The Spectral Properties Of Several Classes Of Operators And The Monotonicity Of Several Operator Functions

In this dissertation,hyponormal operators,A(k)operators and*-A(k)operators are extended to several new operators,and the spectral properties of these operators are dis-cussed and some operator monotone functions are constructed.The contents are summa-rized in the following.Chapter Ⅰ is introduction.We mainly introduce the research background and research target.Besides,we also introduce some basical knowledge referred in this dissertation.Chapter Ⅱ is relative properties of*-A(n)operators and Weyl’s theorem.In this chapter,we prove that*-A(n)operator is normaloid and obtain that algebraically*-A(n) operator on a complex Hilbert space has SVEP and it is polaroid.As an application,if T or T*is an algebraically*-A(n)operator,then Weyl’s theorem holds for f(T) for every f∈H(σ(T)).Chapter Ⅲ is a note on quasi-*-A(k) operators.In this chapter,we introduce quasi-*-A(k) operators and obtain the conclusions as follows:(ⅰ)If T is quasi-*-A(k) for0<k≤1, then the spectral mapping theorem holds for the essential approximate point spectrum.(ⅱ) If T is quasi-*-A(k) for0<k≤1,then σja(T)\{0}=σa(T)\{0}.Besides,we consider the property of the tensor product for*-A(k) operators.In Chapter Ⅳ,we mainly studies quasi-A(k) operators.First,we present quasi-A(k) operators,quasi-absolute-k-paranormal operators and prove their inclusion relation.In addition,we also construct two classes of operator monotone functions and obtain the conclusions as follows:(i)If T is quasi-A(k) for k>0,then F(l)=T*(T*｜T｜2kT)1/l+1T is increasing for l≥k>0.(ii)If T is quasi-absolute-k-paranormal for k>0,then f(l)=‖｜T｜lT2x‖1/l+1‖Tx‖l/l+1is increasing for l≥k>0.Finally,several examples are given to show that proper inclusion relation between the two classes.