Abstract

A bounded linear operator on a Hilbert space is trace class if its singular values are summable. The trace class operators on form an operator ideal and in the case that is finite-dimensional, the trace tr of is given by for any matrix representation of . In applications of trace class operators to scattering theory and representation theory, the subject is complicated by the fact that if is an integral kernel of the operator on the Hilbert space with a -finite measure, then may not be defined, because the diagonal may be a set of -measure zero. The present note describes a class of linear operators acting on a Banach function space which forms a lattice ideal of operators on , rather than an operator ideal, but coincides with the collection of hermitian positive trace class operators in the case of .

1. Introduction

A trace class operator on a separable Hilbert space is a compact operator whose singular values , , satisfy The decreasing sequence consists of eigenvalues of . Equivalently, is trace class if and only if, for any orthonormal basis of , the sum is finite. The number is called the trace of and is independent of the orthonormal basis of . Lidskii’s equality asserts that is actually the sum of the eigenvalues of the compact operator [1, Theorem 3.7].

We refer to [1] for properties of trace class operators. The collection of trace class operators on is an operator ideal and Banach space with the norm . The following facts are worth noting in the case of the Hilbert space with respect to Lebesgue measure on the interval .(a)If is a trace class linear operator, then there exist , , with and  where a.e.. In particular, is regular and has an integral kernel . Moreover, (b)Suppose that is a regular linear operator defined by formula (3) for a continuous function . If is trace class, then , and [2, Theorem ].(c)Suppose that the function is continuous and positive definite; that is, for all and , , and any . Then for all . If , then there exists a unique trace class operator defined by formula (3) [1, Theorem 2.12].

Let be a measure space. The projective tensor product is the set of all sums: The norm of is given by where the infimum is taken over all sums for which the representation (5) holds. The Banach space is actually the completion of the algebraic tensor product with respect to the projective tensor product norm [3, Section 6.1].

There is a one-to-one correspondence between the space of trace class operators acting on and , so that the trace class operator has an integral kernel . If the integral kernel given by (5) has the property that for all such that the sum is finite, then the equality holds. Because the diagonal may be a set of -measure zero in , it may be difficult to determine whether or not a given integral kernel has such a distinguished representation.

The difficulty is addressed by Brislawn [4, 5], [1, Appendix ] who shows that, for a trace class operator with integral kernel , the equality holds. The measure is supposed in [5] to be a -finite Borel measure on a second countable topological space and the regularised kernel is defined from by averaging with respect to the product measure . Extending the result (c) of M. Duflo given above, Brislawn [5, Theorem 4.3] shows that a hermitian positive Hilbert-Schmidt operator is a trace class operator if and only if .

The present paper examines the space of absolute integral operators defined on a Banach function space for which . Elements of are called lattice trace operators because is a lattice ideal in the Banach lattice of regular operators on , whereas the collection of trace class operators on a Hilbert space is an operator ideal in the Banach algebra of all bounded linear operators on . The intersections of and with the hermitian positive operators on are equal for locally square integrable kernels; see Proposition 4.

The regularised kernel of an absolute integral operator is defined by adapting the method of Brislawn [5] to positive operators with an integral kernel. The generalised trace may be viewed alternatively as a bilinear integral with respect to the measure , . Lattice trace operators are employed in the proof of the Cwikel-Lieb-Rosenblum inequality for dominated semigroups [6].

The basic definitions of Banach function spaces and operators with an integral kernel which act upon them are set out in Section 2. The martingale regularisation of the integral kernel of an operator between Banach function spaces is set out in Section 3 and the connection with trace class operators on is set out in Section 4.

2. Banach Function Spaces and Regular Operators

Let be a second countable topological space with Borel -algebra . The diagonal is a closed subset of the Cartesian product . Because the Borel -algebra of is equal to , the diagonal belongs to the -algebra .

We suppose that is a -finite measure space. The space of all -equivalence classes of Borel measurable scalar functions is denoted by . It is equipped with the topology of convergence in -measure over sets of finite measure and vector operations pointwise -almost everywhere. Any Banach space that is a subspace of with the properties that(i) is an order ideal of , that is, if , , and -a.e., then and(ii)if and -a.e., then ,is called a Banach function space (based on ). The Banach function space is necessarily Dedekind complete; that is, every order bounded set has a sup and an inf [7, page 116]. The set of with -a.e. is written as .

We suppose that contains the characteristic functions of sets of finite measure and , , is -additive in on sets of finite measure; for example, is -order continuous; see [8, Corollary 3.6]. If is reflexive and is finite and nonatomic, then it follows from [8, Corollary 3.23] that the values of the variation of are either zero or infinity. In particular, this is the case for with .

Following the account of Brislawn [5], we extend the mapping from the space of trace class linear operators to a larger class of regular operators by representing by a “regularised” kernel, so that the collection of regular operators for which is a vector sublattice of the Riesz space of regular operators—a property not necessarily enjoyed by the trace class operators.

Let be a Banach function space based on the -finite measure space as above. A continuous linear operator is called positive if . The collection of all positive continuous linear operators on is written as . If the real and imaginary parts of a continuous linear operator can be written as the difference of two positive operators, it is said to be regular. The modulus of a regular operator is defined by The collection of all regular operators is written as and it is given the norm , under which it becomes a Banach lattice [7, Proposition ].

A continuous linear operator has an integral kernel if is a Borel measurable function such that for the operator given by for each , in the sense that for -almost all and is an element of . Then is -integrable on any product set with finite measure. If , then -a.e. on [7, Theorem ].

A continuous linear operator is an absolute integral operator if it has an integral kernel for which is a bounded linear operator on . Then [7, Theorem ]. The collection of all absolute integral operators is a lattice ideal in [7, Theorem ].

