Fredholmoperátorok

Fredholmoperátorok are linear operators between Banach spaces that possess specific properties related to their index. An operator T: X -> Y is a Fredholm operator if its null space (kernel) is finite-dimensional and its image (range) is closed and has finite codimension. The codimension of the image is the dimension of the quotient space Y / ran(T).

The Fredholm index of such an operator T is defined as ind(T) = dim(ker(T)) - codim(ran(T)). This index is a topological invariant, meaning that it remains constant under small perturbations of the operator. Fredholm operators are fundamental in the study of partial differential equations, functional analysis, and geometric topology.

A crucial property of Fredholm operators is that the set of Fredholm operators is open in the space of all bounded linear operators between two fixed Banach spaces. Furthermore, if T and S are Fredholm operators and S can be "approximated" by T in a certain sense (e.g., S = T + K where K is compact), then T and S have the same Fredholm index. This property is essential for index theorems like the Atiyah-Singer index theorem.

Compact operators are closely related to Fredholm operators. If T is a Fredholm operator, then T + K is also a Fredholm operator with the same index for any compact operator K. This relationship highlights the role of compact operators as "perturbations" that do not alter the index.