On a coefficient in trace formulas for Wiener-Hopf operators

TL;DR

This paper investigates the coefficient in Widom's trace formula for Wiener-Hopf operators when the function g is non-smooth, focusing on specific functions like |t|^γ, and analyzes its properties.

Contribution

The paper extends Widom's trace formula to non-smooth functions g, providing explicit analysis of the coefficient for such cases with real-valued symbols a.

Findings

Analysis of the coefficient (a; g) for non-smooth g

Explicit results for g(t) = |t|^ with   (0,1]

Insights into the behavior of trace formulas for Wiener-Hopf operators

Abstract

Let $a = a(\xi), \xi\in\mathbb R,$ be a smooth function quickly decreasing at infinity. For the Wiener-Hopf operator $W(a)$ with the symbol $a$, and a smooth function $g:\mathbb C\to~\mathbb C$, H. Widom in 1982 established the following trace formula: \[ {\rm tr}\bigl(g\bigl(W(a)\bigr) - W(g\circ a)\bigr) = \mathcal B(a; g), \] where $\mathcal B(a; g)$ is given explicitly in terms of the functions $a$ and $g$. The paper analyses the coefficient $\mathcal B(a; g)$ for a class of non-smooth functions $g$ assuming that $a$ is real-valued. A representative example of one such function is $g(t) = |t|^{\gamma}$ with some $\gamma\in (0, 1]$.