The A-model with mutually equal model parameters can lead to a Hilbert space model

TL;DR

This paper shows that the A-model with equal parameters can still form a Hilbert space model when extensions are described by linear relations, expanding the understanding of model structures in singular perturbations.

Contribution

It introduces a method to obtain Hilbert space models from the A-model with equal parameters using linear relations, challenging previous assumptions.

Findings

Equal parameter A-models can be Hilbert space models with linear relations

Extensions in the model space are key to the model's structure

Expands the class of models that can be represented as Hilbert spaces

Abstract

It is known that the A-model for higher order singular perturbations can be considered as a Hilbert space model if the model parameters are mutually distinct, and that it is necessarily a Pontryagin space model if otherwise. In this note we demonstrate that the A-model with mutually equal model parameters can nonetheless lead to a Hilbert space model if the extensions in the model space are instead described by suitable linear relations.