BCL index and Fredholm tuples

Fredholm index and spectral flow

for all k. This shows that the sequence of functions is uniformly equicontinuous. Let now t ∈ R and assume that the sequence (|fk(t)|) is unbounded. Then it follows from the estimate that the sequence (min[t−1,t+1] |fk|) is unbounded and therefore also that (‖fk‖L2) is unbounded, contradiction. We conclude the the sequence ‖fk(t)‖ is bounded, for every t ∈ R. Let now T > 0. Then it follows by A...

Fredholm Operators and the Generalized Index

One of the most fundamental problems in mathematics is to solve linear equations of the form Tf = g, where T is a linear transformation, g is known, and f is some unknown quantity. The simplest example of this comes from elementary linear algebra, which deals with solutions to matrix-vector equations of the form Ax = b. More generally, if V,W are vector spaces (or, in particular, Hilbert or Ban...

Fredholm Operators and the Family Index

Stability of index for semi-Fredholm chains

We extend the recent stability results of Ambrozie for Fredholm essential complexes to the semi-Fredholm case. Let X,Y be Banach spaces. By an operator we always mean a bounded linear operator. The set of all operators from X to Y will be denoted by L(X,Y ). Denote by N(T ) and R(T ) the kernel and range of an operator T ∈ L(X, Y ). Recall that an operator T : X → Y is called semi-Fredholm if i...

Homeomorphisms and Fredholm Theory for Perturbations of Nonlinear Fredholm Maps of Index Zero with Applications

We develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps of index zero is also studied. We apply these results to the unique and finite solvability of ...