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
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 two
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (TVSs).
Preliminaries and notation
Throughout let ''X'',''Y'', and ''Z'' be
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (TVSs) and ''L'' : ''X'' → ''Y'' be a linear operator (no assumption of continuity is made unless otherwise stated).
* 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 two
locally convex TVSs ''X'' and ''Y'' is denoted by
and the completion of this space will be denoted by
.
* ''L'' : ''X'' → ''Y'' is a
topological homomorphism In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces (TVSs).
This concept is of considerable importance in functional ana ...
or homomorphism, if it is linear, continuous, and
is an
open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, a ...
, where
, the image of ''L'', has the subspace topology induced by ''Y''.
** If ''S'' is a subspace of ''X'' then both the quotient map ''X'' → ''X''/''S'' and the canonical injection ''S'' → ''X'' are homomorphisms.
* The set of continuous linear maps ''X'' → ''Z'' (resp. continuous bilinear maps
) will be denoted by L(''X'', ''Z'') (resp. B(''X'', ''Y''; ''Z'')) where if ''Z'' is the underlying scalar field then we may instead write L(''X'') (resp. B(''X'', ''Y'')).
* Any linear map
can be canonically decomposed as follows:
where
defines a bijection called the canonical bijection associated with ''L''.
* ''X''* or
will denote the continuous dual space of ''X''.
** To increase the clarity of the exposition, we use the common convention of writing elements of
with a prime following the symbol (e.g.
denotes an element of
and not, say, a derivative and the variables ''x'' and
need not be related in any way).
*
will denote the
algebraic dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of ''X'' (which is the vector space of all linear functionals on ''X'', whether continuous or not).
* A linear map ''L'' : ''H'' → ''H'' from a Hilbert space into itself is called positive if
for every
. In this case, there is a unique positive map ''r'' : ''H'' → ''H'', called the square-root of ''L'', such that
.
** If
is any continuous linear map between Hilbert spaces, then
is always positive. Now let ''R'' : ''H'' → ''H'' denote its positive square-root, which is called the absolute value of ''L''. Define
first on
by setting
for
and extending
continuously to
, and then define ''U'' on
by setting
for
and extend this map linearly to all of
. The map
is a surjective isometry and
.
* A linear map
is called compact or completely continuous if there is a neighborhood ''U'' of the origin in ''X'' such that
is
precompact in ''Y''.
**
In a Hilbert space, positive compact linear operators, say ''L'' : ''H'' → ''H'' have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:
There is a sequence of positive numbers, decreasing and either finite or else converging to 0, and a sequence of nonzero finite dimensional subspaces of ''H'' (i = 1, 2, ) with the following properties: (1) the subspaces are pairwise orthogonal; (2) for every ''i'' and every , ; and (3) the orthogonal of the subspace spanned by is equal to the kernel of ''L''.
Notation for topologies
*
σ(X, X′) denotes the
coarsest topology
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Definition
A topology on a set may be defined as the ...
on ''X'' making every map in X′ continuous and
or
denotes
''X'' endowed with this topology.
*
σ(X′, X) denotes
weak-* topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
on X* and
or
denotes
X′ endowed with this topology.
** Note that every
induces a map
defined by
. ''σ''(X′, X) is the coarsest topology on X′ making all such maps continuous.
*
b(X, X′) denotes the topology of bounded convergence on ''X'' and
or
denotes
''X'' endowed with this topology.
*
b(X′, X) denotes the topology of bounded convergence on X′ or the strong dual topology on X′ and
or
denotes
X′ endowed with this topology.
** As usual, if X* is considered as a topological vector space but it has not been made clear what topology it is endowed with, then the topology will be assumed to be b(X′, X).
A canonical tensor product as a subspace of the dual of Bi(X, Y)
Let ''X'' and ''Y'' be vector spaces (no topology is needed yet) and let Bi(''X'', ''Y'') be the space of all
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
s defined on
and going into the underlying scalar field.
For every
, let
be the canonical linear form on Bi(''X'', ''Y'') defined by
for every ''u'' ∈ Bi(''X'', ''Y'').
This induces a canonical map
defined by
, where
denotes the
algebraic dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of Bi(''X'', ''Y'').
If we denote the span of the range of ''𝜒'' by ''X'' ⊗ ''Y'' then it can be shown that ''X'' ⊗ ''Y'' together with ''𝜒'' forms a
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of ''X'' and ''Y'' (where ''x'' ⊗ ''y'' := ''𝜒''(''x'', ''y'')).
