Informing the structure of complex Hadamard matrix spaces using a flow
Stable sets of planar homeomorphisms with translation pseudo-arcs
On a semigroup problem
Viorel Nitica 1,2,, and Andrei Török 3,2,
Department of Mathematics, West Chester University of Pennsylvania, West Chester, PA 19383, USA
Institute of Mathematics, Romanian Academy, P.O. Box 1-764, RO-70700 Bucharest, Romania
Department of Mathematics, University of Houston, Houston, TX 77204-3308, USA
* Corresponding author: Viorel Nitica
Received July 2016 Revised October 2017 Published January 2019
Fund Project: VN was partially supported by Simons Foundation Grant 208729. AT was partially supported by Simons Foundation Grant 239583
If $ S $ is a semigroup in $ \mathbb{R}^n $ that is not separated by a linear functional, then it is known that the closure of $ S $ is a group. We investigate a similar statement in an infinite dimensional topological vector space $ X $. We show that if $ X $ is an infinite dimensional Banach space, then there exists a semigroup $ S\subset X $, not separated by the continuous functionals supported by the closed linear span of $ S $, for which the closure of the semigroup is not a group. If $ X $ is an infinite dimensional Fréchet space, then the closure of a semigroup that is not separated is always a group if and only if $ X $ is $ \mathbb{R}^{\omega} $, the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as $ \mathbb{R}^{\infty} $, the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper.
Keywords: Semigroup, group, Banach space, Fréchet space, separation by functionals, topological transitivity, extension, hyperbolic dynamical system, infinite direct product of lines, infinite direct sum of lines.
Mathematics Subject Classification: Primary: 37B, 37D; Secondary: 58B.
Citation: Viorel Nitica, Andrei Török. On a semigroup problem. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019148
C. R. Adams, The space of functions of bounded variation and certain general spaces, Trans. Amer. Math. Soc., 40 (1936), 421-438. doi: 10.1090/S0002-9947-1936-1501882-8. Google Scholar
R. D. Anderson, Hibert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc., 72 (1966), 515-519. doi: 10.1090/S0002-9904-1966-11524-0. Google Scholar
C. Bargetz, Completing the Valdivia-Vogt tables of sequence-space representations of spaces of smooth functions and distributions, Monatshefte für Mathematik, 177 (2015), 1-14. doi: 10.1007/s00605-014-0650-2. Google Scholar
C. Bessaga, A. Pełczyński and S. Rolewicz, On diametral approximative dimension and linear homogeneity of $F$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 9 (1961), 677-683. Google Scholar
P. A. Borodin, Density of a semigroup in a Banach space, Izvestiya: Mathematics, 78 (2014), 1079-1104. doi: 10.1070/im2014v078n06abeh002721. Google Scholar
R. Bowen, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35. American Mathematical Society Providence, R.I. 1978. Google Scholar
R. Brown, P. J. Higgins and S. A. Morris, Countable products and sums of lines and circles: Their closed subgroups, quotients and duality properties, Math. Proc. Camb. Phil. Soc., 78 (1975), 19-32. doi: 10.1017/S0305004100051483. Google Scholar
H. Jarchow, Locally Convex Spaces, Springer, 1981. Google Scholar
N. Kalton, Normalization properties of Schauder bases, Proc. London Math.. Soc., 22 (1971), 91-105. doi: 10.1112/plms/s3-22.1.91. Google Scholar
N. Kalton, The metric linear spaces $ L_p $ for $ 0<p<1 $, Contemporary Mathematics, 52 (1986), 55-69. doi: 10.1090/conm/052/840695. Google Scholar
N. Kalton, The basic sequence problem, Studia Mathematica, 116 (1995), 168-187. doi: 10.4064/sm-116-2-167-187. Google Scholar
J. Lindestrauss and L. Tzafriri, Classic Banach Spaces Ⅰ, Ⅱ, Springer-Verlag, 1977. Google Scholar
K. Lui, V. Nitica and S. Venkatesh, The semigroup problem for central semidirect product of $ \mathbb{R} ^n $ with $ \mathbb{R} ^m $, Topology Proceedings, 45 (2015), 9-29. Google Scholar
P. Mankiewicz, On topological, Lipschitz, and uniform classification of LF-spaces, Studia Math., 54 (1974), 109-142. doi: 10.4064/sm-52-2-109-142. Google Scholar
I. Melbourne, V. Nitica and A. Török, Stable transitivity of certain noncompact extensions of hyperbolic systems, Annales Henri Poincaré, 6 (2005), 725-746. doi: 10.1007/s00023-005-0221-0. Google Scholar
I. Melbourne, V. Nitica and A. Török, A note about stable transitivity of noncompact extensions of hyperbolic systems, Contin. Dynam. Systems, 14 (2006), 355-363. doi: 10.3934/dcds.2006.14.355. Google Scholar
