Fredholmoperátorokat

Fredholmoperátorok, sometimes referred to as Fredholm operators, are a class of linear operators between Banach spaces that are fundamental in functional analysis and have significant applications in the study of differential equations and index theory. An operator T from a Banach space X to a Banach space Y is called a Fredholm operator if it satisfies three conditions: its kernel (null space) is finite-dimensional, its image (range) is closed, and the codimension of its image is finite-dimensional. The codimension of the image of T is defined as the dimension of the quotient space Y/Im(T).

The key property of Fredholm operators is their index. The index of a Fredholm operator T is defined as the difference between the dimension of its kernel and the codimension of its image: index(T) = dim(ker(T)) - codim(Im(T)). This index is a topological invariant. A crucial theorem states that the index is preserved under small perturbations of the operator in the space of bounded linear operators with the operator norm. That is, if T is a Fredholm operator and S is a bounded linear operator such that the norm of (T-S) is sufficiently small, then S is also a Fredholm operator and index(T) = index(S). This invariance property is central to many of their applications.

Fredholm operators are also characterized by the fact that they are exactly the operators for which there exists a linear topological simplectomorphism, which is a type of generalized inverse. This concept of a generalized inverse is important for understanding solvability properties of linear equations involving Fredholm operators. The set of Fredholm operators forms an open subset in the space of all bounded linear operators between two given Banach spaces.