Suppose that has an integral kernel , that is, an -valued simple function with . Then it is natural to view as a bilinear integral. Our aim is to extend the integral to a wider class of absolute integral operators.

3. Martingale Regularisation

Let be a countable base for the topology of . An increasing family of countable partitions , , is defined recursively by setting equal to a partition of into Borel sets of finite -measure and for .. For each , let be the -algebra for all countable unions of elements of .

Suppose that is a Borel measurable function defined on that is integrable on every set of finite -measure.

For each , the set is the unique element of the partition containing . For each , the conditional expectation can be represented for -almost all as Let be the set of all for which there exists such that . Then for all because is a refinement of if . Moreover is -null because . If -a.e., then for all . In particular, for all . Although diag may be a set of -measure zero, the application of the conditional expectation operators , , has the effect of regularising . By an appeal to the martingale convergence theorem, converges -a.e. to as .

Let for all and we set If , then , , is a finite measure. For a regular operator with positive and negative parts , we set if one of the integrals on the right-hand side of the equation is finite. The integral is defined by linearity for each regular operator . It is clear from the construction that the collection of absolute integral operators such that is a vector sublattice of the space of regular operators on . We call elements of lattice trace operators.

Theorem 1. The space is a lattice ideal in ; that is, if  , and , then . Moreover, is a Dedekind complete Banach lattice with the norm The map is a positive continuous linear function on .

Proof. If and , then is an absolute integral operator by [7, Theorem ]. If is the integral kernel of and is the integral kernel of , then by [7, Theorem ], the inequality holds -a.e.. Then for -almost all , so that Hence, .
To show that is complete in its norm, suppose that for . Then in the space of regular operators on . The inequality ensures that is an absolute integral operator with kernel by [7, Theorem ] and -a.e..
Suppose first that is a real Banach function space. Each positive part of , , has an integral kernel such that By monotone convergence, there exists a set of full -measures on which for each .. Taking the limsup and applying the monotone convergence theorem pointwise and under the sum show that for -almost all and . Applying the same argument to and then the real and imaginary parts of ensures that and Dedekind completeness is inherited from [7, Theorem ] and [7, Example , page 9]. The bound defines a positive continuous linear function on .

Example 2 (see [4, Example 3.2]). There exist lattice-positive, compact linear operators such that is finite but is not a trace class linear operator on the Hilbert space .
In particular, the Volterra operator is defined by The (lattice) positive linear map is a Hilbert-Schmidt operator but not trace class. Nevertheless,.

4. Trace Class Operators

Proposition 3 (see [5, Theorem 3.1]). If   is a trace class linear operator, then, for any function such that  , where with  , the equalities hold.

If is continuous almost everywhere along the diagonal , then for -almost all [5, Theorem 2.4].

For positive operators in the Hilbert space sense, we have the following sufficient condition for traceability. The operator , , for a Borel set , is denoted by .

Proposition 4. Let   be an absolute integral operator whose integral kernel is square integrable on any set of finite -measure. If   for all , then is trace class if and only if   is finite, and in this case

Proof. The case where is assumed to be trace class is covered by Proposition 3 above. Suppose that is an absolute integral operator such that for all and is finite.
If the integral kernel of is square integrable on any set of finite -measure, then for any Borel set with , the operator is a positive Hilbert-Schmidt operator. If as , then by monotone convergence, so By choosing for for , we have for all , so as . According to [5, Theorem 4.3], is trace class and For every , the inequality By polarisation, in the weak operator topology as , so for very finite rank operator . By [1, Theorem 2.14], is a trace class operator and an appeal to Proposition 3 gives (28).

Proposition 5. If   is an atomic measure space with countably many atoms, then and

5. Lattice Properties

Let be the diagonal embedding , . Let . If converges pointwise -a.e. and in , then there exist scalars and Borel sets such that and we can write for every such that and for -almost all ; see [9].

Proposition 6. Let   be a positive kernel operator. For any nonnegative -measurable functions ,  , the equalities of extended real numbers hold.
For any essentially bounded -measurable function ,

Proof. If the kernel of has the representation (39), then for any sets , we have is equal to for -almost all . The result follows by linearity and approximating and by simple functions.

It is well known that if is a trace class operator on a Hilbert space and is any bounded linear operator on then and are also trace class operators (i.e., is an operator ideal) and [1, Corollary 3.8] By contrast, the space is a lattice ideal in . For and , the operator may not even be a kernel operator, but we have the following trace property.

Proposition 7. Let  , , be positive kernel operators. Then the equalities of extended real numbers hold.

Proof. Suppose that the kernels of , , have the representation (39).
If , , is an increasing sequence of sub--algebras of such that the -algebra generated by is contained in for , then By the Fubini-Tonelli Theorem this is equal to

We also note that a bilinear version of the Fubini-Tonelli Theorem holds.

Let be a -finite measure space. For any function such that , we say that is -integrable if for each , , the scalar function is -integrable and there exists such that for all , . Then we set Because is a lattice ideal, for each , there exists a positive operator such that for all , .

Remark 8. For each , , the tensor product and are continuous linear functionals on , so it is natural to assume that both (48) and (49) hold.

The following statement is a consequence of the definitions.

Proposition 9. Let be a positive operator valued function such that is -integrable.
Then for -almost all , the scalar valued function is -integrable, and the equalities hold. Moreover, for every .

Proof. Equation (52) is the definition of and (53) is a reformulation of assumption (48). For -almost all , we can find a martingale and a regularisation , , of the kernel associated with such that for all and , . Then, for each , we have by the scalar Fubini-Tonelli Theorem.

The following result follows from the observation in Theorem 1 that is a lattice ideal and an application of monotone convergence.

Proposition 10. Let be a positive operator valued measure on a measurable space . If  , then the set function , , is a finite measure such that for all -measurable .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.