Study On The Spectrum Of Operator Matrices

Recently, the study of spectral theory of operator follows the law of the relation between the whole and the part. And there has been heated research on the spectral theory of operator matrix around the world, in particular,the 2*2 upper operator matrix. Under this context, we systematically study the filling holes problem and the problems on the perturbations of spectrum of operator matrix. And these studies have led to some rich and profound results in this article.Firstly, A scientific classification of spectrums is given according to filling holes of spectrum (see Definition 0.0.3) and a related example is given (see Example 3.2.10), which corrects some mistakes that appear in some references. And according to the profound study of Drazin inverse. and the generalized Drazin inverse, which are both widely used in other subjects, we conclude that there are 14 more kinds of spectrums which have the property of the filling holes of spectrum (see§3.2 and§3.3), including Drazin spectrum and generalized Drazin spectrum (see Theorem 1.3.8 and Theorem 12.1.5).Secondly, we study the problems on perturbation of spectrum. The two majorconcerns of comparingσ_*(M_C) withσ_*(?)(in most cases it is equal toσ_*(A)∪σ_*(B)) are as follows: One is the invariability of perturbation of spectrum, that is,σ_*(A)∪σ_*(B) =σ_*(M_C) holds under certain conditions. The other one is the representation of(?)σ_*(M_C). For the latter, we only give partly answer forsome spectrums under certain conditions (see Throrem2.2.3,Theorem4.4.4,Theorem4.4 and Theorem1.1.4).In addition, a feature of this article is that we construct lots of examples, which are of great help for the study of the spectrum of operator matrices.