Abstract
We construct a convolution algebra of admissible homomorphisms defined on a ‘test space’ to demonstrate the fundamental role of convolution in the study of intertwined evolution operators of linear ordinary differential equations in Banach spaces and probability theory. The choice of test space makes the framework we present quite versatile. The applications include semigroups of linear operators, empathy, integrated semigroups and empathies and the convolution semigroups of probability theory.
Similar content being viewed by others
References
Arendt, W.: Vector-valued Laplace transforms and Cauchy problems. Israel J. Math. 59, 327–352 (1987)
Bobrowski, A.: Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, New York (2005)
Bridges, D.S.: Foundations of Real and Abstract Analysis. Graduate Texts in Mathematics. Springer, New York (1997)
Brown, T.J., Sauer, N.: Double families of integrated evolution operators. J. Evol. Equ. 4, 567–590 (2004)
Dunford, N., Schwartz, J.T.: Linear Operators Part I: General Theory. Interscience Publishers Inc, New York (1958)
Fadell, A.G.: Vector Calculus and Differential Equations. D. Van Nostrand, New York (1968)
Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley series in probability and mathematical statistics, vol. II. Wiley, New York (1966)
Hewitt, E., Ross, K.A.: Abstract harmonic Analysis Volume I : Structure of Topological Groups, Integration Theory, Group Representations. Springer, New York (1963)
Hille, E., Phillips, R.S.: Functional Analysis and Semi groups, vol. 31. American Mathematical Society Colloquium Publications, Rhode Island (2000)
Kisynski, J.: The Widder spaces, representations of the convolution algebra \(L^{1}({\mathbb{R}})^{+}\) and one parameter semigroups of operators. Preprint no. 588, Institute of Mathematics, Polish Academy of Sciences, Warsaw, (1998)
Palmer, T. W.: Banach Algebras and The General Theory of *-Algebras; Volume I : Algebras and Banach Algebras. Encyclopaedia of Mathematics and its Applications, Cambridge University Press, New York (1994)
Sauer, N.: Linear evolution equations in two Banach spaces. Proc. Royal Soc. Edinburgh 91A, 287–303 (1982)
Sauer, N., Singleton, J.E.: Evolution operators in empathy with a semigroup. Semigroup Forum 39, 85–94 (1989)
Sauer, N.: Linear Operators. Empathy theory and the laplace transform, vol. 38, pp. 325–338. Banach Center Publications, Warszawa (1997)
Sauer, N.: Causality and causation: what we learn from mathematical dynamic systems theory. Trans. Roy. Soc. S Afr. 65, 65–68 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jerome A. Goldstein.
Rights and permissions
About this article
Cite this article
Lee, WS., Sauer, N. Intertwined evolution operators. Semigroup Forum 94, 204–228 (2017). https://doi.org/10.1007/s00233-016-9796-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-016-9796-7