Standard Models of Abstract Intersection Theory for Operators in Hilbert Space

• Autorzy: Grzegorz Banaszak , Yoichi Uetake
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47A10: Spectrum, resolvent
• 11M26: Nonreal zeros of ζ ( s ) and L ( s , χ ) ; Riemann and other hypotheses

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: 63
• Numer: 2
• Strony: 149-175

Daty

wydano

2015

Twórcy

autor

Grzegorz Banaszak

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

autor

Yoichi Uetake

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

Identyfikatory

DOI

10.4064/ba63-2-5