On the nuclear trace of Fourier integral operators

Mondal

2022

Monatsh Math

Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on the compact homogeneous space G/H characterizing the Schatten-von Neumann classes Sr(L 2 (G/H)) for all 0 < r ≤ ∞. We go on to provide a symbolic characterization for r-nuclear, 0 < r ≤ 1, pseudo-differential operators on L p (G/H) with applications to adjoint, product and trace formulae. The criteria here are given in terms of matrix-valued symbols defined on noncommutative analogue of phase space G/H × G/H. Finally, we present an application of aforementioned results in the context of the heat kernels.

Section: Introductionmentioning

confidence: 99%

Schatten class and nuclear pseudo-differential operators on homogeneous spaces of compact groups

Mondal

2022

Monatsh Math

“…Also, the mapping properties for these operators acting on

ℓ^{p} false(Z^{n} false)

can be found in Molahajloo [45], Rodriguez [47] and in the paper of the first author [18]. The nuclearity of Fourier integral operators on

R^{n}

Z^{n}

, compact Lie groups and compact homogeneous manifolds has been investigated in [19].…”

Section: Introductionmentioning

confidence: 99%

The nuclear trace of periodic vector‐valued pseudo‐differential operators with applications to index theory

Cardona

Mathematische Nachrichten

2021

Self Cite

In this paper we investigate the nuclear trace of vector‐valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo‐differential operators. First we characterise the nuclearity of pseudo‐differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector‐valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector‐valued Fourier multiplier in terms of its operator‐valued symbol and then we use this formula for expressing the index of certain elliptic operators belonging to periodic Hörmander classes.

“…The study of pseudodifferential operators is first introduced by Kahn and Nirenberg [25] and later used by Hörmander [23] for problems in partial differential equations. Pseudo-differential operators on different classes of groups are broadly studied by several authors with their applications [5,6,7,8,11,12,20,26,27,28,29,37,39,40,41,42,43].…”

Section: Introductionmentioning

confidence: 99%

Nuclearity of operators related to finite measure spaces

J. Pseudo-Differ. Oper. Appl.

Mondal

2020

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.