This gives us a canonical tensor product of ''X'' and ''Y''.
If ''Z'' is any other vector space then the mapping Li(''X'' ⊗ ''Y''; ''Z'') → Bi(''X'', ''Y''; ''Z'') given by ''u'' ↦ ''u'' ∘ ''𝜒'' is an isomorphism of vector spaces.
In particular, this allows us to identify the
algebraic dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of ''X'' ⊗ ''Y'' with the space of bilinear forms on ''X'' × ''Y''.
Moreover, if ''X'' and ''Y'' are locally convex
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (TVSs) and if ''X'' ⊗ ''Y'' is given the 𝜋-topology then for every locally convex TVS ''Z'', this map restricts to a vector space isomorphism
from the space of ''continuous'' linear mappings onto the space of ''continuous'' bilinear mappings.
In particular, the continuous dual of ''X'' ⊗ ''Y'' can be canonically identified with the space B(''X'', ''Y'') of continuous bilinear forms on ''X'' × ''Y'';
furthermore, under this identification the
equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
subsets of B(''X'', ''Y'') are the same as the equicontinuous subsets of
.
Nuclear operators between Banach spaces
There is a canonical vector space embedding
defined by sending
to the map
:
Assuming that ''X'' and ''Y'' are Banach spaces, then the map
has norm
(to see that the norm is
, note that
so that
). Thus it has a continuous extension to a map
, where it is known that this map is not necessarily injective. The range of this map is denoted by
and its elements are called nuclear operators.
is TVS-isomorphic to
and the norm on this quotient space, when transferred to elements of
via the induced map
, is called the trace-norm and is denoted by
. Explicitly, if
is a nuclear operator then
.
Characterization
Suppose that ''X'' and ''Y'' are Banach spaces and that
is a continuous linear operator.
* The following are equivalent:
*#
is nuclear.
*# There exists an sequence
in the closed unit ball of
, a sequence
in the closed unit ball of
, and a complex sequence
such that
and
is equal to the mapping:
for all
. Furthermore, the trace-norm
is equal to the infimum of the numbers
over the set of all representations of
as such a series.
* If ''Y'' is
reflexive then
is a nuclear if and only if
is nuclear, in which case
.
Properties
Let ''X'' and ''Y'' be Banach spaces and let
be a continuous linear operator.
* If
is a nuclear map then its transpose
is a continuous nuclear map (when the dual spaces carry their strong dual topologies) and
.
Nuclear operators between Hilbert spaces
Nuclear
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s of 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 ...
are called
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 ...
operators.
Let ''X'' and ''Y'' be Hilbert spaces and let ''N'' : ''X'' → ''Y'' be a continuous linear map. Suppose that
where ''R'' : ''X'' → ''X'' is the square-root of
and ''U'' : ''X'' → ''Y'' is such that
is a surjective isometry. Then ''N'' is a nuclear map if and only if ''R'' is a nuclear map;
hence, to study nuclear maps between Hilbert spaces it suffices to restrict one's attention to positive self-adjoint operators.
Characterizations
Let ''X'' and ''Y'' be Hilbert spaces and let ''N'' : ''X'' → ''Y'' be a continuous linear map whose absolute value is ''R'' : ''X'' → ''X''.
The following are equivalent:
#''N'' : ''X'' → ''Y'' is nuclear.
#''R'' : ''X'' → ''X'' is nuclear.
#''R'' : ''X'' → ''X'' is compact and
is finite, in which case
.
#* Here,
is the trace of ''R'' and it is defined as follows: Since ''R'' is a continuous compact positive operator, there exists a (possibly finite) sequence
of positive numbers with corresponding non-trivial finite-dimensional and mutually orthogonal vector spaces
such that the orthogonal (in ''H'') of
is equal to
(and hence also to
) and for all ''k'',
for all
; the trace is defined as
.
#
is nuclear, in which case
.
#There are two orthogonal sequences
in ''X'' and
in ''Y'', and a sequence
in
such that for all
,
.
#''N'' : ''X'' → ''Y'' is an
integral map
An integral bilinear form is a bilinear functional that belongs to the continuous dual space of X \widehat_ Y, the injective tensor product of the locally convex topological vector spaces (TVSs) ''X'' and ''Y''. An integral linear operator is a c ...
.