S. A. Morris, Locally compact abelian groups and the variety of topological groups generated by the reals, Proc. Amer. Math. Soc., 34 (1972), 290-292. doi: 10.1090/S0002-9939-1972-0294560-4. Google Scholar
V. Nitica and A. Török, Open and dense topological transitivity of extensions by non-compact fiber of hyperbolic systems: a review, Axioms, 4 (2015), 84-101. Google Scholar
V. Nitica and A. Török, Stable transitivity of Heisenberg group extensions of hyperbolic systems, Nonlinearity, 27 (2014), 661-683. doi: 10.1088/0951-7715/27/4/661. Google Scholar
V. Nitica and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 25 (2005), 257-269. doi: 10.1017/S0143385704000471. Google Scholar
A. Pietsch, Nuclear Locally Convex Spaces, Ergebnisse Der Mathematick und Ihrer Grensgebiete, Volume 66, Springer-Verlag, New York/Heidelberg/Berlin, 1972. Google Scholar
Z. Rosengarten and A. Reich, Transitivity of infinite dimensional extensions of Anosov diffeomorphisms, 2012, arXiv: 1209.2183v1.Google Scholar
W. Rudin, Functional Analysis, second edition, McGraw-Hill, 1991. Google Scholar
J. H. Shapiro, On the weak basis theorem in $F$-spaces, Canadian J. Math., 26 (1974), 1294-1300. doi: 10.4153/CJM-1974-124-5. Google Scholar
E. A. Sidorov, Topologically transitive cylindrical cascades (Russian), Mat. Zametki, 14 (1973), 441-452. Google Scholar
T. Silverman and S. M. Miller, Hilbert extensions of Anosov diffeomorpisms, 2013, http://www.math.psu.edu/mass/reu/2013/mathfest/HilbertCounterexample.pdfGoogle Scholar
[28] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, London, 1967.
J. Wengenroth, Acyclic inductive spectra of Fréchet spaces, Studia Mathematica, 120 (19966), 247-258. doi: 10.4064/sm-120-3-247-258. Google Scholar
Joachim Escher, Rossen Ivanov, Boris Kolev. Euler equations on a semi-direct product of the diffeomorphisms group by itself. Journal of Geometric Mechanics, 2011, 3 (3) : 313-322. doi: 10.3934/jgm.2011.3.313
J. Leonel Rocha, Sandra M. Aleixo. An extension of Gompertzian growth dynamics: Weibull and Fréchet models. Mathematical Biosciences & Engineering, 2013, 10 (2) : 379-398. doi: 10.3934/mbe.2013.10.379
Yuk L. Yung, Cameron Taketa, Ross Cheung, Run-Lie Shia. Infinite sum of the product of exponential and logarithmic functions, its analytic continuation, and application. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 229-248. doi: 10.3934/dcdsb.2010.13.229
Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829
Liping Wang, Juncheng Wei. Infinite multiplicity for an inhomogeneous supercritical problem in entire space. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1243-1257. doi: 10.3934/cpaa.2013.12.1243
N. U. Ahmed. Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach. Evolution Equations & Control Theory, 2012, 1 (2) : 235-250. doi: 10.3934/eect.2012.1.235
Éder Rítis Aragão Costa. An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 845-868. doi: 10.3934/cpaa.2019041
George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042
Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial & Management Optimization, 2012, 8 (1) : 179-187. doi: 10.3934/jimo.2012.8.179
Andrew E.B. Lim, John B. Moore. A path following algorithm for infinite quadratic programming on a Hilbert space. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 653-670. doi: 10.3934/dcds.1998.4.653
H.T. Banks, S. Dediu, H.K. Nguyen. Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space. Mathematical Biosciences & Engineering, 2007, 4 (3) : 403-430. doi: 10.3934/mbe.2007.4.403
Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for multistrain models with infinite delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 507-536. doi: 10.3934/dcdsb.2017025
Bo You. Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2283-2298. doi: 10.3934/cpaa.2019103
J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731
J. Alberto Conejero, Marko Kostić, Pedro J. Miana, Marina Murillo-Arcila. Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1915-1939. doi: 10.3934/cpaa.2016022
Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627
Hans Koch, Héctor E. Lomelí. On Hamiltonian flows whose orbits are straight lines. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2091-2104. doi: 10.3934/dcds.2014.34.2091
Daniel Alpay, Mihai Putinar, Victor Vinnikov. A Hilbert space approach to bounded analytic extension in the ball. Communications on Pure & Applied Analysis, 2003, 2 (2) : 139-145. doi: 10.3934/cpaa.2003.2.139
W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893
Viorel Nitica Andrei Török