Fourier multipliers in Hilbert spaces

This is a survey on a notion of invariant operators, or Fourier multipliers on Hilbert spaces. This concept is defined with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. In particular this notion can be applied to the important case of $L^2(M)$ where $M$ is a compact manifold $M$ endowed with a positive measure. The partition in this case comes from the spectral properties of a a fixed elliptic operator $E$.Comment: These notes are based on our paper arXiv:1404.6479 (to appear in J. Anal. Math.) and have been prepared for the instructional volume associated to the Summer School on Fourier Integral Operators held in Ouagadougou, Burkina Faso, where the authors took part during 14-26 September 201

Kernel and symbol criteria for Schatten classes and rr-nuclearity on compact manifolds

In this note we present criteria on both symbols and integral kernels ensuring that the corresponding operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given. A notion of an invariant operator and its full symbol associated to an elliptic operator are introduced. Some applications to the study of $r$-nuclearity on $L^{p}$ spaces are also obtained.Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1404.647

Lp-Nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups

Given a compact Lie group $G$, in this paper we give symbolic criteria for operators to be nuclear and r-nuclear on Lp-spaces, with applications to distribution of eigenvalues and trace formulae. Since criteria in terms of kernels are often not effective in view of Carleman's example, in this paper we adopt the symbolic point of view. The criteria here are given in terms of the concept of matrix symbols defined on the non-commutative analogue of the phase space $G\times\hat{G}$, where $\hat{G}$ is the unitary dual of $G$.Comment: This is the second half of our paper arXiv:1303.3914 so there is overlap in the preliminaries sectio

Schatten classes on compact manifolds: Kernel conditions

In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given.Comment: 22 page

The bounded approximation property of variable Lebesgue spaces and nuclearity

In this paper we prove the bounded approximation property for variable exponent Lebesgue spaces, study the concept of nuclearity on such spaces and apply it to trace formulae such as the Grothendieck-Lidskii formula. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ in terms of global symbols.Comment: 18 pages. The paper has been updated by adding a remark (on page 6) on a link between bounded and metric approximation properties. This should be the final version to appear in Math Scan

A model of host response to a multi-stage pathogen

We model the immune surveillance of a pathogen which passes through $n$ immunologically distinct stages. The biological parameters of this system induce a partial order on the stages, and this, in turn, determines which stages will be subject to immune regulation. This corresponds to the system's unique asymptotically stable fixed point.Comment: 22 pages, no figure

Voluntary Pollution Abatement and Regulation

Altruism, Voluntary Pollution Abatement, Regulation, Environmental Economics and Policy, Q52, Q58, K32,