Nuclear operator

Nuclear operator

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a nuclear operator is roughly a compact operator
Compact operator
In functional analysis, a branch of mathematics, a compact 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...

for which a trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

may be defined, such that the trace is finite and independent of the choice of basis (at least on well behaved spaces; there are some spaces on which nuclear operators do not have a trace).
Nuclear operators are essentially the same as trace class operators
Trace class
In mathematics, 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....

, though most authors reserve the term "trace class operator" for the special case of
nuclear operators on Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

s. The general definition for Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s was given by Grothendieck. This article concentrates on the general case of nuclear operators on Banach spaces; for the important special case of nuclear (=trace class) operators on Hilbert space see the article on trace class operator
Trace class
In mathematics, 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....

s.

Compact operator

An operator on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

is said to be a compact operator
Compact operator
In functional analysis, a branch of mathematics, a compact 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...

if it can be written in the form

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.

Nuclear operator

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

Properties

A nuclear operator on a Hilbert space has the important property that its trace
Trace class
In mathematics, 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....

may be defined so that it is finite and is independent of the basis. Given any orthonormal basis for the Hilbert space, one may define the trace as

since the sum converges absolutely 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
Fredholm kernel
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are one of the objects of study in Fredholm theory....

.

The definition of trace-class operator was extended to Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...

in 1955.

Let A and B be Banach spaces, and A be the dual of A, that is, the set of all continuous or (equivalently) bounded linear functionals on A with the usual norm. Then an operator

is said to be nuclear of order q if there exist sequences of vectors with , functionals with and complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s with

such that the operator may be written as

with the sum converging in the operator norm.

With additional steps, a trace may be defined for such operators when A = B.

Operators that are nuclear of order 1 are called nuclear operators: these are the ones for which the series ∑ρn is absolutely convergent.

More generally, an operator from a locally convex topological vector space
Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces which generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of ...

A to a Banach space B is called nuclear if it satisfies the condition above with all fn* bounded by 1 on some fixed neighborhood of 0.