| This paper is to study Furuta type operator inequalities and some classes of opera-tors including hyponormal operators, such as p-hyponormal operators, class wF(p,r,q),class A(n), n-paranormal operators, and so on. They are further generalizations of hy-ponormal operators.In chapter 1, the Furuta inequality and some classes of operators are introduced.In chapter 2, a new Furuta type inequality is proved, we call it the complete formof Furuta inequality. It is a refinement of Furuta's result ( [44]) and a new developmentof Furuta inequality ( [43]).Chapter 3 is to show the complete form of generalized Furuta inequalities. Thecomplete form of generalized Furuta inequalities are improvement of generalized Fu-ruta inequalities, especially the complete form of grand Furuta inequality ( [47]) isshowed.In chapter 4, some applications of the Furuta inequality and its complete form top-hyponormal operators are obtained. A kind of order structure on powers of opera-tors is introduced and some results are obtained by using Furuta inequality, this is afurther development of the problem of powers of hyponormal operators (Problem 209,Halmos [65]); some results on the Aluthge transformation on p-hyponormal operatorsare provided which are extensions of the related results in [70].In chapter 5, spectrum of class wF(p,r,q) is considered. Emphases are put onthe Riesz idempotent with respect to nonzero isolated point spectrum, single valuedextension property, Bishop's property (β) and Weyl's theorem. This chapter is a gener-alization of the related results in [66,87].In chapter 6, class A(n) and n-paranormal operators are introduced and somespectral properties of these operators are discussed. The spectral mapping theorem on Weyl spectrum of a n-paranormal operator is proved, and the nonzero (approxi-mate) point spectrum and nonzero joint (approximate) point spectrum of class A(n)are identical. These results are generalizations of the related results in [111].
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