Calkin images of Fourier convolution operators with slowly oscillating symbols

TL;DR

This paper characterizes the Calkin images of Fourier convolution operators with slowly oscillating symbols on Banach function spaces, showing their intersection with multiplication operators is trivial.

Contribution

It provides a precise description of the Calkin images for a class of Fourier convolution operators with slowly oscillating symbols, extending understanding of their algebraic structure.

Findings

Intersection of Calkin images is generated by scalar multiplication operators.

Calkin image of convolution operators with slowly oscillating symbols is explicitly characterized.

Results contribute to the theory of Fourier multipliers and operator algebras.

Abstract

Let $\Phi$ be a $C^*$-subalgebra of $L^\infty(\mathbb{R})$ and $SO_{X(\mathbb{R})}^\diamond$ be the Banach algebra of slowly oscillating Fourier multipliers on a Banach function space $X(\mathbb{R})$. We show that the intersection of the Calkin image of the algebra generated by the operators of multiplication $aI$ by functions $a\in\Phi$ and the Calkin image of the algebra generated by the Fourier convolution operators $W^0(b)$ with symbols in $SO_{X(\mathbb{R})}^\diamond$ coincides with the Calkin image of the algebra generated by the operators of multiplication by constants.