Representations of closed quadratic forms associated with Stieltjes and inverse Stieltjes holomorphic families of linear relations

TL;DR

This paper investigates holomorphic families of linear relations in the Stieltjes and inverse Stieltjes classes, revealing their sectorial properties and connections to holomorphic forms, using linear fractional transforms and contractive operators.

Contribution

It establishes the sectorial nature of these families and characterizes their associated forms as holomorphic families of type (B) per Kato's classification.

Findings

Values are maximal sectorial up to rotation in their domain

Associated forms are holomorphic families of type (B)

Uses linear fractional transforms and contractive operators for proofs

Abstract

In this paper holomorphic families of linear relations which belong to the Stieltjes or inverse Stieltjes class are studied. It is shown that in their domain of holomorphy ${\mathbb C}\setminus{\mathbb R}_+$ the values of Stieltjes and inverse Stieltjes families are, up to a rotation, maximal sectorial. This leads to a study of the associated closed sesquilinear forms and their representations. In particular, it is shown that the closed forms associated with the Stieltjes and inverse Stieltjes families of linear relations are holomorphic families of type (B) in the sense of Kato. These results are proved by using linear fractional transforms which connect these families to holomorphic functions that belong to a combined Nevanlinna-Schur class and a key tool then relies on a specific structure of contractive operators.