Fredholm operator

In functional analysis , a branch of mathematics , the class of Fredholm operators (according to EI Fredholm ) is a certain class of linear operators that can be "almost" inverted. Each Fredholm operator is assigned an integer, this is called the Fredholm index , analytical index or index for short .

definition

A bounded linear operator between two Banach spaces and is called the Fredholm operator, or in short: " is Fredholm", if ${\displaystyle A\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle A}$

• ${\displaystyle \ker A}$has finite dimension and
• ${\displaystyle \mathrm {ran} \,A}$has finite codimension in . ${\displaystyle Y}$

Here is the core of , i.e. the set, and is the image of , i.e. the subset . ${\displaystyle \ker A}$ ${\displaystyle A}$ ${\displaystyle \{x\in X:Ax=0\}}$ ${\displaystyle \mathrm {ran} \,A}$ ${\displaystyle A}$ ${\displaystyle \{Ax\mid x\in X\}\subseteq Y}$

The number

${\displaystyle \mathrm {ind} (A)=\dim(\ker A)-\mathrm {codim} (\mathrm {ran} \,A,Y)\in \mathbb {Z} }$

is called the Fredholm index of . ${\displaystyle A}$

properties

Image is a closed subspace

The image of a Fredholm operator is a closed subspace. ${\displaystyle \mathrm {ran} \;(A)}$

structure

If a Fredholm operator, then the finite-dimensional subspace has a closed complementary space in , i.e. i.e., it applies . The restriction of is then obviously a bijective operator, the inverse of which is also bounded according to the theorem of the continuous inverse . The operator is thus continuously invertible "except for a finite number of dimensions". It can be used to prove many of the following properties. ${\displaystyle A\colon X\to Y}$ ${\displaystyle \ker(A)}$ ${\displaystyle W}$ ${\displaystyle X}$ ${\displaystyle X=\ker(A)\oplus W}$ ${\displaystyle {\widetilde {A}}\colon W\to \operatorname {ran} (A)}$ ${\displaystyle A}$ ${\displaystyle A}$

composition

The composition of two Fredholm operators and is again a Fredholm operator and applies to the index ${\displaystyle A\circ B}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle \operatorname {ind} (A\circ B)=\operatorname {ind} (A)+\operatorname {ind} (B)}$.

Dual operator

Let be the operator dual to the Fredholm operator . Then and . Hence it is also a Fredholm operator and holds for its index . ${\displaystyle A'\colon Y'\to X'}$ ${\displaystyle A}$ ${\displaystyle \dim(\ker A')=\operatorname {codim} (\mathrm {ran} (A))}$ ${\displaystyle \operatorname {codim} (\mathrm {ran} (A'))=\dim(\ker A)}$ ${\displaystyle A'}$ ${\displaystyle \operatorname {ind} (T')=-\operatorname {ind} (T)}$

Atkinson's theorem

According to Atkinson's theorem , an operator is a Fredholm operator if and only if there are operators and compact operators such that and holds, that is, if modulo compact operators are invertible. In particular, a bounded operator is a Fredholm operator if and only if its class is invertible in Calkin algebra . ${\displaystyle A\colon X\to Y}$ ${\displaystyle B_{1},B_{2}}$ ${\displaystyle K_{1},K_{2}}$ ${\displaystyle AB_{1}=I_{Y}-K_{1}}$ ${\displaystyle B_{2}A=I_{X}-K_{2}}$ ${\displaystyle A}$ ${\displaystyle A\colon X\to X}$ ${\displaystyle [A]_{{\mathcal {C}}(X)}}$ ${\displaystyle {\mathcal {B}}(X)/{\mathcal {C}}(X)}$

Compact disruption

For every Fredholm operator and every compact operator there is also a Fredholm operator with the same Fredholm index as . Hence the index of a Fredholm operator is said to be invariant under compact perturbations. In particular, every compact perturbation of identity , i.e. every operator of the form for a compact operator, is a Fredholm operator with index 0. ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle A+K}$ ${\displaystyle A}$ ${\displaystyle I+K}$ ${\displaystyle K}$

Properties of the Fredholm Index

The set of Fredholm operators between the Banach spaces and is open in the set of restricted operators . On each connected component of the index is constant: for all . In fact, the mapping is bijective. This immediately results in the following properties of the index: ${\displaystyle \Phi (X,Y)}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle L(X,Y)}$ ${\displaystyle U}$ ${\displaystyle \Phi (X,Y)}$ ${\displaystyle \operatorname {ind} (A)=n_{U}\in \mathbb {Z} }$ ${\displaystyle A\in U}$ ${\displaystyle U\mapsto n_{U}}$

