The degree of ill-posedness for some composition governed by the Cesaro operator

TL;DR

This paper investigates the singular value decay of a composition of the integration and Cesàro operators in Hilbert spaces, revealing that the Cesàro operator increases the ill-posedness degree by one.

Contribution

It establishes the precise degree of ill-posedness for the composition of the integration and Cesàro operators, showing how the Cesàro operator affects ill-posedness.

Findings

The composition's ill-posedness degree is two.

The Cesàro operator increases ill-posedness by one.

Singular value asymptotics are characterized for the composition.

Abstract

In this article, we consider the singular value asymptotics of compositions of compact linear operators mapping in the real Hilbert space of quadratically integrable functions over the unit interval. Specifically, the composition is given by the compact simple integration operator followed by the non-compact Ces`aro operator possessing a non-closed range. We show that the degree of ill-posedness of that composition is two, which means that the Ces`aro operator increases the degree of illposedness by the amount of one compared to the simple integration operator.