Nuclear operators between locally convex spaces
Suppose that ''U'' is a convex balanced closed neighborhood of the origin in ''X'' and ''B'' is a convex balanced bounded
Banach disk
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.
One method is used if the disk D is bounded: in this case, the ...
in ''Y'' with both ''X'' and ''Y'' locally convex spaces. Let
and let
be the canonical projection. One can define the
auxiliary Banach space with the canonical map
whose image,
, is dense in
as well as the auxiliary space
normed by
and with a canonical map
being the (continuous) canonical injection.
Given any continuous linear map
one obtains through composition the continuous linear map
; thus we have an injection
and we henceforth use this map to identify
as a subspace of
.
Definition: Let ''X'' and ''Y'' be Hausdorff locally convex spaces. The union of all
as ''U'' ranges over all closed convex balanced neighborhoods of the origin in ''X'' and ''B'' ranges over all bounded
Banach disk
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.
One method is used if the disk D is bounded: in this case, the ...
s in ''Y'', is denoted by
and its elements are call nuclear mappings of ''X'' into ''Y''.
When ''X'' and ''Y'' are Banach spaces, then this new definition of ''nuclear mapping'' is consistent with the original one given for the special case where ''X'' and ''Y'' are Banach spaces.
Sufficient conditions for nuclearity
* Let ''W'', ''X'', ''Y'', and ''Z'' be Hausdorff locally convex spaces,
a nuclear map, and
and
be continuous linear maps. Then
,
, and
are nuclear and if in addition ''W'', ''X'', ''Y'', and ''Z'' are all Banach spaces then
.
* If
is a nuclear map between two Hausdorff locally convex spaces, then its transpose
is a continuous nuclear map (when the dual spaces carry their strong dual topologies).
** If in addition ''X'' and ''Y'' are Banach spaces, then
.
* If
is a nuclear map between two Hausdorff locally convex spaces and if
is a completion of ''X'', then the unique continuous extension
of ''N'' is nuclear.
Characterizations
Let ''X'' and ''Y'' be Hausdorff locally convex spaces and let
be a continuous linear operator.
* The following are equivalent:
*#
is nuclear.
*# (Definition) There exists a convex balanced neighborhood ''U'' of the origin in ''X'' and a bounded
Banach disk
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.
One method is used if the disk D is bounded: in this case, the ...
''B'' in ''Y'' such that
and the induced map
is nuclear, where
is the unique continuous extension of
, which is the unique map satisfying
where
is the natural inclusion and
is the canonical projection.
*# There exist Banach spaces
and
and continuous linear maps
,
, and
such that
is nuclear and
.
*# There exists an equicontinuous sequence
in
, a bounded
Banach disk
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.
One method is used if the disk D is bounded: in this case, the ...
, a sequence
in ''B'', and a complex sequence
such that
and
is equal to the mapping:
for all
.
* If ''X'' is barreled and ''Y'' is
quasi-complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete.
This concept is of considerable importance for non- metrizable TVSs.
Properties
* Eve ...
, then ''N'' is nuclear if and only if ''N'' has a representation of the form
with
bounded in
,
bounded in ''Y'' and
.
Properties
The following is a type of ''
Hahn-Banach theorem'' for extending nuclear maps:
* If
is a TVS-embedding and
is a nuclear map then there exists a nuclear map
such that
. Furthermore, when ''X'' and ''Y'' are Banach spaces and ''E'' is an isometry then for any
,
can be picked so that
.
* Suppose that
is a TVS-embedding whose image is closed in ''Z'' and let
be the canonical projection. Suppose all that every compact disk in
is the image under
of a bounded Banach disk in ''Z'' (this is true, for instance, if ''X'' and ''Z'' are both Fréchet spaces, or if ''Z'' is the strong dual of a Fréchet space and
is weakly closed in ''Z''). Then for every nuclear map
there exists a nuclear map
such that
.
** Furthermore, when ''X'' and ''Z'' are Banach spaces and ''E'' is an isometry then for any
,
can be picked so that
.
Let ''X'' and ''Y'' be Hausdorff locally convex spaces and let
be a continuous linear operator.
* Any nuclear map is compact.
* For every topology of uniform convergence on
, the nuclear maps are contained in the closure of
(when
is viewed as a subspace of
).
See also
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References
Bibliography
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External links
Nuclear space at ncatlab
{{TopologicalTensorProductsAndNuclearSpaces
Topological vector spaces
Tensors