Bi-continuous semigroups on sequentially complete random Saks spaces

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Abstract

In this paper, we first introduced the Riemann integral for an abstract-valued function from a finite closed real interval to a sequentially complete random Saks space and gave the fundamental theorem of calculus for an L⁰-Lipschitz function. Then we investigated some important properties peculiar to bi-continuous semigroups on a sequentially complete random Saks space. Finally, based on the above work, we established the Hille-Yosida generation theorem for such bi-continuous semigroups, which extends and improves several known results.

Keywords

random Saks spaces / bi-continuous semigroups / Hille-Yosida generation theorem / (ε, λ)-topology

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References

[1] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations , Springer, 1983. https://doi.org/10.1007/978-1-4612-5561-1

[2] K. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, Semigroup Forum, 63 (2001), 278-280. https://doi.org/10.1007/s002330010042

[3] F. Kühnemund, A Hille-Yosida theorem for bi-continuous semigroups, Semigroup Forum, 67 (2003), 205-225. https://doi.org/10.1007/s00233-002-5000-3

[4] J. B. Cooper, Saks Spaces and Applications to Functional Analysis , Elsevier, 1978.

[5] K. Kruse, F. L. Schwenninger, On equicontinuity and tightness of bi-continuous semigroups, J. Math. Anal. Appl., 509 (2022), 125985. https://doi.org/10.1016/j.jmaa.2021.125985

[6] A. A. Albanese, E. Mangino, Trotter-Kato theorems for bi-continuous semigroups and applications to Feller semigroups, J. Math. Anal. Appl., 289 (2004), 477-492. https://doi.org/10.1016/j.jmaa.2003.08.032

[7] C. Budde, S. Wegner, A Lumer-Phillips type generation theorem for bi-continuous semigroups, Z. Anal. Anwend., 41 (2022), 65-80. https://doi.org/10.4171/ZAA/1695

[8] B. Farkas, Perturbations of bi-continuous semigroups, Studia Math., 161 (2004), 147-161.

[9] C. Budde, B. Farkas, A Desch-Schappacher perturbation theorem for bi-continuous semigroups, Math. Nachr., 293 (2020), 1053-1073. https://doi.org/10.1002/mana.201800534

[10] N. Gigli, Nonsmooth Differential Geometry-An Approach Tailored for Spaces with Ricci Curvature Bounded from Below , American Mathematical Society, 2018. http://doi.org/10.1090/memo/1196

[11] T. Guo, Relations between some basic results derived from two kinds of topologies for a random locally convex module, J. Funct. Anal., 258 (2010), 3024-3047. https://doi.org/10.1016/j.jfa.2010.02.002

[12] T. Guo, E. Zhang, Y. Wang, Z. Guo, Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations, J. Math. Anal. Appl., 483 (2020), 123644. https://doi.org/10.1016/j.jmaa.2019.123644

[13] M. Liu, X. Zhang, L. Dai, Trotter-Kato approximations of impulsive neutral SPDEs in Hilbert spaces, Acta Math. Sin. (Engl. Ser.), 40 (2024), 1229-1243. https://doi.org/10.1007/s10114-023-1553-8

[14] M. Liu, X. Zhang, L. Dai, Trotter-Kato approximations of semilinear stochastic evolution equations in Hilbert spaces, J. Math. Phys., 64 (2023), 043506. https://doi.org/10.1063/5.0137515

[15] M. Lučić, E. Pasqualetto, I. Vojnović, On the reflexivity properties of Banach bundles and Banach modules, Banach J. Math. Anal., 18 (2024), 7. https://doi.org/10.1007/s43037-023-00315-9

[16] D. H. Thang, T. C. Son, N. Thinh, Semigroups of continuous module homomorphisms on complex complete random normed modules, Lith. Math. J., 59 (2019), 229-250. https://doi.org/10.1007/s10986-019-09442-z

[17] T. Guo, X. Zhang, Stone’s representation theorem of a group of random unitary operators on complete complex random inner product modules, Sci. Sin. Math., 42 (2012), 181-202.

[18] X. Zhang, H. Zhang, B. Wang, M. Liu, The Cauchy initial value problem in complete random normed modules, Math. Methods Appl. Sci., 43 (2020), 4007-4015. https://doi.org/10.1002/mma.6169

[19] N. Dunford, J. T. Schwartz, Linear Operators , Interscience, 1958.

[20] X. Zhang, M. Liu, On almost surely bounded semigroups of random linear operators, J. Math. Phys., 54 (2013), 053517. https://doi.org/10.1063/1.4805049

[21] T. Guo, S. Peng, A characterization for an L(µ, K)-topological module to admit enough canonical module homomorphisms , J. Math. Anal. Appl., 263 (2001), 580-599. https://doi.org/10.1006/jmaa.2001.7637