In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. Fredholm kernels are named for Ivar Fredholm. Much of the theory of Fredholm kernels was developed by Alexander Grothendieck and published in 1955. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In analysis, consider an integral transform T which transforms a function f into a function Tf given by the integral formula The function k(x,y) that appears in this formula is the kernel of the operator T. See also: Dirichlet kernel convolution kernel trick Categories: Stub | Mathematical analysis ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a nuclear operator or a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. ... Erik Ivar Fredholm (April 7, 1866 - August 17, 1927) was a Swedish mathematician who established the modern theory of integral equations. ... Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. ... Jump to: navigation, search 1955 is a common year starting on Saturday of the Gregorian calendar. ...

Definition

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 has a completion under the norm In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Jump to: navigation, search In functional analysis (a branch of mathematics), a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ...

where the infimum is taken over all finite representations In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ...

The completion, under this norm, is often denoted as

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

Properties

Every Fredholm kernel has a representation in the form

with and such that and

Associated with each such kernel is a linear operator

which has the canonical representation

Associated with every Fredholm kernel is a trace, defined as

p-summable kernels

A Fredholm kernel is said to be p-summable if

A Fredholm kernel is said to be of order q if q is the infimum of all for all p for which it is p-summable. In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ...

Nuclear operators on Banach spaces

An operator is said to be a nuclear operator if there exists an such that . 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 , then there is a unique trace, as given by a theorem of Grothendieck. In mathematics, a nuclear operator or a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. ...

Grothendieck's theorem

If is an operator of order then a trace may be defined, with

where ρ_i are the eigenvalues of . Furthermore, the Fredholm determinant In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... A complex analytic function which generalizes the characteristic polynomial of a matrix. ...

is an entire function of z. The formula In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ...

holds as well. Finally, if is parameterized by some complex-valued parameter w, that is, , and the parameterization is holomorphic on some domain, then In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

is holomorphic on the same domain.

Examples

An important example is the Banach space of holomorphic functions over a domain . In this space, every nuclear operator is of order zero, and is thus of trace-class. In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ...

Nuclear spaces

The idea of a nuclear operator can be adapted to to Fréchet spaces. A nuclear space is a Fréchet space where every bounded map of the space to an arbitrary Banach space is nuclear. This article deals with Fréchet spaces in functional analysis. ...

References

• A. Grothendieck, Produits tensoriels topologiques et espace nuclieares, (1955) Mem. Am. Math.Soc. 16.
• A. Grothendieck, La theorie de Fredholm, (1956) Bull. Soc. Math. France, 84:319-384.
• Maurice Fréchet, On the Behavior of the nth Iterate of a Fredholm Kernel as n Becomes Infinite (1932) Proc Natl Acad Sci U S A. 18(11): 671–673.