The Markov-Stieltjes transform as an operator

TL;DR

This paper investigates the properties of the Markov-Stieltjes transform, establishing its boundedness, non-compactness, and operator characteristics on Hardy and Lebesgue spaces, along with norm estimates and inverse formulas.

Contribution

It provides a comprehensive analysis of the Markov-Stieltjes transform as an operator, including boundedness, norm estimates, and operational properties on various function spaces.

Findings

The transform is a bounded non-compact Hankel operator on Hardy spaces.

It is bounded on Lebesgue spaces $L^p[0,1]$ for $p eq 1, ext{infinity}$.

On $L^2(0,1)$, it is unitarily equivalent to the transform on $H^2$.

Abstract

We prove that the Markov-Stieltjes transform is a bounded non compact Hankel operator on Hardy space $H^p$ with Hilbert matrix with respect to the standard Schauder basis of $H^p$ and a bounded non compact operator on Lebesgue space $L^p[0,1]$ for $p\in(1,\infty)$ and obtain estimates for its norm in this spaces. It is shown that the Markov-Stieltjes transform on $L^2(0,1)$ is unitary equivalent to the Markov-Stieltjes transform on $H^2$. Inverse formulas and operational properties for this transform are obtained.