Fredholm operator
None
In
mathematics
,
Fredholm operators
are certain
operators
that arise in the
Fredholm theory
of
integral equations
. They are named in honour of
Erik Ivar Fredholm
. By definition, a Fredholm operator is a
bounded linear operator
T
:
X
→
Y
between two
Banach spaces
with finite-dimensional
kernel
and finite-dimensional (algebraic)
cokernel
, and with closed
range
. The last condition is actually redundant.
[1]
The index of a Fredholm operator is the integer
or in other words,
Properties
Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator T : X → Y between Banach spaces X and Y is Fredholm if and only if it is invertible modulo compact operators , i.e., if there exists a bounded linear operator
such that
are compact operators on X and Y respectively.
If a Fredholm operator is modified slightly, it stays Fredholm and its index remains the same. Formally: The set of Fredholm operators from X to Y is open in the Banach space L( X , Y ) of bounded linear operators, equipped with the operator norm , and the index is locally constant. More precisely, if T 0 is Fredholm from X to Y , there exists ε > 0 such that every T in L( X , Y ) with || T − T 0 || < ε is Fredholm, with the same index as that of T 0 .
When
T
is Fredholm from
X
to
Y
and
U
Fredholm from
Y
to
Z
, then the composition
is Fredholm from
X
to
Z
and
When T is Fredholm, the transpose (or adjoint) operator T ′ is Fredholm from Y ′ to X ′ , and ind( T ′ ) = − ind( T ) . When X and Y are Hilbert spaces , the same conclusion holds for the Hermitian adjoint T ∗ .
When T is Fredholm and K a compact operator, then T + K is Fredholm. The index of T remains unchanged under such a compact perturbations of T . This follows from the fact that the index i ( s ) of T + s K is an integer defined for every s in [0, 1], and i ( s ) is locally constant, hence i (1) = i (0).
Invariance by perturbation is true for larger classes than the class of compact operators. For example, when
U
is Fredholm and
T
a
strictly singular operator
, then
T
+
U
is Fredholm with the same index.
[2]
The class of
inessential operators
, which properly contains the class of strictly singular operators, is the "perturbation class" for Fredholm operators. This means an operator
is inessential if and only if
T+U
is Fredholm for every Fredholm operator
.
Examples
Let
be a
Hilbert space
with an orthonormal basis
indexed by the non negative integers. The (right)
shift operator
S
on
H
is defined by
This operator
S
is injective (actually, isometric) and has a closed range of codimension 1, hence
S
is Fredholm with
. The powers
,
, are Fredholm with index
. The adjoint
S*
is the left shift,
The left shift S* is Fredholm with index 1.
If
H
is the classical
Hardy space
on the unit circle
T
in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials
is the multiplication operator
M
φ
with the function
. More generally, let
φ
be a complex continuous function on
T
that does not vanish on
, and let
T
φ
denote the
Toeplitz operator
with symbol
φ
, equal to multiplication by
φ
followed by the orthogonal projection
:
Then
T
φ
is a Fredholm operator on
, with index related to the
winding number
around 0 of the closed path
: the index of
T
φ
, as defined in this article, is the opposite of this winding number.
Applications
Any elliptic operator can be extended to a Fredholm operator. The use of Fredholm operators in partial differential equations is an abstract form of the parametrix method.
The Atiyah-Singer index theorem gives a topological characterization of the index of certain operators on manifolds.
The Atiyah-Jänich theorem identifies the K-theory K ( X ) of a compact topological space X with the set of homotopy classes of continuous maps from X to the space of Fredholm operators H → H , where H is the separable Hilbert space and the set of these operators carries the operator norm.
Generalizations
B-Fredholm operators
For each integer
, define
to be the restriction of
to
viewed as a map from
into
( in particular
).
If for some integer
the space
is closed and
is a Fredholm operator, then
is called a
B-Fredholm operator
. The index of a B-Fredholm operator
is defined as the index of the Fredholm operator
. It is shown that the index is independent of the integer
.
B-Fredholm operators were introduced by M. Berkani in 1999 as a generalization of Fredholm operators.
[3]
Semi-Fredholm operators
A bounded linear operator
T
is called
semi-Fredholm
if its range is closed and at least one of
,
is finite-dimensional. For a semi-Fredholm operator, the index is defined by
Unbounded operators
One may also define unbounded Fredholm operators. Let X and Y be two Banach spaces.
-
The
closed linear operator
is called Fredholm if its domain
is dense in
, its range is closed, and both kernel and cokernel of T are finite-dimensional.
-
is called semi-Fredholm if its domain
is dense in
, its range is closed, and either kernel or cokernel of T (or both) is finite-dimensional.
As it was noted above, the range of a closed operator is closed as long as the cokernel is finite-dimensional (Edmunds and Evans, Theorem I.3.2).
Notes
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png)
- ↑ Abramovich, Yuri A.; Aliprantis, Charalambos D. (2002). An Invitation to Operator Theory . Graduate Studies in Mathematics. Vol. 50. American Mathematical Society. p. 156. ISBN 978-0-8218-2146-6 .
- ↑ Kato, Tosio (1958). "Perturbation theory for the nullity deficiency and other quantities of linear operators" . Journal d'Analyse Mathématique . 6 : 273–322. doi : 10.1007/BF02790238 .
- ↑ Berkani, Mohammed (1999). "On a class of quasi-Fredholm operators" . Integral Equations and Operator Theory . 35 (2): 244–249. doi : 10.1007/BF01236475 .
References
- D.E. Edmunds and W.D. Evans (1987), Spectral theory and differential operators, Oxford University Press. ISBN 0-19-853542-2 .
- A. G. Ramm, " A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators ", American Mathematical Monthly , 108 (2001) p. 855 (NB: In this paper the word "Fredholm operator" refers to "Fredholm operator of index 0").
- Weisstein, Eric W. "Fredholm's Theorem" . MathWorld .
- B.V. Khvedelidze (2001) [1994], "Fredholm theorems" , Encyclopedia of Mathematics , EMS Press
- Bruce K. Driver, " Compact and Fredholm Operators and the Spectral Theorem ", Analysis Tools with Applications , Chapter 35, pp. 579–600.
- Robert C. McOwen, " Fredholm theory of partial differential equations on complete Riemannian manifolds ", Pacific J. Math. 87 , no. 1 (1980), 169–185.
- Tomasz Mrowka, A Brief Introduction to Linear Analysis: Fredholm Operators , Geometry of Manifolds, Fall 2004 (Massachusetts Institute of Technology: MIT OpenCouseWare)
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