Uniform lamda-adjustment and mu-approximation in Banach spaces

TL;DR

This paper introduces new perturbation concepts called uniform lambda-adjustment and uniform mu-approximation in Banach spaces, analyzing their effects on Fredholm properties and stability of subspaces, operators, and complexes.

Contribution

It defines these novel perturbation notions, compares them with classical ones, and establishes stability theorems for Fredholm properties under these perturbations in Banach spaces.

Findings

Uniform lambda-adjustment is weaker than small gap and norm perturbations.

Fredholm stability theorems hold under uniform lambda-adjustment in certain Banach spaces.

Uniform mu-approximation is stronger than lambda-adjustment and preserves Fredholm properties.

Abstract

We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform lambda-adjustment which is weaker than perturbations by small gap, operator norm, q-norm, and K2-approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform lambda-adjustment, while other fail. However, uniformly lambda-adjusted subspaces and linear operators retain their (semi--)Fredholm properties in a Banach space which dual is Fr\'{e}chet-Urysohn in weak* topology. We also introduce another concept of perturbation called uniform mu-approximation which is weaker than perturbations by small gap, norm, and compact convergence, yet stronger than uniform lambda-adjustment. We present Fredholm stability theorems for uniform mu-approximation in arbitrary Banach spaces and a theorem on stability of Riesz kernels and ranges for commuting closed essentially Kato operators. Finally, we define the new concepts of a tuple of subspaces and of a complex of subspaces in Banach spaces, and present stability theorems for index and defect numbers of Fredholm tuples and complexes under uniform lambda-adjustment and uniform mu-approximation.