• The index mapping is continuous. ${\displaystyle \operatorname {ind} \colon \Phi (X,Y)\to \mathbb {Z} }$
• The index is invariant under small perturbations, that is, there is so for all of the following applies: . ${\displaystyle A\in \Phi (X,Y)}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle S\in L(X,Y)}$ ${\displaystyle \|S\|<\varepsilon }$ ${\displaystyle \operatorname {ind} (A+S)=\operatorname {ind} (A)}$
• The index is a homotopy-invariant number .: is continuous, then and have the same index. ${\displaystyle \psi \colon [0,1]\to \Phi (X,Y)}$ ${\displaystyle \psi (0)}$ ${\displaystyle \psi (1)}$

Fredholm index surjectivity

The Fredholm index, as a mapping of the set of Fredholm operators into the set of integers, is surjective .

Punctured Neighborhood Theorem

If a Fredholm operator is, then according to the Punctured Neighborhood Theorem there is a , so that for everyone with ${\displaystyle A\colon X\to X}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \lambda \in \mathbb {C} }$ ${\displaystyle 0<|\lambda |<\varepsilon }$

1. ${\displaystyle \dim \ker(A-\lambda I)\equiv \mathrm {const} \leq \dim \ker A}$ and
2. ${\displaystyle \mathrm {codim} \,\mathrm {ran} (A-\lambda I)\equiv \mathrm {const} \leq \mathrm {codim} \,\mathrm {ran} A}$

applies. In particular, is a Fredholm operator. Since the Fredholm index is continuous, it follows . The Punctured Neighborhood Theorem was proven by Israel Gohberg . ${\displaystyle A-\lambda I}$ ${\displaystyle \mathrm {ind} (A-\lambda I)=\mathrm {ind} (A)}$

Elliptic operators

Every uniformly elliptic differential operator is a Fredholm operator.

Be and an area with a Lipschitz border . Then the weak elliptic differential operator with homogeneous Neumann boundary conditions is defined by ${\displaystyle n\geq 1}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle A\colon H^{1,2}(\Omega )\to H^{1,2}(\Omega )'}$

${\displaystyle A(u)(v):=\int _{\Omega }\sum _{i,j}\partial _{i}va_{ij}\partial _{j}u}$

for a Fredholm operator. ${\displaystyle u,v\in H^{1,2}(\Omega )}$

Examples

Shift operator

→ Main article : Shift operator

Integral operator

A classic example of a Fredholm operator is the operator

${\displaystyle A:=I+T}$,

where the identity operator and is a compact operator . On the Banach space of continuous functions or on that of square-integrable functions , the operator is of the form ${\displaystyle I}$ ${\displaystyle T}$ ${\displaystyle C([a,b])}$ ${\displaystyle L^{2}([a,b])}$ ${\displaystyle A}$

${\displaystyle (A\phi )(x)=\phi (x)+\int _{a}^{b}k(x,y)\phi (y)\mathrm {d} y}$,

where the integral kernel is a continuous or square-integrable function. This Fredholm operator has the index 0. In the Fredholm theory , equations of the type are examined. The Fredholm alternative as a central result of the Fredholm theory gives an answer to the conditions under which equations of this type can be solved. ${\displaystyle k}$ ${\displaystyle A\phi (x)=f(x)}$

Laplace operator

→ Main article : Laplace operator

The Laplace operator

${\displaystyle \Delta f=\sum _{k=1}^{n}{\partial ^{2}f \over \partial x_{k}^{2}}}$

defined on the Sobolev space of the twice weakly differentiable quadratic integrable functions is a continuous elliptic operator. Therefore he is also a Fredholm operator. Since he is also self adjoint , he has the Fredholm index 0. ${\displaystyle H^{2}(\mathbb {R} ^{n})}$

Considering the Laplace operator in the distributional sense on , he is not a continuous operator and thus no Fredholm operator with respect to the above definition. In terms of unbounded operators, as will be explained later in the article, it is still a Fredholm operator. ${\displaystyle L^{2}(\mathbb {R} ^{n})}$

Elliptic operator on a manifold

The circle (as thought) can be understood as a one-dimensional closed manifold . A continuous elliptic differential operator of the first order on the smooth functions from the circle into the complex numbers is through ${\displaystyle \mathbb {R} /\mathbb {Z} }$

