一类算子矩阵的扇形性及复指数幂

研究扇形算子在数乘运算下的扇形性,对数乘算子的谱集及预解算子进行刻画,并探讨了它的纯虚数幂的有界性。以此为基础,采用迭代的方式,构造了Jordan块J(A)的预解式及其复指数幂的表达式;然后,借助矩阵的Jordan标准型,证明了当矩阵M的特征值集中在右半平面的一个扇形区域时,算子A的扇形性及纯虚数幂的有界性将在算子矩阵M(A)中得到保持。该研究成果可以应用于带有散度型或非散度型椭圆算子的偏微分方程组的研究。

Sectorial Property and Complex Exponential Powers of a Class of Operator Matrices

The sectorial property and the expression of the complex powers for a type of operator matrices made of sectorial operators are studied. Firstly, the sectorial property of the operator multiplied by a complex number is investigated, the resolventof the scalar multiplication is described, and the boundedness of the imaginary powers is studied. Based on these results, the resolvents and complex powers of the Jordan block are constructed, by means of iteration. Then with the aid of the Jordan canonical forms of matrices, It is proved that sectorial property and boundedness of the imaginary powers of an operator can be inherited by the operator matrix , in case that all the eigenvalues of are concentrated in a sector on the right plane. Therefore, the findings can be applied in the study of partial differential equation teams driven by the elliptic operators in divergence or non-divergence form.