Study On The Perturbations Of Spectrum Of Operator Matrices And Drazin Operators

In this paper, we mainly study the problem of the perturbations of spectrum of operator matrix and the problem of the Drazin invertibility and Drazin inverses of linear combinations of two idempotents, leading to some new results about these thesises.Firstly, we investigate some related spectrum issues for the continuous spectrumσc, point spectrumσp and residual spectrumσr of the 2×2 upper operator matrix. Then the sets∩C∈B(Y,X)σc(Mc),∩C∈B(K,H)σpp(Mc) and∩C∈B(K,H)σr(Mc) are, re-spectively, characterized completely(see theorem 1.2.4,1.2.7 and 1.2.10). We also give an affirmative answer for the attainment problem of the continuous spectrumσc(see proposition 1.2.5).Secondly, three new classes of operators—Kato-Weyl operator, Kato-left Weyl operator and Kato-right Weyl operator—are defined in separable Hilbert spaces(see definitions 2.1.2-2.1.4). Then it follows some sufficient or necessary conditions for the 2×2 upper operator matrix to be Kato-Weyl operator, Kato-left Weyl operator, Kato-right Weyl operator, respectively. And we obtain some stabilities of perturbations of the newly defined operators(see theorem 2.3.1,2.3.2 and 2.3.3).Finally, by virtue of the thought and methods of block of operator matrix, we have some more generalized results of the Drazin invertibility and Drazin inverses of linear combinations of two idempotents under some conditions in separable Hilbert space(see theorem 3.2.3,3.2.5 and 3.2.7). Also the characterizations of eigenprojections at zero of Drazin invertible operators are extended from Hilbert space into Banach space.In addition, this article has two features:one is that we construct some examples, which are of great help for the study of the spectrum of operator matrices; the other one is that we define three new classes of operators and the corresponding spectrumσ*∈{σω