In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a nuclear operator is a
compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
for which a
trace
Trace may refer to:
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* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
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, though most authors reserve the term "trace-class operator" for the special case of nuclear operators on
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s.
The general definition for
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s was given by
Grothendieck. This article presents both cases but concentrates on the general case of nuclear operators on Banach spaces; for more details about the important special case of nuclear (= trace-class) operators on Hilbert space, see the article
Trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a Trace (linear algebra), trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the tra ...
.
Compact operator
An operator
on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
if it can be written in the form
where
and
and
are (not necessarily complete) orthonormal sets. Here
is a set of real numbers, the set of
singular value In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self ...
s 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.
An operator that is compact as defined above is said to be or if
Properties
A nuclear operator on a Hilbert space has the important property that a
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
operation may be defined. Given an orthonormal basis
for the Hilbert space, the trace is defined as
Obviously, the sum converges absolutely, and it can be proven that the result is independent of the basis. It can be shown that this trace is identical to the sum of the eigenvalues of
(counted with multiplicity).
On Banach spaces
The definition of trace-class operator was extended to
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s by
Alexander Grothendieck in 1955.
Let
and
be Banach spaces, and
be the
dual of
that is, the set of all
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
or (equivalently)
bounded linear functional
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector s ...
s on
with the usual norm.
There is a canonical evaluation map
(from the
projective tensor product
The strongest locally convex topological vector space (TVS) topology on X \otimes Y, the tensor product of two locally convex TVSs, making the canonical map \cdot \otimes \cdot : X \times Y \to X \otimes Y (defined by sending (x, y) \in X \times ...
of
and
to the Banach space of continuous linear maps from
to
).
It is determined by sending
and
to the linear map
An operator
is called if it is in the image of this evaluation map.
-nuclear operators
An operator
is said to be if there exist sequences of vectors
with
functionals
with
and
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s
with
such that the operator may be written as
with the sum converging in the operator norm.
Operators that are nuclear of order 1 are called : these are the ones for which the series
is absolutely convergent.
Nuclear operators of order 2 are called
Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm
\, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^2_ ...
s.
Relation to trace-class operators
With additional steps, a trace may be defined for such operators when
Generalizations
More generally, an operator from a
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
to a Banach space
is called if it satisfies the condition above with all
bounded by 1 on some fixed neighborhood of 0.
An extension of the concept of nuclear maps to arbitrary
monoidal categories
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...
is given by .
A monoidal category can be thought of as a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
equipped with a suitable notion of a tensor product.
An example of a monoidal category is the category of Banach spaces or alternatively the category of locally convex, complete, Hausdorff spaces; both equipped with the projective tensor product.
A map
in a monoidal category is called if it can be written as a composition
for an appropriate object
and maps
where
is the monoidal unit.
In the monoidal category of Banach spaces, equipped with the projective tensor product, a map is thick if and only if it is nuclear.
Examples
Suppose that
and
are
Hilbert-Schmidt operators between Hilbert spaces. Then the composition
is a
nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spac ...
.
See also
*
*
*
References
* A. Grothendieck (1955), Produits tensoriels topologiques et espace nucléaires,''Mem. Am. Math.Soc.'' 16.
* A. Grothendieck (1956), La theorie de Fredholm, ''Bull. Soc. Math. France'', 84:319–384.
* A. Hinrichs and A. Pietsch (2010), ''p''-nuclear operators in the sense of Grothendieck, ''Mathematische Nachrichen'' 283: 232–261. .
*
*
*
{{TopologicalTensorProductsAndNuclearSpaces
Operator theory