A hypercyclic finite rank perturbation of a unitary operator

A unitary operator $V$ and a rank $2$ operator $R$ acting on a Hilbert space \H are constructed such that $V+R$ is hypercyclic. This answers affirmatively a question of Salas whether a finite rank perturbation of a hyponormal operator can be supercyclic.Comment: published in Mathematische Annale

Recurrence properties of hypercyclic operators

[EN] We generalize the notions of hypercyclic operators, U-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics, namely A-hypercyclicity. We then state an A-hypercyclicity criterion, inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion, and we show that this criterion characterizes the A-hypercyclicity for weighted shifts. We also investigate which density properties can the sets N(x, U) = {n is an element of N; T-n x is an element of U} have for a given hypercyclic operator, and we study the new notion of reiteratively hypercyclic operators.This work is supported in part by MEC and FEDER, Project MTM2013-47093-P, and by GVA, Projects PROMETEOII/2013/013 and ACOMP/2015/005. The second author was a postdoctoral researcher of the Belgian FNRS.Bès, JP.; Menet, Q.; Peris Manguillot, A.; Puig-De Dios, Y. (2016). Recurrence properties of hypercyclic operators. Mathematische Annalen. 366(1):545-572. https://doi.org/10.1007/s00208-015-1336-3S5455723661Badea, C., Grivaux, S.: Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211, 766–793 (2007)Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358, 5083–5117 (2006)Bayart, F., Grivaux, S.: Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94, 181–210 (2007)Bayart, F., Matheron, É.: Dynamics of linear operators, Cambridge Tracts in Mathematics, 179. Cambridge University Press, Cambridge (2009)Bayart, F., Matheron, É.: (Non-)weakly mixing operators and hypercyclicity sets. Ann. Inst. Fourier 59, 1–35 (2009)Bayart, F., Ruzsa, I.: Difference sets and frequently hypercyclic weighted shifts. Ergodic Theory Dynam. Syst. 35, 691–709 (2015)Bergelson, V.: Ergodic Ramsey Theory- an update, Ergodic Theory of $$\mathbb{Z}^d$$ Z d -actions. Lond. Math. Soc. Lecture Note Ser. 28, 1–61 (1996)Bernal-González, L., Grosse-Erdmann, K.-G.: The Hypercyclicity Criterion for sequences of operators. Studia Math. 157, 17–32 (2003)Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors. Ergodic Theory Dynam. Syst. 27, 383–404 (2007)Bonilla, A., Grosse-Erdmann, K.-G.: Erratum: Ergodic Theory Dynam. Systems 29, 1993–1994 (2009)Chan, K., Seceleanu, I.: Hypercyclicity of shifts as a zero-one law of orbital limit points. J. Oper. Theory 67, 257–277 (2012)Costakis, G., Sambarino, M.: Topologically mixing hypercyclic operators. Proc. Amer. Math. Soc. 132, 385–389 (2004)Furstenberg, H.: Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, Princeton (1981)Giuliano, R., Grekos, G., Mišík, L.: Open problems on densities II, Diophantine Analysis and Related Fields 2010. AIP Conf. Proc. 1264, 114–128 (2010)Grosse-Erdmann, K.-G.: Hypercyclic and chaotic weighted shifts. Studia Math. 139, 47–68 (2000)Grosse-Erdmann, K.-G., Peris, A.: Frequently dense orbits. C. R. Math. Acad. Sci. Paris 341, 123–128 (2005)Grosse-Erdmann, K.G., Peris, A.: Weakly mixing operators on topological vector spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 104, 413–426 (2010)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear chaos, Universitext. Springer, London (2011)Menet, Q.: Linear chaos and frequent hypercyclicity. Trans. Amer. Math. Soc. arXiv:1410.7173Puig, Y.: Linear dynamics and recurrence properties defined via essential idempotents of $$\beta {\mathbb{N}}$$ β N (2014) arXiv:1411.7729 (preprint)Salas, H.N.: Hypercyclic weighted shifts. Trans. Amer. Math. Soc. 347, 993–1004 (1995)Salat, T., Toma, V.: A classical Olivier’s theorem and statistical convergence. Ann. Math. Blaise Pascal 10, 305–313 (2003)Shkarin, S.: On the spectrum of frequently hypercyclic operators. Proc. Am. Math. Soc. 137, 123–134 (2009

