Uniform λ−Adjustment and μ−Approximation in Banach Spaces

We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform λ−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 λ−adjustment, while other fail. However, uniformly λ−adjusted subspaces and linear operators retain their (semi )Fredholm properties in a Banach space which dual is Fréchet-Urysohn in weak∗ topology. We also introduce another concept of perturbation called uniform μ−approximation which is weaker than perturbations by small gap, norm, and compact convergence, yet stronger than uniform λ−adjustment. We present Fredholm stability theorems for uniform μ−approximation in arbitrary Banach spaces and a theorem on stability of Riesz kernels and ranges for commuting closed essentially Kato operators. Finally, we de ne 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 λ−adjustment and uniform μ−approximation. 1 ar X iv :0 80 4. 28 32 v1 [ m at h. FA ] 1 7 A pr 2 00 8