Perturbation analysis for the linear operator equation using pseudospectrum and condition pseudospectrum

In this article, we consider the linear operator equation in a Banach space. The relative perturbation of the solution x corresponding to the perturbation of y, the perturbation of A and the perturbation of both A, y are characterized from the pseudospectrum and the condition pseudospectrum of A. Certain examples are given to illustrate the results. A relation between the pseudospectrum and the condition pseudospectrum of an operator are established. The distance to instability and the distance to singularity of an operator are also found from the condition pseudospectrum of the operator.