导数Hardy空间上的乘法算子

导数Hardy空间S^2(D)是学者们感兴趣的一个函数空间。2015年，Allen, Heller和Pons研究了S^2(D)上的等距乘法算子，但S^2(D)上的m-等距乘法算子还没有学者研究，m > 1。另外，m-等距算子理论和Toeplitz算子，经典函数论，常微分方程及其他数学分支都有联系，所以我们希望研究S^2(D)上的m-等距乘法算子。.S^2(D)空间与m-等距算子有密切的联系。通过在S^2(D)上定义一个等价的范数，我们发现在此范数下，M_z是S^2(D)上的3-等距算子，且M_B也是S^2(D)上的3-等距算子，其中B是一个有限的Blaschke乘积。因为这一新的发现，所以我们将在本项目中研究S^2(D)上两个不同范数下的m-等距乘法算子。由于m-等距算子是研究约化子空间的重要工具，我们也将研究S^2(D)上M_B的约化子空间，其中B是一个二阶的Blaschke乘积。

Many people are interested in the derivative Hardy space S^2(D). In 2015, Allen, Heller and Pons studied the isometric multiplication operators on S^2(D), but there is no study about m-isometric multiplication operators on S^2(D), m > 1. Moreover, the theory for m-isometries has connections to Toeplitz operators, classical function theory, ordinary differential equations and other areas of mathematics, so we want to study the m-isometric multiplication operators on S^2(D)..The derivative Hardy space S^2(D) has a close relationship with m-isomteries. By defining an equivalent norm on S^2(D), we find out that M_z is a 3-isometry on S^2(D), so is M_B, where B is a finite Blaschke product. Because of this new discovery, we will study the m-isometric multiplication operators on S^2(D) under two different norms. Since m-isometry is an important tool in studying the reducing subspaces, we will also study the reducing subspaces of M_B on S^2(D), where B is a finite Blaschke product of order 2.