Perturbations of spectra and property (R) for upper triangular operator matrices

Let H and K be two complex infinite dimensional separable Hilbert spaces. For T is an element of B(H), T is said to satisfy property (R) if the complement in the approximate point spectrum of the Browder essential approximate point spectrum coincides with the isolated eigenvalues of finite multiplicity. In this paper, we mainly give the sufficient and necessary conditions for the 2 x 2 upper triangular operator matrices such that they satisfy property (R) using the features of the elements on the diagonal. (c) 2022 Elsevier Inc. All rights reserved.