Lattice Trace Operators

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


Introduction
A trace class operator  on a separable Hilbert space H is a compact operator whose singular values   (),  = 1, 2, . .., satisfy The decreasing sequence {  ()} ∞ =1 consists of eigenvalues of ( * ) is called the trace of  and is independent of the orthonormal basis {ℎ  } ∞ =1 of H. Lidskii's equality asserts that tr () 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 C 1 (H) of trace class operators on H is an operator ideal and Banach space with the norm ‖ ⋅ ‖ 1 .The following facts are worth noting in the case of the Hilbert space  2 ([0, 1]) with respect to Lebesgue measure on the interval [0, 1].
The norm of  ∈  2 ()⊗   2 () is given by where the infimum is taken over all sums for which the representation (5) holds.The Banach space  2 ()⊗   2 () is actually the completion of the algebraic tensor product  2 () ⊗  2 () 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  2 () and  2 ()⊗   2 (), so that the trace class operator   has an integral kernel  ∈  2 ()⊗   2 ().If the integral kernel  given by ( 5) has the property that for all ,  ∈ Σ such that the sum holds.Because the diagonal {(, ) :  ∈ Σ} may be a set of ( ⊗ )-measure zero in Σ × Σ, it may be difficult to determine whether or not a given integral kernel  : Σ × Σ → C has such a distinguished representation.
The difficulty is addressed by Brislawn [4,5], [1, Appendix D] who shows that, for a trace class operator   :  2 () →  2 () with integral kernel , the equality tr holds.The measure  is supposed in [5] to be a -finite Borel measure on a second countable topological space Σ and the regularised kernel k is defined from  by averaging with respect to the product measure  ⊗ .Extending the result (c) of M. The regularised kernel k : Σ × Σ → C of an absolute integral operator   is defined by adapting the method of Brislawn [5] to positive operators with an integral kernel.The generalised trace ∫ Σ k(, ) () may be viewed alternatively as a bilinear integral ∫ Σ ⟨  , ⟩ with respect to the measure  :   →   ,  ∈ B. 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  2 () is set out in Section 4.

Banach Function Spaces and Regular Operators
Let Σ be a second countable topological space with Borel - We suppose that (Σ, B, ) is a -finite measure space.The space of all -equivalence classes of Borel measurable scalar functions is denoted by  0 ().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  0 () with the properties that We suppose that  contains the characteristic functions of sets of finite measure and  :   →   ,  ∈ S, 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  =   ([0, 1]) with 1 <  < ∞.
Following the account of Brislawn [5], we extend the mapping   → ∫ Σ ⟨, ⟩ from the space C 1 ( 2 ()) 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 ∫ Σ ⟨||, ⟩ < ∞ 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 (Σ, B, ) as above.A continuous linear operator  :  →  is called positive if  :  + →  + .The collection of all positive continuous linear operators on  is written as L + ().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  Suppose that  ∈ L() has an integral kernel  = ∑  =1      , 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.
An increasing family of countable partitions P  ,  = 1, 2, . .., is defined recursively by setting P 1 equal to a partition of Σ into Borel sets of finite -measure and for  = 1, 2, . .... For each  = 1, 2, . .., let E  be the -algebra for all countable unions of elements of P  .Suppose that  ≥ 0 is a Borel measurable function defined on Σ×Σ that is integrable on every set of finite (⊗)-measure.
For each  ∈ Σ, the set   () is the unique element of the partition P  containing .For each  = 1, 2, . .., (13) Let N be the set of all  ∈ Σ for which there exists  = 1, 2, . . .such that (  ()) = 0. Then (  ()) = 0 for all  >  because P  is a refinement of for all (, ) ∈ N  × N  .In particular, for all  ∈ N  .Although diag (Σ × Σ) may be a set of ( ⊗ )measure zero, the application of the conditional expectation operators If ∫ Σ ⟨, ⟩ < ∞, then   → ∫  ⟨, ⟩ = ∫  k(, x) (),  ∈ B, 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 ∫ Σ ⟨, ⟩ is defined by linearity for each regular operator  :  → .It is clear from the construction that the collection of absolute integral operators  such that ∫ Σ ⟨||, ⟩ < ∞ is a vector sublattice C 1 () of the space of regular operators on  2 ().We call elements of C 1 () lattice trace operators.
By monotone convergence, there exists a set of full measures on which for each  = 1, 2, . .... Taking the limsup and applying the monotone convergence theorem pointwise and under the sum show that for -almost all  ∈ Σ and ∫ Σ k+ (, ) < ∞.Applying the same argument to  − and then the real and imaginary parts of  ensures that  ∈ C 1 () and defines a positive continuous linear function on C 1 ().
For positive operators in the Hilbert space sense, we have the following sufficient condition for traceability.The operator   →   ,  ∈ for very finite rank operator .By [1, Theorem 2.14],  is a trace class operator and an appeal to Proposition 3 gives (28).

Lattice Properties
for -almost all  ∈ Σ.The result follows by linearity and approximating  1 and  2 by simple functions.
It is well known that if  is a trace class operator on a Hilbert space H and  is any bounded linear operator on H then  and  are also trace class operators (i.e., C 1 (H) is an operator ideal) and [1,Corollary 3.8] tr () = tr () . (44) By contrast, the space C 1 ( 2 ()) is a lattice ideal in L  ( 2 ()).
For  ∈ C 1 ( 2 ())) and  ∈ L( 2 ()), the operator  may not even be a kernel operator, but we have the following trace property.
1/2 .Equivalently,  is trace class if and only if, for any orthonormal basis {ℎ  } ∞ =1 of H, the sum ∑ ∞ =1 |(ℎ  , ℎ  )| is finite.The number tr The present paper examines the space C 1 () of absolute integral operators   :  →  defined on a Banach function space for which ∫ Σ | k(, )| () < ∞.Elements of C 1 () are called lattice trace operators because C 1 () is a lattice ideal in the Banach lattice of regular operators on , whereas the collection C 1 (H) of trace class operators on a Hilbert space H is an operator ideal in the Banach algebra L(H) of all bounded linear operators on H.The intersections of C 1 () and C 1 ( 2 ()) with the hermitian positive operators on  2 () are equal for locally square integrable kernels; see Proposition 4.
and it is given the norm   → ‖||‖,  ∈ L  () under which it becomes a Banach lattice [7, Proposition 1.3.6].A continuous linear operator  :  →  has an integral kernel  if  : Σ×Σ → C is a Borel measurable function such that  =   for the operator given by