Abel ergodic theorems for $\alpha$-times integrated semigroups


  • Fatih Barki Ibn Zohr University, EST-Dakhla, Morocco
  • Abdelaziz Tajmouati Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez, Morocco




Let $\{T(t)\}_{t\geq 0}$ be an $\alpha$-times integrated semigroup of bounded linear operators on the Banach space $\mathcal{X}$ and let $A$ be their generator. In this paper, we study the uniform convergence of the Abel averages $\mathcal{A} (\lambda) = \lambda^{\alpha +1} \int_{0}^{\infty} e^{-\lambda t} T(t)\,dt$ as $\lambda \to 0^+$, with $\alpha \geq 0$. More precisely, we show that the following conditions are equivalent: (i) $T(t)$ is uniformly Abel ergodic; (ii) $\mathcal{X}= \mathcal{R}(A)\oplus \mathcal{N}(A)$, with $\mathcal{R}(A)$ closed; (iii) $\|\lambda^2 R(\lambda,A)\| \longrightarrow 0$ as $\lambda\to 0^+$, and $\mathcal{R}(A^k)$ is closed for some integer $k$; (iv) $A$ is $a$-Drazin invertible and $\mathcal{R}(A^k)$ is closed for some $k\geq 1$; where $\mathcal{N}(A)$, $\mathcal{R}(A)$ and $R(\lambda,A)$ are the kernel, the range, and the resolvent function of $A$, respectively. Additionally, we show that if $T(t)$ satisfies $\lim_{t\to \infty}\|T(t)\|/ t^{\alpha+1}=0$, then $T(t)$ is uniformly Abel ergodic if and only if $\frac{1}{t^{\alpha+1}}\int_{0}^{t}T(s)\,ds$ converges uniformly as $t \to +\infty$. Finally, we examine simultaneously this theory with the uniform power convergence of the Abel averages $\mathcal{A} (\lambda)$ for some $\lambda>0$.


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W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math. 59 no. 3 (1987), 327–352.  DOI  MR  Zbl

W. Arendt and H. Kellermann, Integrated solutions of Volterra integrodifferential equations and applications, in Volterra Integrodifferential Equations in Banach Spaces and Applications (Trento, 1987), Pitman Res. Notes Math. Ser. 190, Longman Sci. Tech., Harlow, 1989, pp. 21–51.  MR  Zbl

F. Barki, Cesàro and Abel ergodic theorems for integrated semigroups, Concr. Oper. 8 no. 1 (2021), 135–149.  DOI  MR  Zbl

F. Barki, A. Tajmouati, and A. El Bakkali, Uniform and mean ergodic theorems for $C_0$-semigroups, Methods Funct. Anal. Topology 27 no. 2 (2021), 130–141.  DOI  MR  Zbl

E. Boasso, Isolated spectral points and Koliha-Drazin invertible elements in quotient Banach algebras and homomorphism ranges, Math. Proc. R. Ir. Acad. 115A no. 2 (2015), 121–135.  DOI  MR  Zbl

P. L. Butzer and J. J. Koliha, The $a$-Drazin inverse and ergodic behaviour of semigroups and cosine operator functions, J. Operator Theory 62 no. 2 (2009), 297–326.  MR  Zbl

M. Elin, S. Reich, and D. Shoikhet, Numerical Range of Holomorphic Mappings and Applications, Birkhäuser/Springer, Cham, 2019.  DOI  MR  Zbl

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer-Verlag, New York, 2000.  DOI  MR  Zbl

A. Gessinger, Connections between the approximation and ergodic behaviour of cosine operators and semigroups, in Proceedings of the Third International Conference on Functional Analysis and Approximation Theory (Acquafredda di Maratea, 1996), Rend. Circ. Mat. Palermo (2) Suppl., no. 52, vol. II, 1998, pp. 475–489.  MR  Zbl

M. Hieber, Laplace transforms and α-times integrated semigroups, Forum Math. 3 no. 6 (1991), 595–612.  DOI  MR  Zbl

E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, rev. ed., American Mathematical Society Colloquium Publications 31, American Mathematical Society, Providence, RI, 1957.  MR  Zbl

J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 no. 3 (1996), 367–381.  DOI  MR  Zbl

Y. Kozitsky, D. Shoikhet, and J. Zemánek, Power convergence of Abel averages, Arch. Math. (Basel) 100 no. 6 (2013), 539–549.  DOI  MR  Zbl

U. Krengel, Ergodic Theorems, De Gruyter Studies in Mathematics 6, De Gruyter, Berlin, 1985.  DOI  MR  Zbl

M. Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43 (1974), 337–340.  DOI  MR  Zbl

M. Lin, On the uniform ergodic theorem. II, Proc. Amer. Math. Soc. 46 (1974), 217–225.  DOI  MR  Zbl

M. Lin, D. Shoikhet, and L. Suciu, Remarks on uniform ergodic theorems, Acta Sci. Math. (Szeged) 81 no. 1-2 (2015), 251–283.  DOI  MR  Zbl

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983.  DOI  MR  Zbl

S.-Y. Shaw, Uniform ergodic theorems for locally integrable semigroups and pseudoresolvents, Proc. Amer. Math. Soc. 98 no. 1 (1986), 61–67.  DOI  MR  Zbl

S.-Y. Shaw, Uniform ergodic theorems for operator semigroups, in Proceedings of the Analysis Conference, Singapore 1986, North-Holland Math. Stud. 150, North-Holland, Amsterdam, 1988, pp. 261–265.  DOI  MR  Zbl

A. Tajmouati, A. El Bakkali, F. Barki, and M. A. Ould Mohamed Baba, On the uniform ergodic for α-times integrated semigroups, Bol. Soc. Parana. Mat. (3) 39 no. 4 (2021), 9–20.  MR  Zbl

A. Tajmouati, M. Karmouni, and F. Barki, Abel ergodic theorem for $C_0$-semigroups, Adv. Oper. Theory 5 no. 4 (2020), 1468–1479.  DOI  MR  Zbl

A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, second ed., Wiley, New York, 1980.  MR  Zbl

H. R. Thieme, “Integrated semigroups” and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl. 152 no. 2 (1990), 416–447.  DOI  MR  Zbl

K. Yosida, Functional Analysis, third ed., Die Grundlehren der mathematischen Wissenschaften 123, Springer-Verlag, New York, 1971.  Zbl