Almost sure-sign convergence of Hardy-type Dirichlet series

[EN] Hartman proved in 1939 that the width of the largest possible strip in the complex plane on which a Dirichlet series is uniformly a.s.- sign convergent (i.e., converges uniformly for almost all sequences of signs epsilon (n) = +/- 1) but does not convergent absolutely, equals 1/2. We study this result from a more modern point of view within the framework of so-called Hardytype Dirichlet series with values in a Banach space.Supported by CONICET-PIP 11220130100329CO, PICT 2015-2299 and UBACyT 20020130100474BA. Supported by MICINN MTM2017-83262-C2-1-P. Supported by MICINN MTM2017-83262-C2-1-P and UPV-SP20120700.Carando, D.; Defant, A.; Sevilla Peris, P. (2018). Almost sure-sign convergence of Hardy-type Dirichlet series. Journal d Analyse Mathématique. 135(1):225-247. https://doi.org/10.1007/s11854-018-0034-yS2252471351A. Aleman, J.-F. Olsen, and E. Saksman, Fourier multipliers for Hardy spaces of Dirichlet series, Int. Math. Res. Not. IMRN 16 (2014), 4368–4378.R. Balasubramanian, B. Calado, and H. Queffélec, The Bohr inequality for ordinary Dirichlet series Studia Math. 175 (2006), 285–304.F. Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136 (2002), 203–236.F. Bayart, A. Defant, L. Frerick, M. Maestre, and P. Sevilla-Peris, Monomial series expansion of Hp-functions and multipliers ofHp-Dirichlet series, Math. Ann. 368 (2017), 837–876.F. Bayart, D. Pellegrino, and J. B. Seoane-Sepúlveda, The Bohr radius of the n-dimensional polydisk is equivalent to $$\sqrt {\left( {\log n} \right)/n}$$ ( log n ) / n , Adv. Math. 264 (2014), 726–746.F. Bayart, H. Queffélec, and K. Seip, Approximation numbers of composition operators on Hp spaces of Dirichlet series, Ann. Inst. Fourier (Grenoble) 66 (2016), 551–588.H. F. Bohnenblust and E. Hille. On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32 (1931), 600–622.H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variablen in der Theorie der Dirichlet–schen Reihen $$\sum {\frac{{{a_n}}}{{{n^s}}}}$$ ∑ a n n s , Nachr. Ges.Wiss. Göttingen, Math. Phys. Kl., 1913, pp. 441–488.D. Carando, A. Defant, and P. Sevilla-Peris, Bohr’s absolute convergence problem for Hp- Dirichlet series in Banach spaces, Anal. PDE 7 (2014), 513–527.D. Carando, A. Defant, and P. Sevilla-Peris, Some polynomial versions of cotype and applications, J. Funct. Anal. 270 (2016), 68–87.B. J. Cole and T. W. Gamelin, Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc. (3) 53 (1986), 112–142.R. de la Bretèche. Sur l’ordre de grandeur des polynômes de Dirichlet, Acta Arith. 134 (2008), 141–148.A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounäies, and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174 (2011), 485–497.A. Defant, D. García, M. Maestre, and D. Pérez-García, Bohr’s strip for vector valued Dirichlet series, Math. Ann. 342 (2008), 533–555.A. Defant, M. Maestre, and U. Schwarting, Bohr radii of vector valued holomorphic functions, Adv. Math. 231 (2012), 2837–2857.A. Defant and A. Pérez, Hardy spaces of vector-valued Dirichlet series, StudiaMath. (to appear), 2018 DOI: 10.4064/sm170303-26-7.A. Defant, U. Schwarting, and P. Sevilla-Peris, Estimates for vector valued Dirichlet polynomials, Monatsh. Math. 175 (2014), 89–116.J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge University Press, Cambridge, 1995.P. Hartman, On Dirichlet series involving random coefficients, Amer. J. Math. 61 (1939), 955–964.H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L2(0, 1), Duke Math. J. 86 (1997), 1–37.A. Hildebrand, and G. Tenenbaum, Integers without large prime factors, J. Thor. Nombres Bordeaux 5 (1993), 411–484.S. V. Konyagin and H. Queffélec, The translation 1/2 in the theory of Dirichlet series, Real Anal. Exchange 27 (2001/02) 155–175.J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Springer-Verlag, Berlin, 1979.B. Maurizi and H. Queffélec, Some remarks on the algebra of bounded Dirichlet series, J. Fourier Anal. Appl. 16 (2010), 676–692.H. Queffélec, H. Bohr’s vision of ordinary Dirichlet series; old and new results, J. Anal. 3 (1995), 43–60.H. Queffélec and M. Queffélec, Diophantine Approximation and Dirichlet Series, Hindustan Book Agency, New Delhi, 2013.G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge, 1995

