Research On Several Special Cyclic Operators

In this paper, we study related problems of skew circulant matrices, ω circulant matrices, block ω circulant matrices, block R circulant matrices and block left circulant matrices, such as the problem of isomorphic operators, the problem of norm estimates. The main contents of the thesis consist of four chapters:The first chapter includes three parts, in the first part, we introduce applied background and the current research situation of skew circulant matrices, ω circulant matrices, block ω circulant matrices, block R circulant matrices and block left circulant matrices in the domestic and overseas. In the second part, we give definitions of these special circulant matrices, a few kinds of weak unitarily invariant norms and an important lemma, which are closely related to the following research contents. In the third part, we simply introduce main work of this thesis.The second chapter includes two parts, the isomorphic operators and functional equa-tions for skew circulant algebra and ω circulant algebra are considered. In the first part, we introduce the idempotent bases of skew circulant algebra. Then the linear entire function is constructed by using the idempotent bases. Based on the properties of this linear entire function, we define the operators which are isomorphic to the algebra of complex skew cir-culant matrices and proposed some properties of these operators. What’s more, other skew circulant algebras and a linear involution are given. In the second part, the research content is an extension of the results of the first part, i.e., the results of ω circulant algebra are extension of skew circulant algebra. The research contents of ω circulant algebra include the idempotent bases, function equations, other ω circulant algebras and a linear involution.The third chapter includes three parts, the norm estimates of block ω circulant operator matrices, block R circulant operator matrices and block left circulant operator matrices are mainly studied, where ω= e1θ (0≤θ≤2π), R is an nonsingular unitary matrix. The oper-ator matrices are defined in C*-algebra. In the first part, because block ω circulant operator matrices are diagonalizable and weak unitarily invariant norms have invariant properties, the norm equalities are obtained. We study the connection between general block operator matrices and block ω circulant operator matrices by constructing an special unitary matrix. Based on the properties of pinching inequality, we get the norm inequalities of block ω cir-culant operator matrices. In the second part, norm estimates of block R circulant operator matrices are considered according to the special structure of block R circulant operator ma-trices, the properties of weak unitarily invariant norms and pinching inequality. In the third part, norm estimates of block left circulant operator matrices are proposed.In the forth chapter, we summarize the main work of the paper, then give some sug-gestions and expectations to study the related contents.