Nuclearity of operators related to finite measure spaces

Abstract

Let (S, B, m) be a finite measure space. The aim of this paper is to give necessary and sufficient conditions on symbols such that the corresponding Z-operator from L p 1 (Z) into L p 2 (Z) and S-operator from L p 1 (S) into L p 2 (S) to be nuclear for 1 ≤ p 1 , p 2 < ∞. We show that the adjoint of the nuclear Z-operator from L p 2 into L p 1 is again a nuclear operator. As applications, we get the symbol of the product of the nuclear operators with bounded operators.

Nulcearity for power series kernels on higher dimensions

“…Kernel criteria on L p spaces has been considered in [16], and the special case of L 2 in [12], [13], [17], [25] for the study of different spectral properties. Symbolic criteria to ensure the nuclearity of pseudodifferential operators on several kind of domains has been studied in [6], [7], [8], [9], [15], [22], [23]. The interesting case of the Fox-Li operator corresponds to a kernel of the form K(x, y) = e iω(x−y) 2 = e iωx 2 e −2iωxy e iωy 2 , where X = Y = [−1, 1] and ω is a positive real number.…”

Nuclearity for power series kernels in higher dimensions

“…Kernel criteria on L p spaces has been considered in [16], and the special case of L 2 in [12], [13], [17], [25] for the study of different spectral properties. Symbolic criteria to ensure the nuclearity of pseudodifferential operators on several kind of domains has been studied in [6], [7], [8], [9], [15], [22], [23]. The interesting case of the Fox-Li operator corresponds to a kernel of the form K(x, y) = e iω(x−y) 2 = e iωx 2 e −2iωxy e iωy 2 , where X = Y = [−1, 1] and ω is a positive real number.…”

Trace class and Hilbert-Schmidt pseudo differential operators on step two nilpotent Lie groups

“…Using this idea, pseudo-differential operators on different classes of groups such as S 1 , Z, finite abelian groups, locally compact abelian groups, affine groups, compact groups, compact Lie groups, homogeneous spaces of compact groups, Heisenberg group, and on general locally compact type I groups have been defined and studied broadly by several researchers. We refer to [2,4,12,13,16,20,30,17,18,26,27] and references therein.…”