Characterizations of nuclear pseudo-differential operators on ℤ with some applications

Iran University of Science and Technology, Iran, Tehran

^* Corresponding author: m_jamalpour@mathdep.iust.ac.ir

Received: 21 August 2017
Accepted: 21 January 2018

Abstract

In this paper, we give necessary and sufficient conditions on the symbols σ, such that the corresponding pseudo-differential operators T_σ from L^p1(ℤ) into L^p2(ℤ), 1 ≤ p1,p2 < ∞, be nuclear. We show that the adjoint operators of the nuclear pseudo-differential operators from L^p′₂(ℤ) into L^p′₁(ℤ) are nuclear and present a necessary and sufficient condition on the symbols of the nuclear pseudo-differential operators from L²(ℤ) into L²(ℤ) to be self-adjoint. As applications, We get the symbol of the product of the nuclear operators with the bounded operators, and a necessary and sufficient condition on the symbols of nuclear operators is given to be normal.