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Definition [ edit ]
Let B be an arbitrary Banach space , and let B * be its dual, that is, the space of bounded linear functionals on B . The tensor product
B
∗
⊗
B
{\displaystyle B^{*}\otimes B}
has a completion under the norm
‖
X
‖
π
=
inf
∑
{
i
}
‖
e
i
∗
‖
‖
e
i
‖
{\displaystyle \Vert X\Vert _{\pi }=\inf \sum _{\{i\}}\Vert e_{i}^{*}\Vert \Vert e_{i}\Vert }
where the infimum is taken over all finite representations
X
=
∑
{
i
}
e
i
∗
⊗
e
i
∈
B
∗
⊗
B
{\displaystyle X=\sum _{\{i\}}e_{i}^{*}\otimes e_{i}\in B^{*}\otimes B}
The completion, under this norm, is often denoted as
B
∗
⊗
^
π
B
{\displaystyle B^{*}{\widehat {\,\otimes \,}}_{\pi }B}
and is called the projective topological tensor product . The elements of this space are called Fredholm kernels .
Properties [ edit ]
Every Fredholm kernel has a representation in the form
X
=
∑
{
i
}
λ
i
e
i
∗
⊗
e
i
{\displaystyle X=\sum _{\{i\}}\lambda _{i}e_{i}^{*}\otimes e_{i}}
with
e
i
∈
B
{\displaystyle e_{i}\in B}
and
e
i
∗
∈
B
∗
{\displaystyle e_{i}^{*}\in B^{*}}
such that
‖
e
i
‖
=
‖
e
i
∗
‖
=
1
{\displaystyle \Vert e_{i}\Vert =\Vert e_{i}^{*}\Vert =1}
and
∑
{
i
}
|
λ
i
|
<
∞
.
{\displaystyle \sum _{\{i\}}\vert \lambda _{i}\vert <\infty .\,}
Associated with each such kernel is a linear operator
L
X
:
B
→
B
{\displaystyle {\mathcal {L}}_{X}:B\to B}
which has the canonical representation
L
X
f
=
∑
{
i
}
λ
i
e
i
∗
(
f
)
e
i
.
{\displaystyle {\mathcal {L}}_{X}f=\sum _{\{i\}}\lambda _{i}e_{i}^{*}(f )e_{i}.\,}
Associated with every Fredholm kernel is a trace, defined as
tr
X
=
∑
{
i
}
λ
i
e
i
∗
(
e
i
)
.
{\displaystyle {\mbox{tr}}X=\sum _{\{i\}}\lambda _{i}e_{i}^{*}(e_{i}).\,}
p -summable kernels[ edit ]
A Fredholm kernel is said to be p -summable if
∑
{
i
}
|
λ
i
|
p
<
∞
{\displaystyle \sum _{\{i\}}\vert \lambda _{i}\vert ^{p}<\infty }
A Fredholm kernel is said to be of order q if q is the infimum of all
0
<
p
≤
1
{\displaystyle 0<p\leq 1}
for all p for which it is p -summable.
Nuclear operators on Banach spaces [ edit ]
An operator L : B →B is said to be a nuclear operator if there exists an
X ∈
B
∗
⊗
^
π
B
{\displaystyle B^{*}{\widehat {\,\otimes \,}}_{\pi }B}
such that L = L X . Such an operator is said to be p -summable and of order q if X is. In general, there may be more than one X associated with such a nuclear operator, and so the trace is not uniquely defined. However, if the order q ≤ 2/3, then there is a unique trace, as given by a theorem of Grothendieck.
Grothendieck's theorem [ edit ]
If
L
:
B
→
B
{\displaystyle {\mathcal {L}}:B\to B}
is an operator of order
q
≤
2
/
3
{\displaystyle q\leq 2/3}
then a trace may be defined, with
Tr
L
=
∑
{
i
}
ρ
i
{\displaystyle {\mbox{Tr}}{\mathcal {L}}=\sum _{\{i\}}\rho _{i}}
where
ρ
i
{\displaystyle \rho _{i}}
are the eigenvalues of
L
{\displaystyle {\mathcal {L}}}
. Furthermore, the Fredholm determinant
det
(
1
−
z
L
)
=
∏
i
(
1
−
ρ
i
z
)
{\displaystyle \det \left(1-z{\mathcal {L}}\right)=\prod _{i}\left(1-\rho _{i}z\right)}
is an entire function of z . The formula
det
(
1
−
z
L
)
=
exp
Tr
log
(
1
−
z
L
)
{\displaystyle \det \left(1-z{\mathcal {L}}\right)=\exp {\mbox{Tr}}\log \left(1-z{\mathcal {L}}\right)}
holds as well. Finally, if
L
{\displaystyle {\mathcal {L}}}
is parameterized by some complex -valued parameter w , that is,
L
=
L
w
{\displaystyle {\mathcal {L}}={\mathcal {L}}_{w}}
, and the parameterization is holomorphic on some domain, then
det
(
1
−
z
L
w
)
{\displaystyle \det \left(1-z{\mathcal {L}}_{w}\right)}
is holomorphic on the same domain.
Examples [ edit ]
An important example is the Banach space of holomorphic functions over a domain
D
⊂
C
k
{\displaystyle D\subset \mathbb {C} ^{k}}
. In this space, every nuclear operator is of order zero, and is thus of trace-class .
Nuclear spaces [ edit ]
The idea of a nuclear operator can be adapted to Fréchet spaces . A nuclear space is a Fréchet space where every bounded map of the space to an arbitrary Banach space is nuclear.
References [ edit ]
t
e
Spaces
Theorems
Operators
Algebras
Open problems
Applications
Advanced topics
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Fredholm_kernel&oldid=1190518925 "
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● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
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