A
linear operator
mapping one
locally convex space
into another, and having a special form of approximation by
operators of finite rank
(that is, by continuous linear operators with finite-dimensional ranges).
A nuclear operator has certain properties inherent in finite-dimensional
operators. In particular, a nuclear operator mapping a space with a
basis into itself has a finite trace (see below), which coincides with
the sum of the series formed from the diagonal elements of
the matrix of this operator relative to an arbitrary basis. Nuclear
operators first appeared in mathematical quantum mechanics and were called
"operator with a traceoperators with a trace"
(see
[1],
[2]).
On a Hilbert space the operators with a
trace are in a one-to-one correspondence with bivalent tensors, and
the trace of an operator coincides with contraction of
the corresponding tensor. By using this correspondence,
A.F. Ruston
[3]
carried over the concept of a nuclear operator to Banach
spaces. Independently, in connection with the theory of nuclear spaces (cf.
Nuclear space),
A. Grothendieck
carried over the concept to locally convex spaces (see
[4],
[5]).
Let
and
be locally convex spaces over the field of real or complex numbers, let
and
be their duals endowed with the strong topology, let
be the vector space of all continuous linear mappings from
into
,
and let
be the space of all weakly continuous mappings from
into
.
Set
and
.
A linear operator
is called
nuclear
if it can be represented in the form
where
is a summable numerical sequence,
is an equicontinuous sequence in
,
is a sequence of elements from a certain complete bounded convex circled set in
(cf.
Topological vector space),
and
denotes the value of the linear functional
at a vector
.
The representation
(1)
can be regarded as an expansion
of the operator as a sum of operators of rank
1 (that is, with a one-dimensional range), and
the corresponding series is absolutely convergent in
in the topology of uniform convergence on bounded
sets. Thus, in this topology, the nuclear operator
is the limit of a sequence of operators of finite rank. If
and
are Banach spaces, then a nuclear operator
can be approximated, in the
nuclear norm,
by operators of finite rank.
The expansion
(1)
is called a
nuclear representation
of
.
Every nuclear operator has a nuclear representation
(1)
such that
,
.
If
is a
barrelled space
and is complete, or at least
quasi-complete
(i.e. closed bounded sets in
are complete), then the expansion
(1)
is nuclear if and only if
and
are bounded.
By changing the conditions on
,
,
and
one can obtain different modifications of the concept of a nuclear operator (see
[4],
[5],
[7]).
If instead of the equicontinuity of
one requires its elements to belong to a complete bounded convex circled set in
,
then the expansion
(1)
defines a
Fredholm operator;
these operators form the natural domain of application of the Fredholm theory (see
[4],
[5]).
Every nuclear operator is a Fredholm operator, and when
is endowed with the
Mackey topology,
any Fredholm operator
is nuclear. A nuclear operator
is called
strongly nuclear
(or a
nuclear operator of order
)
if it admits a nuclear representation
(1)
in which
is a rapidly decreasing sequence, that is,
for all
.
Integral operators (in particular, Fredholm integral operators) provide
many examples of nuclear operators and their modifications (see
[4],
[5],
[7],
[8]).
Properties of nuclear operators.
Every nuclear operator
is compact, that is, it maps a neighbourhood of zero in
into a set with compact closure in
.
Thus, every nuclear operator is continuous, and
every Fredholm operator is weakly continuous. The product (in
any order) of a nuclear operator and a continuous linear
operator is a nuclear operator. In particular, the set of
all nuclear operators is an ideal in the algebra
;
correspondingly, the Fredholm operators form an ideal in
.
The strongly nuclear operators also form an ideal in
.
Every nuclear operator
has a unique extension
,
where
is the completion of
and
is nuclear. If
is a Fredholm operator, then the dual mapping
is a nuclear operator. For any nuclear operator
one can find Banach spaces
and
,
compact operators
and
,
and a nuclear operator
,
such that
.
