Estimates for vector valued Dirichlet polynomials

[EN] We estimate the -norm of finite Dirichlet polynomials with coefficients in a Banach space. Our estimates quantify several recent results on Bohr's strips of uniform but non absolute convergence of Dirichlet series in Banach spaces.A. Defant and P. Sevilla-Peris were supported by MICINN Project MTM2011-22417.Defant, A.; Schwarting, U.; Sevilla Peris, P. (2014). Estimates for vector valued Dirichlet polynomials. Monatshefte f�r Mathematik. 175(1):89-116. https://doi.org/10.1007/s00605-013-0600-4S891161751Balasubramanian, 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)Bennett, G.: Inclusion mappings between $$l^{p}$$ l p spaces. J. Funct. Anal. 13, 20–27 (1973)Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. (2) 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., Heft 4, 441–488 (1913)Bohr, H.: Über die gleichmäßige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math. 143, 203–211 (1913)Carl, B.: Absolut- $$(p,\,1)$$ ( p , 1 ) -summierende identische Operatoren von $$l_{u}$$ l u in $$l_{v}$$ l v . Math. Nachr. 63, 353–360 (1974)Carlson, F.: Contributions à la théorie des séries de Dirichlet. Note i. Ark. fö”r Mat., Astron. och Fys. 16(18), 1–19 (1922)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. (2) 174(1), 485–497 (2011)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.: Bohr’s strips for Dirichlet series in Banach spaces. Funct. Approx. Comment. Math. 44(part 2), 165–189 (2011)Defant, A., Maestre, M., Schwarting, U.: Bohr radii of vector valued holomorphic functions. Adv. Math. 231(5), 2837–2857 (2012)Defant, A., Popa, D., Schwarting, U.: Coordinatewise multiple summing operators in Banach spaces. J. Funct. Anal. 259(1), 220–242 (2010)Defant, A., Sevilla-Peris, P.: Convergence of Dirichlet polynomials in Banach spaces. Trans. Am. Math. Soc. 363(2), 681–697 (2011)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics, vol. 43. Cambridge University Press, Cambridge (1995)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)Kahane, J.-P.: Some Random Series of Functions. Cambridge Studies in Advanced Mathematics, vol. 5, 2nd 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. Exch. 27(1):155–175 (2001/2002)Kwapień, S.: Some remarks on $$(p,\, q)$$ ( p , q ) -absolutely summing operators in $$l_{p}$$ l p -spaces. Studia Math. 29, 327–337 (1968)Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes, reprint of the 1991 edn. Classics in Mathematics. Springer, Berlin (2011)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. I. Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92. Springer, Berlin (1977)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. II, Function Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97. Springer, Berlin (1979)Maurizi, B., Queffélec, H.: Some remarks on the algebra of bounded Dirichlet series. J. Fourier Anal. Appl. 16, 676–692 (2010)Prachar, K.: Primzahlverteilung. Springer, Berlin (1957)Queffélec, H.: H. Bohr’s vision of ordinary Dirichlet series; old and new results. J. Anal. 3, 43–60 (1995)Tomczak-Jaegermann, N.: Banach–Mazur Distances and Finite-Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38. Longman Scientific & Technical, Harlow (1989

Factorization of Operators Through Orlicz Spaces

[EN] We study factorization of operators between quasi-Banach spaces. We prove the equivalence between certain vector norm inequalities and the factorization of operators through Orlicz spaces. As a consequence, we obtain the Maurey-Rosenthal factorization of operators into L-p-spaces. We give several applications. In particular, we prove a variant of Maurey's Extension Theorem.The research of the first author was supported by the National Science Centre (NCN), Poland, Grant No. 2011/01/B/ST1/06243. The research of the second author was supported by Ministerio de Economia y Competitividad, Spain, under project #MTM2012-36740-C02-02Mastylo, M.; Sánchez Pérez, EA. (2017). Factorization of Operators Through Orlicz Spaces. Bulletin of the Malaysian Mathematical Sciences Society. 40(4):1653-1675. https://doi.org/10.1007/s40840-015-0158-5S16531675404Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)Davis, W.J., Garling, D.J.H., Tomczak-Jaegermann, N.: The complex convexity of quasi-normed linear spaces. J. Funct. Anal. 55, 110–150 (1984)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Mastyło, M., Michels, C.: Orlicz norm estimates for eigenvalues of matrices. Isr. J. Math. 132, 45–59 (2002)Defant, A., Sánchez Pérez, E.A.: Maurey–Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Defant, A., Sánchez Pérez, E.A.: Domination of operators on function spaces. Math. Proc. Camb. Phil. Soc. 146, 57–66 (2009)Diestel, J.: Sequences and Series in Banach Spaces. Springer, Berlin (1984)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Dilworth, S.J.: Special Banach lattices and their applications. In: Handbook of the Geometry of Banach Spaces, vol. 1. Elsevier, Amsterdam (2001)Figiel, T., Pisier, G.: Séries alétoires dans les espaces uniformément convexes ou uniformément lisses. Comptes Rendus de l’Académie des Sciences, Paris, Série A 279, 611–614 (1974)Kalton, N.J., Montgomery-Smith, S.J.: Set-functions and factorization. Arch. Math. (Basel) 61(2), 183–200 (1993)Kamińska, A., Mastyło, M.: Abstract duality Sawyer formula and its applications. Monatsh. Math. 151(3), 223–245 (2007)Kantorovich, L.V., Akilov, G.P.: Functional Analysis, 2nd edn. Pergamon Press, Oxford (1982)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices IV, Sibirsk. Mat. Z. 14, 140–155 (1973) (in Russian); English transl.: Siberian. Math. J. 14, 97–108 (1973)Lozanovskii, G.Ya.:Transformations of ideal Banach spaces by means of concave functions. In: Qualitative and Approximate Methods for the Investigation of Operator Equations, Yaroslavl, vol. 3, pp. 122–147 (1978) (Russian)Mastyło, M., Szwedek, R.: Interpolative constructions and factorization of operators. J. Math. Anal. Appl. 401, 198–208 (2013)Nikišin, E.M.: Resonance theorems and superlinear operators. Usp. Mat. Nauk 25, 129–191 (1970) (Russian)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conference Series in Mathematics, vol. 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1986)Reisner, S.: On two theorems of Lozanovskii concerning intermediate Banach lattices, geometric aspects of functional analysis (1986/87). Lecture Notes in Math., vol. 1317, pp. 67–83. Springer, Berlin (1988)Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge University Press, Cambridge (1991

