About stability of continuous invertibility of linear operators in Banach space

Author: jemal rogava
Co-authors: Jemal Rogava
Keywords: linear operators, continuous invertibility
Annotation:

Applying schemes for proofs for "symmetric" theorems about closure and self-adjointness stability for linear operators, from widely-known book " Perturbation Theory for Linear Operators " by T. Kato, we prove the following: Let A and B be densely defined linear operators in Banach space X . Let D(A) is a subset of D(B) and let the following inequalities hold for any vector u in D(A): ||Bu||≤c||Au|| and ||Bu||≤q||Au||+||(A+B)u||, where c and q are positive constants and q<1. Then operator A is continuously invertible if and only if (A+B) is.