The Weyl Type Theorems Of Operators And The P-fusion Frames

One of the most important problems in the spectral theory of operators is the modification and extension of Weyl type theorems,which is a hot topic in recent years.Fruitful results have been achieved.The concept of frame in Hilbert spaces was first introduced by Duffin and Schacffer,it is a generalization of standard or-thogonal basis.After the fundamental work of Daubechies,Grossman and Meyer,the frame theory and its applications has obtained many domestic and foreign schol-ars's interests and concerns,and it has been widely studied and developed rapidly.This thesis is based on this international frontier research.It focus on the spectral properties(Mw)and(mv)and the equivalent description of the p-fusion frames.Spectral properties(Mw)and(mw)are associated with the Weyl type theorem.This paper is divided into three chapters.The first chapter gives the definitions of two spectral properties(Mw)and(mw)and studies them.It mainly discusses the relationship among the spectral properties(Mw),(mw)and other spectral proper-ties,including(z)?(az)?(aW)?(aB)?(W)?(B)?(qaW)and(qaB).The stabilities of(Mw)and(mw)under all kinds of operator perturbations and under the direct sum operation have also been discussed.By using the results of the relations be-tween the topological uniform descent operators and the Weyl type theorems,the second chapter achieves some results of the relations between the quasi-Fredholm operators and the Weyl type theorems,and also achieves some results of the rela-tions between the quasi-Fredholm operators and the properties(R)and(gR).The third chapter studies the p fusion frames.It gives the definitions of p fusion Bessel sequences,p fusion frames and q fusion Riesz bases in Banach spaces.It describes a close relation between the p fusion frames and the p-frames and provide the equiva-lent descriptions of the p fusion Bessel sequences,p fusion frames and q fusion Riesz bases by using the analysis operators and the synthesis operators.