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

Extension and splitting theorems for Fréchet spaces of type 2

Sin resume

Factorizing Kernel Operators

Consider an operator T : X--->Y between Banach function spaces having adequate order continuity and Fatou properties. Assume that T can be factorized through a Banach space as T =SR, where R and the adjoint of S are p-th power and q-th power factorable, respectively. Then a canonical factorization scheme can be given for T. We show that it provides a tool for analyzing T that becomes specially useful for the case of kernel operators. In particular, we show that this square factorization scheme for T is equivalent to some inequalities for the bilinear form defined by T. Kernel operators are studied from this point of view.Support of the Ministerio de Economia y Competitividad under project # MTM2012-36740-C02-02 (Spain) is gratefully acknowledged.Galdames, O.; Sánchez Pérez, EA. (2013). Factorizing Kernel Operators. Integral Equations and Operator Theory. 75(1):13-29. https://doi.org/10.1007/s00020-012-2019-zS1329751Bennett C., Sharpley R.: Interpolation of Operators. Academic Press, Boston (1988)Calabuig J.M., Delgado O., Sánchez-Pérez E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Defant A.: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant A., Sánchez Pérez E.A.: Maurey-Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys, vol. 15. American Mathematical Society, Providence (1977)Fernández A., Mayoral F., Naranjo F., Sáez C., Sánchez-Pérez E.A.: Spaces of integrable functions with respect to a vector measure and factorizations through L p and Hilbert spaces. J. Math. Anal. Appl. 330, 1249–1263 (2007)Galdames O., Sánchez Pérez E.A.: Optimal range theorems for operators with p-th power factorable adjoints. Banach J. Math. Anal. 6(1), 63–71 (2012)Lindenstrauss J., Tzafriri L.: Classical Banach Spaces II. Springer, Berlin (1979)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Zaanen, A.C.: Integration, 2nd revised edn. North Holland, Amsterdam; Interscience, New York (1967

Traced tensor norms and multiple summing multilinear operators

[EN] Using a general tensor norm approach, our aim is to show that some distinguished classes of summing operators can be characterized by means of an 'order reduction' procedure for multiple summing multilinear operators, which becomes the keystone of our arguments and can be considered our main result. We work in a tensor product framework involving traced tensor norms and the representation theorem for maximal operator ideals. Several applications are given not only to multi-ideals, but also to linear operator ideals. In particular, we get applications to multiple p-summing bilinear operators, (p, q)-factorable linear operators, tau(p)-summing linear operators and absolutely p-summing linear operators, providing a characterization of this later class whenever the absolutely p-summing linear operators take values in an L-p-space.This work was supported by the Ministerio de Economia y Competitividad (Spain) [grant number MTM2011-22417], [grant number MTM2012-36740-C02-02].Rueda, P.; Sánchez Pérez, EA.; Tallab, A. (2017). Traced tensor norms and multiple summing multilinear operators. Linear and Multilinear Algebra. 65(4):768-786. https://doi.org/10.1080/03081087.2016.1202186S76878665

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

Some polynomial versions of cotype and applications

[EN] We introduce non-linear versions of the classical cotype of Banach spaces. We show that spaces with l.u.st. and cotype, and spaces having Fourier cotype enjoy our non-linear cotype. We apply these concepts to get results on convergence of vector-valued power series in infinite many variables and on l(1)-multipliers of vector-valued Dirichlet series. Finally we introduce cotype with respect to indexing sets, an idea that includes our previous definitions. (C) 2015 Elsevier Inc. All rights reserved.The first author was partially supported by CONICET-PIP 11220130100329CO, UBACyT 20020130100474BA and ANPCyT PICT 2011-1456. The third author was also supported by UPV-SP20120700. All three authors were supported by project MTM2014-57838-C2-2-P.Carando, D.; Defant, A.; Sevilla Peris, P. (2016). Some polynomial versions of cotype and applications. Journal of Functional Analysis. 270(1):68-87. https://doi.org/10.1016/j.jfa.2015.09.017S6887270

The geometric measure of entanglement for a symmetric pure state with positive amplitudes

In this paper for a class of symmetric multiparty pure states we consider a conjecture related to the geometric measure of entanglement: 'for a symmetric pure state, the closest product state in terms of the fidelity can be chosen as a symmetric product state'. We show that this conjecture is true for symmetric pure states whose amplitudes are all non-negative in a computational basis. The more general conjecture is still open.Comment: Similar results have been obtained independently and with different methods by T-C. Wei and S. Severini, see arXiv:0905.0012v