Fredholm-Operator

In the functional analysis, a branch of mathematics, is the class of Fredholm operators (after EI Fredholm ) a certain class of linear operators that can be "almost" invert. Each Fredholm operator assigns it to an integer, this is called the Fredholm index, analytical index or index shortly.

Definition

A bounded linear operator between two Banach spaces and is called a Fredholm operator, or you can just say, " is Fredholm " when

• Has finite dimension and
• Finite codimension has.

The core is, therefore, the amount and the picture is from, so the subset.

The number

Is called Fredholm index.

Properties

• Is a closed subspace.
• The figure

• By the theorem of Atkinson is an operator if and only a Fredholm operator if there are operators and compact operators, and so holds, ie if modulo compact operators is invertible. In particular, a bounded operator if and only a Fredholm operator if its class is invertible in the Calkin algebra.
• Is also for every Fredholm operator and every compact operator is a Fredholm operator with same Fredholm index as. In particular, any compact perturbation of the identity, so any operator of the form of a compact operator is a Fredholm operator of index 0
• Is a Fredholm operator, then there is an after Punctured Neighbourhood Theorem, such that for all with:   is a Fredholm operator;
• ;
• ;
• .
• Each uniformly elliptic differential operator is a Fredholm operator.
• Be a Lipschitz field. Then the weak elliptic differential operator with homogeneous Neumann boundary conditions is defined by a Fredholm operator.