Annals of Functional Analysis

Actions of arithmetic functions on matrices and corresponding representations

Abstract

In this paper, we study a class of representations of arithmetic functions, and corresponding operator-theoretic and free probabilistic properties. We associate given arithmetic functions $f$ to certain matrices $\alpha _{n}(f).$

Article information

Source
Ann. Funct. Anal., Volume 5, Number 2 (2014), 90-117.

Dates
First available in Project Euclid: 7 April 2014

https://projecteuclid.org/euclid.afa/1396833506

Digital Object Identifier
doi:10.15352/afa/1396833506

Mathematical Reviews number (MathSciNet)
MR3192013

Zentralblatt MATH identifier
1309.46037

Citation

Cho, Ilwoo; Jorgensen, Palle. Actions of arithmetic functions on matrices and corresponding representations. Ann. Funct. Anal. 5 (2014), no. 2, 90--117. doi:10.15352/afa/1396833506. https://projecteuclid.org/euclid.afa/1396833506

References

• D. Alpay, A. Dijksma, J. van der Ploeg, and H. S. V. de Snoo,Holomorphic Operators Between Krein Spaces and the Number of Squares of Associated Kernel, Oper. Theory Adv. Appl. 59 (1992), 11–29.
• D. Alpay, and I. Lewkowicz,An easy-to-compute factorization of rational generalized positive functions, Systems Control Lett. 59 (2010), no. 9, 517–521.
• D. Alpay,Some remarks on reproducing kernel Krein spaces, Rocky Mountain J. Math.21 (1991), no. 4, 1189–1205.
• D. Alpay, Some Krein Spaces of Analytic Functions and an Inverse Scattering Problem, Michigan Math. J. 34 (1987), no. 3, 349–359.
• T. Ando,Linear Operators on Krein Spaces, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo, 1979.
• T. M. Apostol,Modular Functions and Dirichilet Series in Number Theory, Second edition. Graduate Texts in Mathematics, 41. Springer-Verlag, New York, 1990.
• A.S. Besicovitch, Almost Periodic Functions, MR Number: 0068029 (16,817a), (1955) Dover Publisher.
• A. S. Besicovitch, On Parseval's Theorem for Dirichlet Series, Proc. London Math. Soc. S2-26 (1927), no. 1, 25–34.
• H. Bohr, Collected Mathematical Works, vol I, Dirichlet Series, The Riemann Zeta Function, vol II, Almost Periodic Functions, vol III, Almost Periodic Functions (Continued), Linear Congruences, Diophantine Approximations, Function Theory, Addition of Convex Curves, Other papers, MR Number: 0057790 (15,276i) (1952) Published by Dansk Matematisk Forening Kobenhavn.
• J. B. Bost, and A. Connes, Hecke Algebras, Type $III$-Factors and Phase Transitions with Spontaneous Symmetry Breaking in Number Theory, Selecta Math. (N.S.) 1 (1995), no. 3, 411–457.
• D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge, 1997.
• I. Cho, Classification on Arithmetic Functions and Corresponding Free-Moment $L$-Functions, Bulletin Korean Math. Soc. (2013), (to appear).
• I. Cho, Operators Induced by Prime Numbers, Methods Appl. Anal. 19 (2012), no. 4, 313–339.
• I. Cho, $p$-adic Banach-space operators and adelic Banach-space operators, Opuscula Math. (2013), (to appear).
• I. Cho, Free distributional data of arithmetic functions and corresponding generating functions determined by gaps between primes, Complex Anal. Oper. Theory 8 (2014), no. 2, 537–570.
• I. Cho and T. GillespieArithmetic functions and corresponding free probability determined by primes, (2013), Submitted to Rocky Mountain J. Math.
• I. Cho and T. Gillespie,Real numbers acting on arithmetic functions, (2012) Submitted to INTEGERS: Elec. J. Combinat. Numb. Theo.
• I. Cho and P.E.T. Jorgensen, Krein-space operators induced by Dirichlet characters, Special Issue: Contemp. Math. (2013) Amer. Math. Soc.
• I. Cho and P.E.T. Jorgensen,Arithmetic functions in harmonic analysis and operator theory, (2013) Submitted to Springer Book Project: Handbook of Operator Theory.
• G. Christner, Application of the extension properties of operators on Krein spaces, Univ. of Virginia, (1993) PhD Thesis.
• J.L. Dalecki and M.G. Krein, Stability of solutions of differential equations in Banach space, (Translated from the Russian by S. Smith), Translations of Math. Monographs, Vol. 43. American Mathematical Society, Providence, R.I., 1974.
• H. Davenport, Multiplicative number theory, 3-rd Ed., Grad. Texts in Math., 74. Springer-Verlag, New York, 2000.
• M.A. Dritchel and J. Rovnyak, Operators on indefinite inner product spaces, Lecture Note, Univ. of Virginia, Dept. of Math., (1996).
• A.J. Hildebrand, Introduction to analytic number theory, Lecture Notes, availble at http://www.math.uiuc.edu/\symbol126 hilderbr/ant, (2006).
• T. Gillespie, superposition of zeroes of automorphic $L$-functions and functoriality, Univ. of Iowa, (2010) PhD Thesis.
• T. Gillespie, Prime number theorems for Rankin-Selberg $L$-functions over number fields, Sci. China Math. 54 (2011), no. 1, 35–46.
• M.G. Krein and H. Langer,Uber einige Fortsetzungs probleme, die eng mit der Theorie hermitescher Operatoren im Raume $\Pi \sb{\kappa }$ zusammenhängen, I. Einige Funktionenklassen und ihre Darstellungen. Math. Nachr. 77 (1977), 187–236.
• J.C. Lagarias, Euler's Constant: Euler's work and modern developments, Bulletin Amer. Math. Soc., 50 (2013), no. 4, 527–628.
• P.D. Lax and R.S. Phillips, Scattering theory for transport Phenomena, Funct. Anal. Proc. Conf. Irvine. Calif. (1966), 119 - 130.
• M.B. Nathanson, Additive number theory, Grad. Text in Math., 164, ISBN: 0-387-94656-X, Springer-Verlag, 1996.
• D.J. Newman, Analytic number theory, Grad. Text in Math., 177, ISBN: 0-387-98308-2, Springer-Verlag, 1998.
• R.S. Phillips, The extension of dual subspaces invariant under an algebra, Proc. Internat. Sympos. Linear Spaces, Jerusalem Acad. Press, (1960), 366–398.
• F. Radulescu, Random matrices, amalgamated free products and subfactors of the $C^{*}$-algebra of a free group of nonsingular index, Invent. Math., 115 (1994), 347–389.
• R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627.
• V.S. Vladimirov, I.V. Volovich and E.I. Zelenov,$p$-adic analysis and mathematical physics, Ser. Soviet & East European Math., vol 1, ISBN: 978-981-02-0880-6, (1994) World Scientific.
• D. Voiculescu, K. Dykemma, and A. Nica, Free random variables, CRM Monograph Series, vol 1, (1992).