Abstract
We reveal three surprising properties of cosine families, distinguishing them from semigroups of operators: (1) A single trajectory of a cosine family is either strongly continuous or not measurable. (2) Pointwise convergence of a sequence of equibounded cosine families implies that the convergence is almost uniform for time in the entire real line; in particular, cosine families cannot be perturbed in a singular way. (3) A non-constant trajectory of a bounded cosine family does not have a limit at infinity; in particular, the rich theory of asymptotic behaviour of semigroups does not have a counterpart for cosine families. In addition, we show that equibounded cosine families that converge strongly and almost uniformly in time may fail to converge uniformly.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arendt W., Batty C.J.K., Hieber M., Neubrander F.: Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel (2001)
Banasiak J., Bobrowski A.: Interplay between degenerate convergence of semigroups and asymptotic analysis: a study of a singularly perturbed abstract telegraph system. J. Evol. Equ. 9, 293–314 (2009)
Bobrowski A.: Degenerate convergence of semigroups. Semigroup Forum 49, 303–327 (1994)
Bobrowski A.: The Widder–Arendt theorem on inverting of the Laplace transform, and its relationships with the theory of semigroups of operators. Methods Funct. Anal. Topology 3, 1–39 (1997)
Bobrowski A.: A note on convergence of semigroups. Ann. Polon. Math. 69, 107–127 (1998)
Bobrowski A.: Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge (2005)
Bobrowski A.: Degenerate convergence of semigroups related to a model of stochastic gene expression. Semigroup Forum 73, 345–366 (2006)
Bobrowski A.: On a semigroup generated by a convex combination of two Feller generators. J. Evol. Equ. 7, 555–565 (2007)
Bobrowski A.: On limitations and insufficiency of the Trotter–Kato theorem. Semigroup Forum 75, 317–336 (2007)
Bobrowski A.: Generation of cosine families via Lord Kelvin’s method of images. J. Evol. Equ. 10, 663–675 (2010)
Bobrowski A.: From diffusions on graphs to Markov chains via asymptotic state lumping. Ann. Henri Poincaré 13, 1501–1510 (2012)
Bochner S.: Abstrakte Fastperiodische Funktionen. Acta Math. 61, 149–184 (1933)
Chander R., Singh H.: On the measurability and continuity properties of the cosine operator. Indian J. Pure Appl. Math. 12, 81–83 (1981)
Corduneanu C.: Almost Periodic Functions. Interscience Publishers, New York (1968)
Engel K.J., Nagel R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Fattorini H.O.: Ordinary differential equations in linear topological spaces. I. J. Differential Equations 5, 72–105 (1969)
Fattorini H.O.: Ordinary differential equations in linear topological spaces. II. J. Differential Equations 6, 50–70 (1969)
Fattorini H.O.: Second order linear differential equations in Banach spaces. North-Holland, Amsterdam (1985)
Goldstein J.A.: On the convergence and approximation of cosine functions. Aequationes Math. 11, 201–205 (1974)
Goldstein J.A.: Semigroups of Linear Operators and Applications. Oxford University Press, New York (1985)
Hille E.: Une généralisation du problème de Cauchy. Ann. Inst. Fourier (Grenoble) 4, 31–48 (1952)
Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. Amer. Math. Soc. Colloq. Publ. 31. Amer. Math. Soc., Providence, R. I. (1957)
Itô, K., McKean, Jr., H.P.: Diffusion Processes and Their Sample Paths. Springer, Berlin (1996). Repr. of the 1974 ed.
Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)
Konishi, Y.: Cosine functions of operators in locally convex spaces. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18, 443–463 (1971/1972)
Kurepa, S.: Semigroups and cosine functions. In: Functional Analysis (Dubrovnik, 1981), Lecture Notes in Math., vol. 948, pp. 47–72. Springer, Berlin (1982)
Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics. Springer (1994)
Piskarev S., Shaw S.Y.: On certain operator families related to cosine operator functions. Taiwanese J. Math. 1, 527–546 (1997)
Vasilev, V.V., Piskarev, S.I.: Differential equations in Banach spaces. II. Theory of cosine operator functions. J. Math. Sci. (N. Y.) 122, 3055–3174 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Jan Kisyński on the occasion of his 80th birthday
Adam Bobrowski is on leave from Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Bobrowski, A., Chojnacki, W. Cosine families and semigroups really differ. J. Evol. Equ. 13, 897–916 (2013). https://doi.org/10.1007/s00028-013-0208-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-013-0208-0