Positive Definite Operator Functions and Sesquilinear Forms

TL;DR

This paper reviews how positive definite operator functions induce sesquilinear forms, unifying existing results and recent advances to deepen understanding in dilation theory.

Contribution

It provides a unified framework for analyzing positive definite operator functions through sesquilinear forms, incorporating recent research developments.

Findings

Connections between operator functions and sesquilinear forms clarified

Recent results integrated into a cohesive theoretical framework

Enhanced understanding of dilation theory implications

Abstract

Due to the fundamental works of T. Ando, W. Szyma\'nski, F. H. Szafraniec, and many others it is well known that sesquilinear forms play an important role in dilation theory. The crucial fact is that every positive definite operator function induces a sesquilinear form in a natural way. The purpose of this survey-like paper is to apply some recent results of Z. Sebesty\'en, Zs. Tarcsay, and the author for such functions. While most of the results are not new, the paper's main contribution is the unified discussion from the viewpoint of sesquilinear forms.