Standard Models of Abstract Intersection Theory for Operators in Hilbert Space

Grzegorz Banaszak, and Yoichi Uetake. "Standard Models of Abstract Intersection Theory for Operators in Hilbert Space." Bulletin of the Polish Academy of Sciences. Mathematics 63.2 (2015): 149-175. <http://eudml.org/doc/281259>.

@article{GrzegorzBanaszak2015,
abstract = { 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. },
author = {Grzegorz Banaszak, Yoichi Uetake},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {abstract intersection theory; Riemann hypothesis; complex field; Dirichlet -function; Dirichlet character; -adic cohomology; finite field; Gelfand-Naimark-Segal representation},
language = {eng},
number = {2},
pages = {149-175},
title = {Standard Models of Abstract Intersection Theory for Operators in Hilbert Space},
url = {http://eudml.org/doc/281259},
volume = {63},
year = {2015},
}

TY - JOUR
AU - Grzegorz Banaszak
AU - Yoichi Uetake
TI - Standard Models of Abstract Intersection Theory for Operators in Hilbert Space
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2015
VL - 63
IS - 2
SP - 149
EP - 175
AB - 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.
LA - eng
KW - abstract intersection theory; Riemann hypothesis; complex field; Dirichlet -function; Dirichlet character; -adic cohomology; finite field; Gelfand-Naimark-Segal representation
UR - http://eudml.org/doc/281259
ER -