Impact of pump phase modulation on system performance of fibre-optical parametric amplifiers

Bit error rate induced in a parametric amplifier has been measured with a 10 Gbit=s line rate. It is shown experimentally that the phase modulation of the pump distorts the mark level of the channel and may cause system penalty. Different phase modulation schemes have been compared

Impact of pump OSNR on noise figure for fiber-optical parametric amplifiers

Electrical measurements of the noise figure of a fiberoptical parametric amplifier are presented and compared with optical measurements. The transfer of pump noise by four-wave mixing was clearly demonstrated. A numerical model was developed to simulate the transfer of pump noise and validated by these measurements. Using this model, we determine, for practical systems, a minimum required pump optical signal-to-noise ratio of 65 dB

A note on abscissas of Dirichlet series

[EN] We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series.A. Defant: Partially supported by MINECO and FEDER MTM2017-83262-C2-1-P. A. Pérez: Supported by La Caixa Foundation, MINECO and FEDER MTM2014-57838-C2-1-P and Fundación Séneca - Región de Murcia (CARM 19368/PI/14). P. Sevilla-Peris: Supported by MINECO and FEDER MTM2017-83262-C2-1-P.Defant, A.; Pérez, A.; Sevilla Peris, P. (2019). A note on abscissas of Dirichlet series. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(3):2639-2653. https://doi.org/10.1007/s13398-019-00647-yS263926531133Bayart, F.: Hardy spaces of Dirichlet series and their composition operators. Mon. Math. 136(3), 203–236 (2002)Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann Math. 32(3), 600–622 (1931)Bohr, H.: Über die Bedeutung der Potenzreihen unendlich vieler Variablen in der Theorie der Dirichlet–schen Reihen $$\sum \,\frac{a_n}{n^s}$$ ∑ a n n s . Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl., pp. 441–488 (1913)Bonet, J.: Abscissas of weak convergence of vector valued Dirichlet series. J. Funct. Anal. 269(12), 3914–3927 (2015)Carando, D., Defant, A., Sevilla-Peris, P.: Bohr’s absolute convergence problem for $$\cal{H}_p$$ H p -Dirichlet series in Banach spaces. Anal. PDE 7(2), 513–527 (2014)Carando, D., Defant, A., Sevilla-Peris, P.: Some polynomial versions of cotype and applications. J. Funct. Anal. 270(1), 68–87 (2016)Defant, A., García, D., Maestre, M., Pérez-García, D.: Bohr’s strip for vector valued Dirichlet series. Math. Ann. 342(3), 533–555 (2008)Defant, A., García, D., Maestre, M., Sevilla–Peris, P.: Dirichlet Series and Holomorphic Funcions in High Dimensions, vol. 37 of New Mathematical Monographs. Cambridge University Press, Cambridge (2019)Defant, A., Pérez, A.: Optimal comparison of the $$p$$ p -norms of Dirichlet polynomials. Israel J. Math. 221(2), 837–852 (2017)Defant, A., Pérez, A.: Hardy spaces of vector-valued Dirichlet series. Studia Math. 243(1), 53–78 (2018)Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators, vol. 43 of Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge (1995)Maurizi, B., Queffélec, H.: Some remarks on the algebra of bounded Dirichlet series. J. Fourier Anal. Appl. 16(5), 676–692 (2010)Queffélec, H., Queffélec, M.: Diophantine approximation and Dirichlet series, vol. 2 of Harish–Chandra research institute lecture notes. Hindustan Book Agency, New Delhi (2013