If
is a strongly nuclear operator, then the sequence of its eigen
values (in general, complex), ordered in
decreasing absolute value, is rapidly decreasing.
Let
be a nuclear space and let
be a complete or quasi-complete space. Then for
the following assertions are equivalent: 1)
is a nuclear operator; 2)
is a
compact operator;
3)
is a
bounded operator;
i.e.
maps a neighbourhood of zero in
into a bounded set in
;
and 4)
is a strongly nuclear operator.
Let
,
and
be Hilbert spaces, and let
and
be Hilbert–Schmidt operators (cf.
Hilbert–Schmidt operator).
Then
is nuclear. Conversely, every nuclear operator is the product
of two operators of Hilbert–Schmidt type.
An arbitrary completely-continuous (compact) operator
is nuclear if and only if the series of eigen values of the positive-definite operator
in the
polar decomposition
converges, where
is an isometric operator mapping the range of
into
(see
[9]).
Operators with a trace.
Let
be an arbitrary locally convex space, and let
be a nuclear (respectively, Fredholm) operator mapping
into itself and admitting a representation of the form
(1).
The series
converges absolutely; if its sum does not depend on
the representation
(1),
then the sum is called the
trace of the nuclear
(Fredholm)
operator
,
and is denoted by
.
In this case the trace is
well defined
(see
[4],
[5]).
The operator is said to be an
operator of finite trace
or a
trace class operator.
If
(1)
contains only a finite number of terms, then
is an operator of finite rank, and
is the same as the trace of the finite-dimensional operator induced in the range of
.
Let
be the
inductive tensor product
of
and
,
that is, the completion of the (algebraic) tensor product
in the strongest
locally convex topology
in which the canonical bilinear mapping
(
goes into
)
is continuous in each variable separately. The composition of
this mapping with any continuous linear form on
gives a bilinear form on
that is continuous in each variable separately, and the correspondence
between forms of this type is one-to-one. In particular, the bilinear form
corresponds to a continuous linear form on
.
The value of this form at a
is denoted by
.
An element
is called a
Fredholm kernel
if it admits an expansion of the form
where
,
and
are the same as in the expansion
(1)
for a
Fredholm operator. The Fredholm kernels form a subspace in
,
denoted by
.
Suppose that the algebra
of weakly continuous operators on
is endowed with the weak
operator topology
defined by semi-norms
,
where
,
and
and
range over
and
,
respectively. The mapping
that sends an element
of the form
(2)
into an operator
of the form
(1),
is well defined, linear and continuous; also
if the trace of the operator
is well defined. If
and
are complete (for example, if
is a
Fréchet space),
then
can be continuously extended to
.
The images of elements of
under this mapping are called
operators with a trace
(see
[4],
[5]).
If
is a Banach space, then every operator with a trace is
nuclear, so that in this case the classes of nuclear operators, of
Fredholm operators and of operators with a trace coincide. There are
operators with a trace that are not Fredholm operators (for
example, in nuclear Fréchet spaces). The non-compactness
of these operators makes their study difficult.
The single-valuedness problem (the
"problème de biunivocité" ).
If the mapping
is one-to-one, or at least if
implies
,
then the trace of
is well defined by
.
This possibility is closely connected with the
approximation property,
which is that
contains a net (cf.
Net (directed set))
of operators of finite rank that converges to the identity operator
in the topology of uniform convergence on all pre-compact sets. If
is a Banach space, then the trace of any nuclear operator
is well defined if and only if the approximation property holds
[4].
A reflexive separable space
without the approximation property (and without a
Schauder basis, thus solving a well-known
problem
of
S. Banach)
has been constructed
[11].
This solves the single-valuedness problem: There is an
such that
but
.
If a locally convex space
has the approximation property, then every nuclear operator has a well defined trace; if
is a net of operators of finite rank that converges to an arbitrary operator
uniformly on all pre-compact (or, at least, on convex balanced compact) sets, then
is valid for any nuclear operator
(see
[12]).