${\displaystyle D={\frac {\mathrm {d} }{\mathrm {d} x}}-2\pi i\lambda }$

given for a complex constant . The kernel of is the space spanned by the terms of the form , if , and 0 in the other cases. The core of the adjoint operator is a similar space, only it is replaced by its complex-conjugate. The Fredholm operator thus has the index 0. This example shows that the kernel and kernel of an elliptic operator can jump discontinuously if the elliptic operator is varied so that the above-mentioned terms are included. Since the jumps in the dimensions of the core and the coke are the same, their difference, the index, changes continuously . ${\displaystyle \lambda }$ ${\displaystyle D}$ ${\displaystyle \exp(i2\pi \,\lambda \,x)}$ ${\displaystyle \lambda \in \mathbb {Z} }$ ${\displaystyle \lambda }$ ${\displaystyle D}$

Unlimited Fredholm operators

So far in this article, Fredholm operators have only been considered as special constrained operators. For example, in the index theory of elliptic operators over non-compact spaces, it makes sense to extend the definition of the Fredholm operator to include unrestricted operators. Except for the required closeness of the operator, the definition is identical to that in the restricted case:

Be and two Banach spaces and a subspace of . An (unbounded) operator is called a Fredholm operator, if ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {D}}(A)}$ ${\displaystyle X}$ ${\displaystyle A\colon X\supset {\mathcal {D}}(A)\to Y}$

• ${\displaystyle A}$ completed is,
• the dimension of the core is finite, ${\displaystyle \operatorname {ker} (A)}$
• the codimension of in is finite. ${\displaystyle \operatorname {ran} (A)}$ ${\displaystyle Y}$

Some authors also demand that the domain be close to , which is obviously completely independent of the actual Fredholm property. As in the case of bounded operators, the Fredholm index is through ${\displaystyle {\mathcal {D}}(A)}$ ${\displaystyle X}$

${\displaystyle \mathrm {ind} (A)=\dim(\ker A)-\mathrm {codim} (\mathrm {ran} (A))\in \mathbb {Z} }$

Are defined.

If the domain of a closed operator is given the so-called graph norm , then it is a Banach space and , considered as an operator from to , a bounded operator. Consequently, an unbounded Fredholm operator can always be reduced to a bounded Fredholm operator. Accordingly, many properties from above also hold for unbounded Fredholm operators. So the concatenation of unbounded Fredholm operators is again a Fredholm operator, for which the above index formula applies; Atkinson's theorem also holds, and the Fredholm index of unbounded Fredholm operators is also invariant under compact perturbations and locally constant (the word "local" here refers to the so-called gap metric). Finally, the Punctured Neighborhood Theorem also holds for unbounded Fredholm operators. However, there is no connection to the Calkin algebra for unlimited Fredholm operators. ${\displaystyle {\mathcal {D}}(A)\subset X}$ ${\displaystyle A\colon X\to Y}$ ${\displaystyle \|x\|_{A}:=\|x\|_{X}+\|Ax\|_{Y}}$ ${\displaystyle X_{A}:=({\mathcal {D}}(A),\|\cdot \|_{A})}$ ${\displaystyle A}$ ${\displaystyle X_{A}}$ ${\displaystyle Y}$

See also

• Essential spectrum

literature

• Dirk Werner : Functional Analysis. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 .

Individual evidence

1. ↑ Vladimir Müller: Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras . Birkhäuser, Basel 2007, ISBN 978-3-7643-8265-0 , p. 159 .
2. ↑ Vladimir Müller: Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras . Birkhäuser, Basel 2007, ISBN 978-3-7643-8265-0 , p. 156 .
3. ↑ Masoud Khalkhali: Basic Noncommutative Geometry . 2nd Edition. EMS, 2013, ISBN 978-3-03719-128-6 , pp. 201 .
4. ↑ Jürgen Appell, Martin Väth : Elements of functional analysis . Springer / Vieweg, 2005, ISBN 3-322-80243-4 , pp. 164-165 .
5. ↑ Vladimir Müller: Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras . Birkhäuser, Basel 2007, ISBN 978-3-7643-8265-0 , p. 171, 293-294 .
6. ↑ Vladimir Müller: Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras . Birkhäuser, Basel 2007, ISBN 978-3-7643-8265-0 , p. 231 .
7. ^ Martin Schechter: Fredholm Operators and the Essential Spectrum . (on-line)