Perturbation of Browder Spectrum of Upper-triangular Operator Matrices

Abstract

Let $M_{C}=\left(\begin{array}{cc}A&C\\0&B\\\end{array} \right)$ is a 2-by-2 upper triangular operator matrix acting on the Banach space $X\oplus Y$ or Hilbert space $H\oplus K$. For the most import spectra such as spectrum, essential spectrum and Weyl spectrum, the characterizations of the perturbations of $M_{C}$ on Banach space have been presented. However, the characterization for the Browder spectrum on the Banach space is still unknown. The goal of this paper is to present some necessary and sufficient conditions for $M_C$ to be Browder for some $C\in B(Y,X)$ by using an alternative approach based on matrix representation of operators and the ghost index theorem. Moreover, in the Hilbert space case the characterizations of the Fredholm and invertible perturbations of Browder spectrum are also given.