Elementary Operator Norm And The P-hyponormal Son Study

The research of this thesis focuses on norms of elementary operators on rank-oneoperators, Riesz idempotent, Weyl's theorem and normality for p-w--hyponormal operators. the research on norms of elementary operators on rank-one operators gives thesufficient and necessary conditions which are different from A.Seddik's conclusion. Moreoverwe study their properties. The research on p-w-yponormal operators proves theconclusions of wyponormal operators holding for p-w-hyponormal operators andobtains some results which T is quasinormal if and only if T is quasinormal by using blockoperator matrices. The article is divided into four chapters.In chapter 1, we mainly introduce the background knowledge and structure of thispaper.In chapter 2, some notations and definitions are introduced and some well-knowntheorems are given. In section 1, we give some notations. In section 2, we introduce thedifinitions of elementary operator, numerical rang, p-w-yponormal operator, Fredholmoperator, Weyl operator and Weyl spectrum and so on. In section 3, we give some well-knowntheorems and proven theorems, such as polar decomposition theorem and Weyl'stheorem and so on.In chapter 3, we discuss some properties of d(R_A,B) which is denoted by the supremumof the norm of R_A,B rank one operators on Hilbert space H. Accordningto the sufficient and necessary conditions for d(Ra,b)(?), the definitetion of the normalized algebraic numerical range and the properties of the norm, someproperties of (1(Ra,b) are studied. And a new sufficient and necessary condition ford(RA,B) = (?) for n=2 and the lower bound of d(?)are given.In chapter 4, we study some properties of p-w-hyponormal operator. Some characterizationsof Riesz idempotent E_λand (?), with respect toλεisoa(T), of T and (?),respetively are given. It is shown that E_λH=(?). Consequently, E_λis self-adjiont,E_λ=(?) and E_λH = ker(T-λ) = ker(?). Moreover, it is shown that Weyl'stheorem holds for T and (?)∈H((?)T),(?). Finally, using block operator matrices, weshow that T is quasinormal if and only if T is quasinormal when T is p-w-hyponormal.Moreover an example that there exists a non subnormal p-w-hyponormal operator Tsuch that (?) is subnormal is given.