On The Spectra Of Upper-triangular Operator Matrices

Operator spectral theory is an important research subject of functional analysis, which is widely used in many fields such as mathematical physics and engineering. Due to the fact that an operator which has invariant subspace can be written as a 2×2 upper-triangular operator matrix, in the last two decades, problems related to spectra of upper triangular matrices has become an important direction of study, and attracted a large number of scholars.In this thesis, we study the relationship between upper-triangular operator matrices and single-valued extension property, the perturbations of Browder spec-trum and the Fredholm perturbation of some kinds of spectra for upper-triangular operator matrix. The main content of it contains the following ones:First, we brief introduce the background and history of operator matrices and point out the goal and significance of such subjects.Second, we establish the relationship between the Browder spectrum. Drazin spectrum, left (right) spectrum, left (right) semi-Browder spectrum, left (right) semi-Fredholm spectrum, left (right) semi-Weyl spectrum and the single valued expansion property of operators.Third, making use of the backward shift and shift multiplicities, we give the characterizations for the bounded perturbations of left semi-Browder spectrum and Browder spectrum. Moreover, the new characterizations for the bounded pertur-bations of left spectrum, left semi-Fredholm spectrum and left semi-Weyl spectrum are given, which are more concise and explicit.In the last part, we present the characterizations for the Fredholm perturba-tions of right spectrum, right semi-Fredholm spectrum, right semi-Weyl spectrum, left spectrum, left semi-Fredholm spectrum, left semi-Weyl spectrum, Browder spec-trum, spectrum, essential spectrum and Weyl spectrum.