On Spectra Of 2×2 Upper Triangular Operator Matrices

Let H and K be Hilbert spaces, B(H), B(K,H) denote the set of all bounded linear operators on H and from H into K, respectively. For given A ∈ B(H), B ∈ B(K), let M_C denote the 2 × 2 upper triangular operator matrix:In recent ten years, the problems on perturbations of spectra of 2 × 2 upper triangular operator matrices have obsorbed many scholars, such as H. K. Du, D. S. Djordjevic, W. Y. Lee, J. K. Han, H. Y. Lee, M. Barraa and so on (see [1-9]). In this article, we continue to study the problems on perturbation of spectra of 2 x 2 upper triangular operator matrices, such as Drazin spectrum, Weyl spectrum and Browder spectrum. Moreover, we give a partial answer to the question that the existence of operator C₀ advanced by H. K. Du and J. Pan in [1].There are three chapters in this article, and the main content as follows:In the first chaper, we point out that if two of the three operators A, B, M_C are Drazin invertible, then the third is Drazin invertible by the research towards Drazin invertibility of upper triangular operator matrix. Similar to the spectra, Drazin spectra of A, B, M_C have following relation: σ_D(M_C) (?) σ_D(A)∪σ_D(B) and (σ_D(A)∪σ_D(B)) \ σ_D(M_C) (?) σ_D(A)∩σ_d(B). Besides, we give a characterization of the intersection.of Drazin spectra ∩_C∈BK,H σ_D(M_C) under a certain condition.In the second chapter, we study the Browder's theorem and perturbations of Weyl spectrum, Browder spectrum. For given operators A ∈ B(H), B ∈ B(K), we investigate the intersection of Weyl spectra ∩_C∈BK,H σ_w(M_C) by analyse the spectrum structure of operators and give a complete characterization of the intersection of Browder spectra ∩_C∈BK,H σ_b(M_C). Moreover, we give another repesentention of ∩_C∈BK,H σ_b(M_C) under a certaion condition.And in the chapter three, we explore the problem advanced by H. K. Du and J. Pan in [1]:For given operators A ∈ B(H), B ∈ B(K), whether there exists an operatorCo G B(K, H) such thata{MCo) = P|