| Since the 1970s,the study of the reducing subspace problem of multiplication operators in the space of analytical functions has yielded fruitful results.The first was that Nordgren,Thomson,Cowen and others became interested in the transposition and reducing subspace structure problems of multiplicative operators on Hardy space,and thus did a lot of research work and achieved a lot of excellent results.Driven by their research,the results of the reducing subspace of multiplicative operators on Hardy space are becoming more and more abundant.Combined with the results of many branches such as analysis,geometry,algebra,and group theory,a series of conclusions about multiplication operators on Hardy space are obtained.People naturally begin to think about what is the reducing subspace of the Toeplitz operator or other operators on other analytic function spaces? In1995,the concept of the slant Toeplitz operator was introduced by HO.on the Lebesgue space,and its properties were studied in detail.This paper mainly studies the reducing subspace of the slant Toeplitz operator on Hardy space.This article is divided into five chapter as follows:The first part introduces some relevant background knowledge and significance,and reviews the research process of the reducing subspace of the multiplication operator on the spatial analysis function and the important conclusions made.The second part,the study of the form of the reducing subspace and the minimalized subspace of the slant Toeplitz operator on the Hardy space of the unit circumference: first divide the entire space into three parts that are orthogonal to each other,verify that these three parts are the reducing subspaces of the slant Toeplitz operator,and then give the specific form and number of miniaturized subspaces in each part.The third part,based on the first part,generalizes the conclusions obtained to the unit double disk Hardy space to get some similar conclusions.At the same time,by defining the N-transparent function,every minimalated subspace can be generated by the function.The fourth part,which studies the reducing subspace of the H-Toeplitz operator on the Hardy space of a unit circumference,gives the concrete form of its very small reducing subspace.The fifth chapter summarizes the main research content of this article. |
