Standard Models of Abstract Intersection Theory for Operators in Hilbert Space

Grzegorz Banaszak¹ ; Yoichi Uetake¹

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) no. 2, pp. 149-175.

Résumé

For an operator in a possibly infinite-dimensional Hilbert space of a certain class, we set down axioms of an abstract intersection theory, from which the Riemann hypothesis regarding the spectrum of that operator follows. In our previous paper (2011) we constructed a GNS (Gelfand–Naimark–Segal) model of abstract intersection theory. In this paper we propose another model, which we call a standard model of abstract intersection theory. We show that there is a standard model of abstract intersection theory for a given operator if and only if the Riemann hypothesis and semisimplicity hold for that operator. (For the definition of semisimplicity of an operator in Hilbert space, see the Introduction.) We show this result under a condition for a given operator which is much weaker than the condition in the previous paper. An operator satisfying this condition can be constructed by using the method of automorphic scattering of Uetake (2009). Combining this with a result from Uetake (2009), we can show that a Dirichlet $L$-function, including the Riemann zeta-function, satisfies the Riemann hypothesis and its all nontrivial zeros are simple if and only if there is a corresponding standard model of abstract intersection theory. Similar results can be proven for GNS models since the same technique of proof for standard models can be applied.

Détail

DOI : 10.4064/ba63-2-5

Keywords: operator possibly infinite dimensional hilbert space certain class set down axioms abstract intersection theory which riemann hypothesis regarding spectrum operator follows previous paper constructed gns gelfand naimark segal model abstract intersection theory paper propose another model which call standard model abstract intersection theory there standard model abstract intersection theory given operator only riemann hypothesis semisimplicity operator definition semisimplicity operator hilbert space see introduction result under condition given operator which much weaker condition previous paper operator satisfying condition constructed using method automorphic scattering uetake combining result uetake dirichlet l function including riemann zeta function satisfies riemann hypothesis its nontrivial zeros simple only there corresponding standard model abstract intersection theory similar results proven gns models since technique proof standard models applied