A categorical approach to operator semigroups


The aim of this paper is to exploit the structure of strongly continuous operator semigroups in order to formulate a categorical framework in which a fresh perspective can be applied to past operator theoretic results. In particular, we investigate the inverse-producing Arens extension for Banach algebras (Trans. Am. Math. Soc. 88:536–548, 1958) adapted for operators and operator semigroups by Batty and Geyer (J. Oper. Theory 78(2):473–500, 2017) in this new framework, asking and answering questions using categorical language. We demonstrate that the Arens extension defines an extension functor in this setting and that it forms an adjunction with the suitably defined forgetful functor. As a by-product of this categorical framework, we also revisit the work on Banach direct sums by Lachowicz and Moszyński (Semigroup Forum 93(1):34–70, 2016). This paper can be considered as a brief exploration of the triple interface between operator semigroups, Banach algebras, and category theory.

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  1. 1.

    In the Hilbert space setting the author has elsewhere [29] generalised the notion of direct sums of \(C_0\)-semigroups, here treated as UB-coproducts, to direct integrals of \(C_0\)-semigroups.

  2. 2.

    The author actually first got the idea of thinking this way from watching https://www.youtube.com/watch?v=qHuUazkUcnU&ab_channel=mlbaker, where the notion of tensor products is explained as a standardising tool for turning multilinear maps into linear maps.


  1. 1.

    Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96, 2nd edn. Birkhäuser/Springer AG, Basel (2011)

    Book  Google Scholar 

  2. 2.

    Ando, T., Ceauşescu, Z., Foiaş, C.: On intertwining dilations II. Acta Sci. Math. (Szeged) 39(1–2), 3–14 (1977)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Arens, R.: Linear topological division algebras. Bull. Am. Math. Soc. 53, 623–630 (1947)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Arens, R.: Inverse-producing extensions of normed algebras. Trans. Am. Math. Soc. 88, 536–548 (1958)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Arens, R.: Ideals in Banach algebra extensions. Stud. Math. 31, 29–34 (1968)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Arens, R., Hoffman, K.: Algebraic extension of normed algebras. Proc. Am. Math. Soc. 7, 203–210 (1956)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Arveson, W.: Dilation theory yesterday and today, Oper. Theory Adv. Appl., Birkhäuser Verlag, Basel, no. 207, 99–123 (2010)

  8. 8.

    Badea, C., Müller, V.: Growth conditions and inverse producing extensions. J. Oper. Theory 54, 415–439 (2005)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Batty, C.J.K., Geyer, F.: Lower bounds for unbounded operators and semigroups. J. Oper. Theory 78(2), 473–500 (2017)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Batty, C.J.K., Yeates, S.B.: Extensions of semigroups of operators. J. Oper. Theory 46(1), 139–157 (2001)

    MathSciNet  Google Scholar 

  11. 11.

    Batty, C.J.K., Duyckaerts, T.: Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8(4), 765–780 (2008)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Batty, C.J.K., Paunonen, L., Seifert, D.: Optimal energy decay in a one-dimensional coupled wave-heat system. J. Evol. Equ. 16(3), 649–664 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Bercovici, H., Foias, C., Kérchy, L., Sz.-Nagy, B.: Harmonic Analysis of Operators on Hilbert Space, 2nd edn., Universitext, Springer, New York (2010)

  14. 14.

    Bergh, J., Löfström, J.: Interpolation Spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)

  15. 15.

    Bollobás, B.: Adjoining inverses to commutative Banach algebras. Trans. Am. Math. Soc. 181, 165–174 (1973)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Bollobás, B.: Best possible bounds of the norms of inverses adjoined to normed algebras. Stud. Math. 51, 87–96 (1974)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Borichev, A., Tomilov, Y.: Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347(2), 455–478 (2010)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Brehmer, S.: Über vetauschbare Kontraktionen des Hilbertschen Raumes. Acta Sci. Math. Szeged 22, 106–111 (1961)

  19. 19.

    Castillo, J.M.F.: The hitchhiker guide to categorical Banach space theory. Part I. Extracta Math. 25(2), 103–149 (2010)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Cooper, J.L.B.: One-parameter semigroups of isometric operators in Hilbert space. Ann. Math. 48(2), 827–842 (1947)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Dorofeev, S., Kleisli, H.: Functorial methods in the theory of group representations, I. Appl. Categ. Struct. 3(2), 151–172 (1995)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Douglas, R.G.: On extending commutative semigroups of isometries. Bull. Lond. Math. Soc. 1, 157–159 (1969)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Engel, K., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)

  24. 24.

    Fackler, S., Glück, J.: A toolkit for constructing dilations on Banach spaces. Proc. Lond. Math. Soc. 118(2), 416–440 (2018)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Ghez, P., Lima, R., Roberts, J.E.: \(W^\ast \)-categories. Pac. J. Math. 1, 79–109 (1985)

    Article  Google Scholar 

  26. 26.

    Itô, T.: On the commutative family of subnormal operators. J. Fac. Sci. Hokkaido Univ. Ser. I(14), 1–15 (1958)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Lachowicz, M., Moszyński, M.: Infinite Banach direct sums and diagonal \(C_0\)-semigroups with applications to a stochastic particle system. Semigroup Forum 93(1), 34–70 (2016)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  29. 29.

    Ng, A.C.S.: Direct integrals of strongly continuous operator semigroups. J. Math. Anal. Appl. 489(2), 124176 (2020)

  30. 30.

    Ng, A.C.S., Seifert, D.: Optimal energy decay in a one-dimensional wave-heat system with infinite heat part. J. Math. Anal. Appl. 482(2), 123563 (2020)

  31. 31.

    Read, C.J.: Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element. Trans. Am. Math. Soc. 286, 1–17 (1984)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Read, C.J.: Extending an operator from a Hilbert space to a larger Hilbert space, so as to reduce its spectrum. Israel J. Math. 57, 375–380 (1987)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Read, C.J.: Spectrum reducing extension for one operator on a Banach space. Trans. Am. Math. Soc. 308, 413–429 (1988)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Rozendaal, J., Seifert, D., Stahn, R.: Optimal rates of decay for operator semigroups on Hilbert spaces. Adv. Math. 346, 359–388 (2019)

    MathSciNet  Article  Google Scholar 

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The author is grateful to many people for their help during the production of this article. To Charles Batty and David Seifert for insightful discussions on the subject of this article, to Joshua Ciappara for making some crucial suggestions from an algebraist’s perspective (such as suggesting that I look at adjunctions), and to the reviewer for numerous useful comments as a result of which this paper is much improved. This work was partially supported by the University of Sydney through the Barker Graduate Scholarship while the author was a doctoral student at the University of Oxford. The revisions were completed when the author was funded by ARC Grant DP180100595.

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Correspondence to Abraham C. S. Ng.

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Communicated by Markus Haase.

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Ng, A.C.S. A categorical approach to operator semigroups. Semigroup Forum 102, 495–516 (2021). https://doi.org/10.1007/s00233-020-10158-7

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  • Category theory
  • Operator theory
  • \(C_0\)-semigroups
  • Extensions