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

[EN] Let H-infinity be the set of all ordinary Dirichlet series D = Sigma(n) a(n)(n-1) ann-s representing bounded holomorphic functions on the right half plane. A completely multiplicative sequence (b(n)) of complex numbers is said to be an l(1)-multiplier for H-infinity whenever Sigma(n vertical bar)a(n)b(n vertical bar) < infinity for every D is an element of H-infinity. We study the problem of describing such sequences (b(n)) in terms of the asymptotic decay of the subsequence (b(pj)), where p(j) denotes the j th prime number. Given a completely multiplicative sequence b = (b(n)) we prove (among other results): b is an l(1)-multiplier for H-infinity provided vertical bar b(pj)vertical bar < 1 for all j and (lim(n)) over bar 1/log(n) Sigma(n)(j=1) b(p j)*(2) < 1, and conversely, if b is an l(1)-multiplier for H-infinity, then vertical bar b(pj)vertical bar < 1 for all j and (lim(n)) over bar 1/log(n) Sigma(n)(j=1) b(p j)*(2) <= 1 (here b* stands for the decreasing rearrangement of b). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences z in the infinite dimensional polydisk D-infinity (the open unit ball of l(infinity)) for which every bounded and holomorphic function f on D-infinity has an absolutely convergent monomial series expansion Sigma(alpha) partial derivative alpha f (0)/alpha! z alpha. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus T-infinity.The second, fourth and fifth authors were supported by MINECO and FEDER Project MTM2014-57838-C2-2-P. The fourth author was also supported by PrometeoII/2013/013. The fifth author was also supported by project SP-UPV20120700.Bayart, F.; Defant, A.; Frerick, L.; Maestre, M.; Sevilla Peris, P. (2017). Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables. Mathematische Annalen. 368(1-2):837-876. https://doi.org/10.1007/s00208-016-1511-1S8378763681-2Aleman, A., Olsen, J.-F., Saksman, E.: Fatou and brother Riesz theorems in the infinite-dimensional polydisc. arXiv:1512.01509Balasubramanian, R., Calado, B., Queffélec, H.: The Bohr inequality for ordinary Dirichlet series. Studia Math. 175(3), 285–304 (2006)Bayart, F.: Hardy spaces of Dirichlet series and their composition operators. Monatsh. Math. 136(3), 203–236 (2002)Bayart, F., Pellegrino, D., Seoane-Sepúlveda, J.B.: The Bohr radius of the $$n$$ n -dimensional polydisk is equivalent to $$\sqrt{(\log n)/n}$$ ( log n ) / n . Adv. Math. 264:726–746 (2014)Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. 32(3), 600–622 (1931)Bohr, H.: Über die Bedeutung der Potenzreihen unendlich vieler Variablen in der Theorie der Dirichlet–schen Reihen $$\sum \,\frac{a_n}{n^s}$$ ∑ a n n s . Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl. 441–488 (1913)Bohr, H.: Über die gleichmäßige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math. 143, 203–211 (1913)Cole, B.J., Gamelin., T.W.: Representing measures and Hardy spaces for the infinite polydisk algebra. Proc. Lond. Math. Soc. 53(1), 112–142 (1986)Davie, A.M., Gamelin, T.W.: A theorem on polynomial-star approximation. Proc. Am. Math. Soc. 106(2), 351–356 (1989)de la Bretèche, R.: Sur l’ordre de grandeur des polynômes de Dirichlet. Acta Arith. 