Develop the theory of the strengthened boundedness condition

Develop a theory for the second path arising from the dissipativity inequalities, namely the study of functions obtained by strengthening the boundedness condition I_m-F(z)F(z)^*\ge 0 in the open right half-plane, including its connections with passive linear systems and operator theory.

Background

The paper identifies three possible directions for refining or combining the positivity and boundedness conditions associated with passive systems. The first direction, strengthening the positivity condition, had been studied in earlier work, while the second direction concerns strengthening the boundedness condition. The authors explicitly state that they are not aware of studies addressing this second path and defer returning to it in future work. This constitutes an unresolved research direction concerning the theory and applications of the strengthened boundedness condition.

References

We are not aware of studies pertaining to the second path and will go back to this question in a forthcoming publication.

— Hyperpositive functions, sector bounded functions and a trace formula  (2609.03403 - Alpay et al., 3 Sep 2026) in Section 1, Introduction, subsection “Prologue”