Fredholm kernel

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 ${\displaystyle B^{*}\otimes B}$ has a completion under the norm

${\displaystyle \Vert X\Vert _{\pi }=\inf \sum _{\{i\}}\Vert e_{i}^{*}\Vert \Vert e_{i}\Vert }$

where the infimum is taken over all finite representations

${\displaystyle X=\sum _{\{i\}}e_{i}^{*}\otimes e_{i}\in B^{*}\otimes B}$

The completion, under this norm, is often denoted as

${\displaystyle B^{*}{\widehat {\,\otimes \,}}_{\pi }B}$

and is called the projective topological tensor product. The elements of this space are called Fredholm kernels.

Every Fredholm kernel has a representation in the form

${\displaystyle X=\sum _{\{i\}}\lambda _{i}e_{i}^{*}\otimes e_{i}}$

with ${\displaystyle e_{i}\in B}$ and ${\displaystyle e_{i}^{*}\in B^{*}}$ such that ${\displaystyle \Vert e_{i}\Vert =\Vert e_{i}^{*}\Vert =1}$ and

${\displaystyle \sum _{\{i\}}\vert \lambda _{i}\vert <\infty .\,}$

Associated with each such kernel is a linear operator

${\displaystyle {\mathcal {L}}_{X}:B\to B}$

which has the canonical representation

${\displaystyle {\mathcal {L}}_{X}f=\sum _{\{i\}}\lambda _{i}e_{i}^{*}(f)e_{i}.\,}$

Associated with every Fredholm kernel is a trace, defined as

${\displaystyle {\mbox{tr}}X=\sum _{\{i\}}\lambda _{i}e_{i}^{*}(e_{i}).\,}$

A Fredholm kernel is said to be p-summable if

${\displaystyle \sum _{\{i\}}\vert \lambda _{i}\vert ^{p}<\infty }$

A Fredholm kernel is said to be of order q if q is the infimum of all ${\displaystyle 0<p\leq 1}$ for all p for which it is p-summable.

An operator L : B→B is said to be a nuclear operator if there exists an X ∈ ${\displaystyle B^{*}{\widehat {\,\otimes \,}}_{\pi }B}$ 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.

If ${\displaystyle {\mathcal {L}}:B\to B}$ is an operator of order ${\displaystyle q\leq 2/3}$ then a trace may be defined, with

${\displaystyle {\mbox{Tr}}{\mathcal {L}}=\sum _{\{i\}}\rho _{i}}$

where ${\displaystyle \rho _{i}}$ are the eigenvalues of ${\displaystyle {\mathcal {L}}}$. Furthermore, the Fredholm determinant

${\displaystyle \det \left(1-z{\mathcal {L}}\right)=\prod _{i}\left(1-\rho _{i}z\right)}$

is an entire function of z. The formula

${\displaystyle \det \left(1-z{\mathcal {L}}\right)=\exp {\mbox{Tr}}\log \left(1-z{\mathcal {L}}\right)}$

holds as well. Finally, if ${\displaystyle {\mathcal {L}}}$ is parameterized by some complex-valued parameter w, that is, ${\displaystyle {\mathcal {L}}={\mathcal {L}}_{w}}$, and the parameterization is holomorphic on some domain, then

${\displaystyle \det \left(1-z{\mathcal {L}}_{w}\right)}$

is holomorphic on the same domain.

An important example is the Banach space of holomorphic functions over a domain ${\displaystyle D\subset \mathbb {C} ^{k}}$. In this space, every nuclear operator is of order zero, and is thus of trace-class.

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.

• Grothendieck A (1955). "Produits tensoriels topologiques et espaces nucléaires". Mem. Amer. Math. Soc. 16.
• Grothendieck A (1956). "La théorie de Fredholm". Bull. Soc. Math. France. 84: 319–84.
• B.V. Khvedelidze, G.L. Litvinov (2001) [1994], "Fredholm kernel", Encyclopedia of Mathematics, EMS Press
• Fréchet M (November 1932). "On the Behavior of the nth Iterate of a Fredholm Kernel as n Becomes Infinite". Proc. Natl. Acad. Sci. U.S.A. 18 (11): 671–3. doi:10.1073/pnas.18.11.671. PMC 1076308. PMID 16577494.