Traces of certain integral operators related to the Riemann hypothesis

Alfredo Sotelo-Pejerrey; Alfredo Sotelo-Pejerrey

doi:10.3934/math.20231274

AIMS Mathematics

2023, Volume 8, Issue 10: 24971-24983. doi: 10.3934/math.20231274

Previous Article Next Article

Research article Special Issues

Traces of certain integral operators related to the Riemann hypothesis

Alfredo Sotelo-Pejerrey ^,

Faculty of Engineering, Universidad Continental, Lima, Peru

Received: 28 May 2023 Revised: 14 August 2023 Accepted: 17 August 2023 Published: 28 August 2023
MSC : 35J62, 35A15, 35J20

We prove the existence of a nontrivial singular trace $ \tau $ defined on an ideal $ \mathcal{J} $ closed with respect to the logarithmic submajorization such that $ \tau(A_\rho(\alpha)) = 0 $, where $ A_\rho(\alpha):L^{2}(0, 1)\to L^{2}(0, 1) $, $ {[A_\rho(\alpha)f](\theta) = \int^{1}_{0}\rho(\alpha\theta/x)f(x)dx} $, $ 0 < \alpha\leq 1 $. We also show that $ \tau(A_\rho(\alpha)) = 0 $ for every $ \tau $ nontrivial singular trace on $ \mathcal{J} $. Finally, we give a recursion formula from which we can evaluate all the traces $ {\mbox{Tr}}\, (A^{r}_{\rho}(\alpha)) $, $ r\in \mathbb{N} $, $ r\geq 2 $.
- singular trace,
- spectral trace,
- modified Fredholm determinant,
- Riemann hypothesis
Citation: Alfredo Sotelo-Pejerrey. Traces of certain integral operators related to the Riemann hypothesis[J]. AIMS Mathematics, 2023, 8(10): 24971-24983. doi: 10.3934/math.20231274

Related Papers:

Abstract

We prove the existence of a nontrivial singular trace $ \tau $ defined on an ideal $ \mathcal{J} $ closed with respect to the logarithmic submajorization such that $ \tau(A_\rho(\alpha)) = 0 $, where $ A_\rho(\alpha):L^{2}(0, 1)\to L^{2}(0, 1) $, $ {[A_\rho(\alpha)f](\theta) = \int^{1}_{0}\rho(\alpha\theta/x)f(x)dx} $, $ 0 < \alpha\leq 1 $. We also show that $ \tau(A_\rho(\alpha)) = 0 $ for every $ \tau $ nontrivial singular trace on $ \mathcal{J} $. Finally, we give a recursion formula from which we can evaluate all the traces $ {\mbox{Tr}}\, (A^{r}_{\rho}(\alpha)) $, $ r\in \mathbb{N} $, $ r\geq 2 $.

