## Functiones et Approximatio Commentarii Mathematici

### Uniform mean ergodicity of $C_0$-semigroups\newline in a class of Fréchet spaces

#### Abstract

Let $(T(t))_{t\geq 0}$ be a strongly continuous $C_0$-semigroup of bounded linear operators on a~Banach space $X$ such that $\lim_{t\to\infty}||T(t)/t||=0$. Characterizations of when $(T(t))_{t\geq 0}$ is uniformly mean ergodic, i.e., of when its Cesàro means $r^{-1}\int_0^r T(s)ds$ converge in operator norm as $r\to\infty$, are known. For instance, this is so if and only if the infinitesimal generator $A$ has closed range in $X$ if and only if $\lim_{\lambda\downarrow 0^+}\lambda R(\lambda, A)$ exists in the operator norm topology (where $R(\lambda,A)$ is the resolvent operator of $A$ at $\lambda$). These characterizations, and others, are shown to remain valid in the class of quojection Fréchet spaces, which includes all Banach spaces, countable products of Banach spaces, and many more. It is shown that the extension fails to hold for all Fréchet spaces. Applications of the results to concrete examples of $C_0$-semigroups in particular Fréchet function and sequence spaces are presented.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 50, Number 2 (2014), 307-349.

Dates
First available in Project Euclid: 26 June 2014

https://projecteuclid.org/euclid.facm/1403811847

Digital Object Identifier
doi:10.7169/facm/2014.50.2.8

Mathematical Reviews number (MathSciNet)
MR3229064

Zentralblatt MATH identifier
1314.47060

#### Citation

Albanese, Angela A.; Bonet, José; Ricker, Werner J. Uniform mean ergodicity of $C_0$-semigroups\newline in a class of Fréchet spaces. Funct. Approx. Comment. Math. 50 (2014), no. 2, 307--349. doi:10.7169/facm/2014.50.2.8. https://projecteuclid.org/euclid.facm/1403811847

