Linear Operators on Polynomials and $K$-Positivity Preserver -- A journey from analysis to algebra and back

TL;DR

This paper explores the theory of linear operators on polynomials and their role in preserving $K$-positivity, bridging concepts from analysis and algebra through extended lecture notes.

Contribution

It provides a comprehensive overview of $K$-positivity preservers and their generators, connecting analytical and algebraic perspectives.

Findings

Characterization of $K$-positivity preserving operators

Connections between analysis and algebra in polynomial operators

Extended lecture notes as a resource for further study

Abstract

These are the extended lecture notes of my lecture about ``Linear Operators on Polynomials, $K$-Positivity Preserver, and their Generators''. The lecture was given at the University of Konstanz in the winter semester 2025/26.