134(2), 141–148 (2008)Defant, A., Frerick, L., Ortega-Cerdà, J., Ounaïes, M., Seip, K.: The Bohnenblust–Hille inequality for homogeneous polynomials is hypercontractive. Ann. Math. 174(1), 485–497 (2011)Defant, A., García, D., Maestre, M.: New strips of convergence for Dirichlet series. Publ. Mat. 54(2), 369–388 (2010)Defant, A., García, D., Maestre, M., Pérez-García, D.: Bohr’s strip for vector valued Dirichlet series. Math. Ann. 342(3), 533–555 (2008)Defant, A., Maestre, M., Prengel, C.: Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables. J. Reine Angew. Math. 634, 13–49 (2009)Dineen, S.: Complex Analysis on Infinite-dimensional Spaces. Springer Monographs in Mathematics. Springer-Verlag London Ltd, London (1999)Floret, K.: Natural norms on symmetric tensor products of normed spaces. Note Mat. 17(153–188), 1997 (1999)Harris, L. A.: Bounds on the derivatives of holomorphic functions of vectors. In: Analyse fonctionnelle et applications (Comptes Rendus Colloq. Analyse, Inst. Mat., Univ. Federal Rio de Janeiro, Rio de Janeiro, 1972), pp. 145–163. Actualités Aci. Indust., No. 1367. Hermann, Paris (1975)Hedenmalm, H., Lindqvist, P., Seip, K.: A Hilbert space of Dirichlet series and systems of dilated functions in $$L^2(0,1)$$ L 2 ( 0 , 1 ) . Duke Math. J. 86(1), 1–37 (1997)Helson, H., Lowdenslager, D.: Prediction theory and Fourier series in several variables. Acta Math. 99, 165–202 (1958)Hibert, D.: Gesammelte Abhandlungen (Band 3). Verlag von Julius Springer, Berlin (1935)Hilbert, D.: Wesen und Ziele einer Analysis der unendlichvielen unabhängigen Variablen. Rend. del Circolo Mat. di Palermo 27, 59–74 (1909)Kahane, J.-P.: Some Random Series of Functions, Volume 5 of Cambridge Studies in Advanced Mathematics, second edn. Cambridge University Press, Cambridge (1985)Konyagin, S.V., Queffélec, H.: The translation $$\frac{1}{2}$$ 1 2 in the theory of Dirichlet series. Real Anal. Exchange 27(1):155–175 (2001/2002)Maurizi, B., Queffélec, H.: Some remarks on the algebra of bounded Dirichlet series. J. Fourier Anal. Appl. 16(5), 676–692 (2010)Queffélec, H.: H. Bohr’s vision of ordinary Dirichlet series; old and new results. J. Anal. 3, 43–60 (1995)Queffélec, H., Queffélec, M.: Diophantine Approximation and Dirichlet Series. HRI Lecture Notes Series, New Delhi (2013)Rudin, W.: Function Theory in Polydisks. W. A. Benjamin Inc, New York (1969)Toeplitz, O.: Über eine bei den Dirichletschen Reihen auftretende Aufgabe aus der Theorie der Potenzreihen von unendlichvielen Veränderlichen. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, pp. 417–432 (1913)Weissler, F.B.: Logarithmic Sobolev inequalities and hypercontractive estimates on the circle. J. Funct. Anal. 37(2), 218–234 (1980)Wojtaszczyk, P.: Banach Spaces for Analysts, Volume 25 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1991

Abstract basins of attraction

Abstract basins appear naturally in different areas of several complex variables. In this survey we want to describe three different topics in which they play an important role, leading to interesting open problems

Nigerian scam e-mails and the charms of capital

So-called '419' or 'advance-fee' e-mail frauds have proved remarkably successful. Global losses to these scams are believed to run to billions of dollars. Although it can be assumed that the promise of personal gain which these e-mails hold out is part of what motivates victims, there is more than greed at issue here. How is it that the seemingly incredible offers given in these unsolicited messages can find an audience willing to treat them as credible? The essay offers a speculative thesis in answer to this question. Firstly, it is argued, these scams are adept at exploiting common presuppositions in British and American culture regarding Africa and the relationships that are assumed to exist between their nations and those in the global south. Secondly, part of the appeal of these e-mails lies in the fact that they appear to reveal the processes by which wealth is created and distributed in the global economy. They thus speak to their readers’ attempts to map or conceptualise the otherwise inscrutable processes of that economy. In the conclusion the essay looks at the contradictions in the official state response to this phenomena