Lattice Trace Operators

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