In mathematics , Fredholm's alternative named after Ivar Fredholm is a result of Fredholm's theory . It can be expressed in different ways: as a theorem of linear algebra , as a theorem of integral equations, or as a theorem of Fredholm operators . In particular, it says that a complex number not equal to 0 in the spectrum of a compact operator is an eigenvalue .
Version of linear algebra
In a -dimensional vector space , exactly one of the following statements applies to a linear mapping :
n
{\ displaystyle n}
V
{\ displaystyle V}
A.
:
V
→
V
{\ displaystyle A \ colon V \ to V}
For every vector in there is a vector in such that . In other words: is surjective.
v
{\ displaystyle v}
V
{\ displaystyle V}
u
{\ displaystyle u}
V
{\ displaystyle V}
A.
u
=
v
{\ displaystyle Au = v}
A.
{\ displaystyle A}
There is an in with , that is: is not injective.
u
≠
0
{\ displaystyle u \ neq 0}
V
{\ displaystyle V}
A.
u
=
0
{\ displaystyle Au = 0}
A.
{\ displaystyle A}
Fredholm integral equations
Be an integral kernel . Consider the homogeneous Fredholm integral equation ,
K
(
x
,
y
)
{\ displaystyle K (x, y)}
λ
ϕ
(
x
)
-
∫
a
b
K
(
x
,
y
)
ϕ
(
y
)
d
y
=
0
{\ displaystyle \ lambda \ phi (x) - \ int _ {a} ^ {b} K (x, y) \ phi (y) \, dy = 0}
,
as well as the inhomogeneous equation
λ
ϕ
(
x
)
-
∫
a
b
K
(
x
,
y
)
ϕ
(
y
)
d
y
=
f
(
x
)
{\ displaystyle \ lambda \ phi (x) - \ int _ {a} ^ {b} K (x, y) \ phi (y) \, dy = f (x)}
.
Fredholm's alternative says that for a complex number , either the first equation has a nontrivial solution, or the second equation has a solution for any right-hand side .
0
≠
λ
∈
C.
{\ displaystyle 0 \ neq \ lambda \ in \ mathbb {C}}
f
(
x
)
{\ displaystyle f (x)}
A sufficient condition for this theorem to be true is the integrability of squares on the rectangle (where a and / or b may also be plus or minus infinity).
K
(
x
,
y
)
{\ displaystyle K (x, y)}
[
a
,
b
]
×
[
a
,
b
]
{\ displaystyle [a, b] \ times [a, b]}
Fredholm's alternative
statement
Be a compact operator on and be with . Then there is a Fredholm operator with Fredholm index 0. The Fredholm alternative is now:
K
∈
K
(
X
)
{\ displaystyle K \ in K (X)}
X
{\ displaystyle X}
λ
∈
C.
{\ displaystyle \ lambda \ in \ mathbb {C}}
λ
≠
0
{\ displaystyle \ lambda \ neq 0}
T
x
: =
λ
x
-
K
x
{\ displaystyle Tx: = \ lambda x-Kx}
Either both have the homogeneous equation
λ
x
-
K
x
=
0
{\ displaystyle \ lambda x-Kx = 0}
as well as the adjoint equation
λ
x
′
-
K
′
x
′
=
0
{\ displaystyle \ lambda x'-K'x '= 0}
only the trivial solution is zero and thus the equations are inhomogeneous
λ
x
-
K
x
=
y
{\ displaystyle \ lambda x-Kx = y}
and
λ
x
′
-
K
′
x
′
=
y
′
{\ displaystyle \ lambda x'-K'x '= y'}
clearly solvable,
or the homogeneous equation
λ
x
-
K
x
=
0
{\ displaystyle \ lambda x-Kx = 0}
and the adjoint equation
λ
x
′
-
K
′
x
′
=
0
{\ displaystyle \ lambda x'-K'x '= 0}
have exactly linearly independent solutions (where the identical figure denotes) and thus the equation would be inhomogeneous
n
=
dim
ker
(
λ
id
-
K
)
<
∞
{\ displaystyle n = \ dim \ ker (\ lambda \ operatorname {id} -K) <\ infty}
id
{\ displaystyle \ operatorname {id}}
λ
x
-
K
x
=
y
{\ displaystyle \ lambda x-Kx = y}
solvable if and only if .
y
∈
(
ker
(
λ
id
-
K
′
)
)
⊥
{\ displaystyle y \ in (\ ker (\ lambda \ operatorname {id} -K ')) ^ {\ bot}}
In connection with the integral equations
Note that the delta distribution is the identity of the convolution . Let be a Banach space , for example, and be a Fredholm operator which through
X
{\ displaystyle X}
X
=
L.
2
(
[
a
,
b
]
)
{\ displaystyle X = L ^ {2} ([a, b])}
T
:
X
→
X
{\ displaystyle T \ colon X \ to X}
T
ϕ
(
x
)
=
∫
a
b
λ
δ
(
x
-
y
)
ϕ
(
y
)
-
k
(
x
,
y
)
ϕ
(
y
)
d
y
=
λ
ϕ
(
x
)
-
∫
a
b
k
(
x
,
y
)
ϕ
(
y
)
d
y
,
{\ displaystyle T \ phi (x) = \ int _ {a} ^ {b} \ lambda \ delta (xy) \ phi (y) -k (x, y) \ phi (y) dy = \ lambda \ phi (x) - \ int _ {a} ^ {b} k (x, y) \ phi (y) dy,}
is defined, where must hold in order to obtain a Fredholm operator. Then there is a compact operator and you can see that this statement generalizes the statement about Fredholm's integral equations.
k
∈
L.
2
(
[
a
,
b
]
)
{\ displaystyle k \ in L ^ {2} ([a, b])}
∫
a
b
k
(
x
,
y
)
ϕ
(
y
)
d
y
{\ displaystyle \ textstyle \ int _ {a} ^ {b} k (x, y) \ phi (y) dy}
Fredholm's alternative can then be formulated as follows: A is either an eigenvalue of or it is part of the resolvent set
λ
≠
0
{\ displaystyle \ lambda \ neq 0}
K
{\ displaystyle K}
ρ
(
K
)
=
{
λ
∈
C.
:
(
λ
id
-
K
)
limited invertible
}
{\ displaystyle \ rho (K) = \ {\ lambda \ in \ mathbb {C}: (\ lambda \ operatorname {id} -K) {\ text {limited invertible}} \}}
.
literature
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