×2 block operator matrices is studied. First, many problems can convert into 2×2 block operator matrices in our study.So it is necessary to study block operator matrices. It is described in this paper that how linear operator semi-group generatation and work. Secondly, a series of sufficient conditions that diagonal block operator matrices can generate Co semigroups is obtained. The results are applied to the Skew-diagonal Block Operator matrix generated by a class of initial valve problem of constant hyperbolic PDEs,, which also proves that diagonal block operator matrices can generate Co semigroups. Moreover, the results have been verified by Hille-Yosida theorem, and the expressions of generated Co semigroups are presented. Finally, the sufficient conditions for Co semigroups generated by 2×2 block operator matrices are also given and instructed by examples.