References

[1]	A. Albeverio, D. Guido, A. Ponosov, S. Scarlatti, Singular traces and compact operators, J. Funct. Anal., 137 (1996), 281–302. https://doi.org/10.1006/jfan.1996.0047 doi: 10.1006/jfan.1996.0047
[2]	J. Alcántara-Bode, An integral equation formulation of the Riemann hypothesis, Integr. Equat. Oper. Th., 17 (1993), 151–168. https://doi.org/10.1007/bf01200216. doi: 10.1007/bf01200216
[3]	J. Alcántara-Bode, An algorithm for the evaluation of certain Fredholm determinants, Integr. Equat. Oper. Th., 39 (2001), 153–158. https://doi.org/10.1007/bf01195814 doi: 10.1007/bf01195814
[4]	J. Alcántara-Bode, A completeness problem related to the Riemann hypotesis, Integr. Equat. Oper. Th., 53 (2005), 301–309. https://doi.org/10.1007/s00020-004-1315-7 doi: 10.1007/s00020-004-1315-7
[5]	J. Alcántara-Bode, An example of two non-unitarily equivalent compact operators with the same traces and kernel, Pro. Math., 23 (2009), 105–111.
[6]	N. Azamov, F. Sukochev, A Lidskii type formula for Dixmier traces, C. R. Math. Acad. Sci. Paris, 340 (2005), 107–112. https://doi.org/10.1016/j.crma.2004.12.005 doi: 10.1016/j.crma.2004.12.005
[7]	A. Beurling, A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. Sci., 41 (1955), 312–314. https://doi.org/10.1073/pnas.41.5.312 doi: 10.1073/pnas.41.5.312
[8]	P. Borwein, T. Erdelyi, The full Muntz theorem in $C[0, 1]$ and $L_1[0, 1]$, J. London Math. Soc., 54 (1996), 102–110. https://doi.org/10.1112/jlms/54.1.102 doi: 10.1112/jlms/54.1.102
[9]	J. Calkin, Two-ideals and congruences in the ring of bounded operators in Hilbert space, Ann. Math., 42 (1941), 839–873. https://doi.org/10.2307/1968771 doi: 10.2307/1968771
[10]	A. Connes, Geometrie non commutative (Inter-Editions), Paris, 1990.
[11]	J. Dixmier, Existences de traces non normales, C. R. Acad. Sci. Paris, 262 (1966). doi: 10.1515/crll.1998.103
[12]	K. J. Dykema, N. J. Kalton, Spectral characterization of sums of commutators Ⅱ, J. Reine Angew. Math., 504 (1998), 127–137. https://doi.org/10.1515/crll.1998.103 doi: 10.1515/crll.1998.103
[13]	I. Gohberg, S. Goldberg, N. Krupnik, Traces and determinants of linear operators, Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, 116 (2000).
[14]	I. Gohberg, S. Goldberg, M. Kaashoek, Basic classes of linear operators, Birkhauser, 2003.
[15]	D. Guido, T. Isola, On the domain of singular traces, J. Funct. Anal., 13 (2002), 667–674. https://doi.org/10.1142/s0129167x02001447 doi: 10.1142/s0129167x02001447
[16]	S. I. Kabanikhin, Inverse and Ⅲ-posed problem: Theory and applications, De Gruyter, 2012. https://doi.org/10.1515/9783110224016
[17]	N. J. Kalton, Unusual traces on operators ideals, Math. Nachr., 134 (1987), 119–130. https://doi.org/10.1002/mana.19871340108 doi: 10.1002/mana.19871340108
[18]	N. J. Kalton, Spectral characterization of sums of commutators I, J. Reine Angew. Math., 504 (1998), 115–125. https://doi.org/10.1515/crll.1998.102 doi: 10.1515/crll.1998.102
[19]	V. Lidskii, Conditions for completeness of a system of root subspaces for non-selfadjoint operators with discrete spectrum, Tr. Mosk. Mat. Obs., 8 (1959), 83–120. https://doi.org/10.1090/trans2/034/08 doi: 10.1090/trans2/034/08
[20]	S. Lord, F. Sukochev, D. Zanin, Singular traces, theory and applications, De Gruyter, 2012.
[21]	A. Pietsch, Traces and shift invariant functionals, Math. Nachr., 145 (1990), 7–43. https://doi.org/10.1002/mana.19901450102 doi: 10.1002/mana.19901450102
[22]	A. Sedaev, F. Sukochev, D. Zanin, Lidskii-type formulae for Dixmier traces, Integr. Equat. Oper. Th., 68 (2010), 551–572. https://doi.org/10.1007/s00020-010-1828-1 doi: 10.1007/s00020-010-1828-1
[23]	A. Sotelo-Pejerrey, Singular traces of an integral operator related to the Riemann hypothesis, Pro. Math., 32 (2022), 55–71. Available from: https://revistas.pucp.edu.pe/index.php/promatematica/article/wiew/25729.
[24]	F. Sukochev, D. Zanin, Which traces are spectral? Adv. Math., 252 (2014), 406–428. https://doi.org/10.1016/j.aim.2013.10.028 doi: 10.1016/j.aim.2013.10.028
[25]	J. V. Varga, Traces on irregular ideals, Proc. Amer. Math. Soc., 107 (1989), 715–723. https://doi.org/10.1090/s0002-9939-1989-0984818-8 doi: 10.1090/s0002-9939-1989-0984818-8
[26]	J. Von Neumann, Mathematische grundlagen der quantenmechanik, Grundlehren Math. Wiss. Einzeldarstellungen, Bd. XXXVIII, Springer-Verlag, Berlin, 1932.
[27]	A. C. Zaanen, Linear analysis, North Holland, Amsterdam, 1984.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)