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. Nuclear operators are usually defined on Hilbert spaces, but the definition can be extended to Banach spaces, the extension having been given by Grothendieck. Hi dustin ... 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. ... In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a... The word trace has several meanings: in linear algebra, the trace of a square matrix A is the sum of its main diagonal elements. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ...

Compact operator

An operator on a Hilbert space In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...

is said to be a compact operator if it can be written in the form In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a...

where and and are (not necessarily complete) orthonormal sets. Here, are a set of real numbers, the singular values of the operator, obeying if . The bracket is the scalar product on the Hilbert space; the sum on the right hand side must converge in norm. In mathematics, in particular functional analysis, singular values, or s-numbers of an bounded operator T acting on a Hilbert space are defined as the eigenvalues of (T*T)1/2. ...

Nuclear operator

An operator that is compact as defined above is said to be nuclear or trace-class if

Properties

A nuclear operator has the important property that its trace may be defined so that it is finite and is independent of the basis. Given any orthonormal basis {ψ_n} for the Hilbert space, one may define the trace as 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. ...

since the sum converges aboslutly and is independent of the basis. Furthermore, this trace is identical to the sum over the eigenvalues of (counted with multiplicity).

On Banach spaces

See main article Fredholm kernel.

The definition of trace-class operator was extended to Banach spaces by Alexander Grothendieck in 1955. In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. ... 1955 is a common year starting on Saturday of the Gregorian calendar. ...

Let B be a Banach space, and B^* be its dual, that is, the set of all continuous bounded linear functionals on B with the usual norm. Then an operator In topology, a continuous function is generally defined as one for which preimages of open sets are open. ...

is said to be nuclear of order q if there exist families and such that and such that the operator may be written as

with the right hand side converging in the operator norm. Here, the {ρ_n} are a sequence of complex numbers, with In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...

With additional steps, a trace may be defined for such operators.

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.