The Perturbations On Spectra Of Operator Matrix

It is well known that operator matrix is a matrix with operators as its elements. In the recent ten years, the problems on perturbations of spectrum of 2 ×2 operator matrix, especially 2×2 upper triangular operator matrix, have absorbed many scholars, such as H.K.Du, D.S.Djordjevic, J.Pan, J.K.Han, H.Y.Lee, W.Y.Lee and so on. In this article, we study the problems on perturbations of spectrum of 2x2 operator matrix, especially 2×2 upper triangular operator matrix, such as left (right) spectrum, left(right) essential spectrum and left (right) Weyl spectrum. Let H. and K. be complex separable Hilbert spaces. Denote B(H) and B(K,H} the Banach spaces of all the bounded linear operators on H and from K into H, respectively. If A ∈ B(H), B ∈ B(K), for C ∈ B(K, H), let MC denote the 2 × 2 upper triangular operator matrix:There are three chapters in this article, and the main content as follows:In the first chapter, we have studied the intersection of left spectrum of MC, when C is restricted on the set of all the operators with closed range Bc(K,H), the set of all the left invertible operators Bl(K, H], the set of all the compact operators K(K, H) and the set of all the operators with finite range F(K,,H), respectively. We give the characterizations of them. As a result, we getand And in the second chapter, we have studied the perturbations of left (right) essential spectrum, and left (right) Weyl spectrum. If A ∈ B(H),B ∈ B(K), we give the complete characterizations of the intersection of left (right) essential spectrum and the intersection of left (right) Weyl spectrumIn Chapter Three, we have studied the perturbations of spectrum of general 2×2 operator matrix MX, whereIf A ∈ B(H),B ∈ B(K),C ∈ Bc(K,H), we characterize the intersection of left (right) essential spectrum of , , and the intersection of left (right) Weyl spectrum of...