Perturbation of operators and applications to frame theory

A celebrated classical result states that an operator U on a Banach space is invertible if it is close enough to the identity operator I in the sense that \\I - U\\ < 1. Here Mle show that U actually is invertible under a much weaker condition. As an application we prove new theorems concerning stability of frames (and frame-like decompositions) under perturbation in both Hilbert spaces and Banach spaces.