On a generalisation of Krein's example

TL;DR

This paper generalizes Krein's classical example by explicitly computing resolvent and spectral projection differences, describing spectral invariants, and linking spectral projection differences to Hankel operators.

Contribution

It extends Krein's example with explicit calculations and a comprehensive analysis of spectral invariants and their connection to Hankel operators.

Findings

Explicit formulas for resolvent and spectral projection differences

Complete description of spectrum and multiplicity

Identified link between spectral projections and Hankel operators

Abstract

We generalise a classical example given by Krein in 1953. We compute the difference of the resolvents and the difference of the spectral projections explicitly. We further give a full description of the unitary invariants, i.e., of the spectrum and the multiplicity. Moreover, we observe a link between the difference of the spectral projections and Hankel operators.