However, there is a locally convex space with the approximation property in
which it is impossible to properly define the trace for all
Fredholm operators. Any Fredholm operator on a locally convex space
has a well-defined trace if
has the
bounded approximation property,
that is, if there is a net of operators of finite rank that
converges to the identity operator in the weak
operator topology, and that is bounded in this topology;
any space with a Schauder basis has this property. If
is a bounded net that converges in
to an arbitrary operator
(for example, if
is an arbitrary countable convergent sequence in
),
then
(3)
holds for any Fredholm operator
provided that the
have a well-defined trace (for example, if the
are operators of finite rank, or if
has the bounded approximation property). If
has the approximation (respectively, bounded approximation) property,
then for any nuclear (respectively, Fredholm) operator
and any
(respectively,
)
one has
(see
[12]).
Matrix trace.
Suppose that a locally convex space
has a Schauder
basis
so that any
can be expanded as
,
where
.
Then
is called the
matrix trace
of the operator
if the series is convergent. This series converges
absolutely if the basis is unconditional. Any Fredholm operator
on a space with a Schauder basis has a
well-defined trace that coincides with its matrix trace, which in this
case does not depend on the choice of the basis
[13].
An arbitrary continuous operator on a Hilbert space is nuclear if and
only if it has a finite matrix trace for any orthonormal basis (see
[2],
[8],
[9]).
Nuclear trace.
Let
be a compact space with a Borel measure
,
let
be the Banach space of continuous functions on
equipped with the topology of uniform convergence, and let
be a continuous function on
.
Then the linear integral operator
on
(a classical Fredholm integral operator) is nuclear
and has a well-defined trace; moreover,
If
is the integral operator with kernel
,
acting on a space of functions on a space
with a measure
,
and if the right-hand side of
(4)
can be given
a reasonable meaning, then this quantity is called the
nuclear trace
of
.
For different classes of integral operators, conditions can be obtained that ensure
the nuclearity of these operators, and enable one to give a meaning to
(4)
(see
[4],
[5],
[8],
[14]).
Spectral trace.
Let
be a locally convex space over the field of complex numbers, and let
be a nuclear operator on
.
The spectrum of
,
as of any compact operator (cf.
Spectrum of an operator),
is either a finite set or is a sequence that
converges to zero, and any non-zero value has finite spectral multiplicity. If the series
formed from the non-zero eigen values of
(each eigen values appears in
(5)
as many
times as its spectral multiplicity) converges absolutely, then its sum is called the
spectral trace
of
,
and is denoted by
.
Every nuclear operator on a Hilbert space has a
spectral trace, which coincides with its matrix trace
[15].
Let
be a
multi-Hilbert space
(a
Hilbertiable space),
that is, the topology in
can be generated by a family of semi-norms each of
which is obtained from a non-negative definite Hermitian form on
;
any
nuclear space
is an example of a multi-Hilbert space. Then any nuclear operator
on
has a well-defined trace and a spectral trace, and
(see
[13]).
A nuclear operator
need not have a matrix trace. A nuclear operator
on a Banach space need not have a spectral trace even when the space has a basis and
is well defined. Also, the equality
can be violated. For example, in the Banach space
of sequences that converge to 0 there is a nuclear operator
such that
and
,
so that
does not have non-zero eigen values, and
.
For a nuclear operator
acting on an arbitrary Banach or locally
convex space (without, perhaps, any approximation properties),
it is possible to give conditions on
under which
and
exist and are equal (see
[4],
[14],
[16],
[17]).
Example.
Let
be a complex Banach space and let
be the algebra of continuous linear operators on
equipped with the usual operator norm. For any
let
denote the greatest lower bound of
when
ranges over the set of all operators in
with
rank
(that is, dimension of the range) not exceeding
.
The set of all
for which
is denoted by
.
Every
is nuclear; if
is a Hilbert space, then
coincides with the set of all nuclear operators on
.
For an arbitrary Banach space
,
each operator
has a trace,
,
and a spectral trace, and
(see
[16],
[17]).