#### References

• A.A. Albanese, A Fréchet space of continuous functions which is a prequojection, Bull. Soc. Roy. Sci. Liége 60 (1991), 409–417,
• A.A. Albanese, J. Bonet, W.J. Ricker, Mean ergodic operators in Fréchet spaces, Ann. Acad. Sci. Fenn. Math. 34 (2009), 401–436.
• A.A. Albanese, J. Bonet, W.J. Ricker, Grothendieck spaces with the Dunford–Pettis property, Positivity 14 (2010), 145–164.
• A.A. Albanese, J. Bonet, W.J. Ricker, On mean ergodic operators, In: Vector Measures, Integration and Related Topics, G.P. Curbera et. al. (Eds), Operator Theory: Advances and Applications 201, Birkhäuser Verlag Basel, 2010, pp. 1–20.
• A.A. Albanese, J. Bonet, W.J. Ricker, $C_0$–semigroups and mean ergodic operators in a class of Fréchet spaces, J. Math. Anal. Appl. 365 (2010), 142–157.
• A.A. Albanese, J. Bonet, W.J. Ricker, Mean ergodic semigroups of operators, Rev. R. Acad. Cien. Serie A, Mat., RACSAM, 106 (2012), 299–319.
• A.A. Albanese, J. Bonet, W.J. Ricker, Montel resolvents and uniformly mean ergodic semigroups of linear operators, Quaest. Math. 36 (2013), 253–290.
• A.A. Albanese, J. Bonet, W.J. Ricker, Convergence of arithmetic means of operators in Fréchet spaces, J. Math. Anal. Appl. 401 (2013), 160–173.
• A.A. Albanese, L. Lorenzi, V. Manco, Mean ergodic theorems for bi–continuous semigroups, Semigroup Forum 82 (2011), 141–171.
• A.A. Albanese, E. Mangino, Some permanence results of the Dunford–Pettis and Grothendieck properties in lcHs, Funct. Approx. Comment. Math. 44 (2011), 243–258.
• W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel–Boston–Berlin, 2001.
• S.F. Bellenot, E. Dubinsky, Fréchet spaces with nuclear Köthe quotients, Trans. Amer. Math. Soc. 273 (1982), 579–594.
• E. Berhens, S. Dierolf, P. Harmand, On a problem of Bellenot and Dubinsky, Math. Ann. 275 (1986), 337–339.
• J. Bonet, M. Maestre, G. Metafune, V.B. Moscatelli, D. Vogt, Every quojection is the quotient of a countable product of Banach spaces, in “Advances in the Theory of Fréchet spaces”, T. Terzioğlu (Ed.), NATO ASI Series, 287, Kluwer Acad. Publ., Dordrecht, 1989, pp. 355-356.
• J. Bonet, B. de Pagter, W.J. Ricker, Mean ergodic operators and reflexive Fréchet lattices, Proc. Roy. Soc. Edinburgh (Sect. A) 141 (2011), 897–920.
• J. Bonet, W.J. Ricker, Schauder decompositions and the Grothendieck and Dunford–Pettis properties in Köthe echelon spaces of infinite order, Positivity 11 (2007), 77–93.
• S. Dierolf, V.B. Moscatelli, A note on quojections, Funct. Approx. Comment. Math. 17 (1987), 131–138.
• S. Dierolf, D.N. Zarnadze, A note on strictly regular Fréchet spaces, Arch. Math. 42 (1984), 549–556.
• P. Domański, Twisted Fréchet spaces of continuous functions, Results Math. 23 (1993), 45–48.
• N. Dunford, J.T. Schwartz, Linear Operators I: General Theory (2nd printing), Wiley–Interscience, New York, 1964.
• W.F. Eberlein, Abtract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240.
• R.E. Edwards, Functional Analysis, Reinhart and Winston, New York, 1965.
• K.-J. Engel, R. Nagel, One–parameter Semigroups for Linear Evolution Equations, Springer Verlag, Berlin-Heidelberg-New York, 2000.
• E. Hille, R.S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc., Providence, 1957.
• I.I. Hirschman, Jr., Infinite Series, Holt, Rinehart and Winston, New York, 1962.
• H. Jarchow, Locally Convex Spaces, B.G. Teubner, Stuttgart, 1981.
• H. Komatsu, Semigroups of operators in locally convex spaces, J. Math. Soc. Japan 16 (1964), 230–262.
• T. Komura, Semigroups of operators in locally convex spaces, J. Funct. Anal. 2 (1968), 258–296.
• G. Köthe, Topological Vector Spaces I, 2nd Rev. Ed., Springer Verlag, Berlin-Heidelberg-New York, 1983.
• G. Köthe, Topological Vector Spaces II, Springer Verlag, Berlin-Heidelberg-New York, 1979.
• U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.
• M. Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43 (1974), 337–340.
• M. Lin, On the uniform ergodic theorem. II, Proc. Amer. Math. Soc. 46 (1974), 217–225.
• M. Lin, Ergodic properties of an operator obtained from a continuous representation, Ann. Inst. Henri Poincaré 13 (1977), 321–331.
• R. Meise, D. Vogt, Introduction to Functional Analysis, Clarendon Press, Oxford, 1997.
• G. Metafune, V.B. Moscatelli, Quojections and prequojections, in “Advances in the Theory of Fréchet spaces”, T. Terzioğlu (Ed.), NATO ASI Series, 287, Kluwer Academic Publishers, Dordrecht, 1989, pp. 235–254.
• G. Metafune, V.B. Moscatelli, Prequojections and their duals, in “Progress in Functional Analysis”, K.D. Bierstedt, J. Bonet, J. Horváth and M. Maestre (Eds.), North Holland, Amsterdam, 1992, pp. 215-232.
• V.B. Moscatelli, Fréchet spaces without norms and without bases, Bull. London Math. Soc. 12 (1980), 63–66.
• V.B. Moscatelli, Strongly non-norming subspaces and prequojections, Studia Math. 95 (1990), 249–254.
• S. Önal, T. Terzioğlu, Unbounded linear operators and nuclear Köthe quotients, Arch. Math. 54 (1990), 576–581.
• S. Ouchi, Semi-groups of operators in locally convex spaces, J. Math. Soc. Japan 25 (1973), 265–276.
• K. Piszczek, Quasi-reflexive Fréchet spaces and mean ergodicity, J. Math. Anal. Appl. 361 (2010), 224–233.
• K. Piszczek, Quasi-reflexive Fréchet spaces and contractively power bounded operators, Arch. Math. 96 (2011), 49–58.
• H.H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin–Heidelberg, 1974.
• D. Vogt, On two problems of Mityagin, Math. Nachr. 141 (1989), 13–25.
• K. Yosida, Functional Analysis, Springer–Verlag, Berlin, 1980.