Abstract
We prove a Hille–Yosida type theorem for relatively uniformly continuous positive semigroups on vector lattices. We introduce the notions of relatively uniformly continuous, differentiable, and integrable functions on ℝ+. These notions allow us to study the generators of relatively uniformly continuous semigroups. Our main result provides sufficient and necessary conditions for an operator to be the generator of an exponentially order bounded, relatively uniformly continuous, positive semigroup.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. D. Aliprantis and R. Tourky, Cones and Duality, Graduate Studies in Mathematics, vol. 84, American Mathematical Society (Providence, RI, 2007).
A. Bátkai, M. Kramar Fijavž, and A. Rhandi, Positive Operator Semigroups: From Finite to Infinite Dimensions, Operator Theory: Advances and Applications, vol. 257, Birkhäuser/Springer (Cham, 2017).
K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag (New York, 2000).
E. Hille, Functional Analysis and Semi-Groups, AMS Colloquium Publications, vol. 31, American Mathematical Society (New York, 1948).
M. Kandić and M. Kaplin, Relatively uniformly continuous semigroups on vector lattices, arXiv:1807.02543 (2018).
W. A. J. Luxemburg and L. C. Moore, Jr, Archimedean quotient Riesz spaces, Duke Math. J., 34 (1967), 725–739.
W. A. J. Luxemburg and A. C. Zaanen, The linear modulus of an order bounded linear transformation. I, Indag. Math., 33 (1971), 422–434.
W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces, vol. I, North-Holland Mathematical Library, North-Holland Publishing Co. (Amsterdam-London), American Elsevier Publishing Co. (New York, 1971).
L. C. Moore, Jr, The relative uniform topology in Riesz spaces, Indag. Math., 30 (1968), 442–447.
A. L. Peressini, Ordered Topological Vector Spaces, Harper & Row, Publishers (New York-London, 1967).
W. Rudin, Principles of Mathematical Analysis, International series in pure and applied mathematics, 3rd edition, McGraw-Hill (1976).
M. A. Taylor and V. G. Troitsky, Bibasic sequences in Banach lattices. J. Funct. Anal., 278 (2020), 108448.
B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, translated from Russian by Leo F. Boron, with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Is´eki, Wolters-Noordhoff Scientific Publications, Ltd. (Groningen, 1967).
K. Yosida, On the differentiability and the representation of one-parameter semi-group of linear operators, J. Math. Soc. Japan, 1 (1948), 15–21.
Acknowledgements
We would like to express our gratitude to Marko Kandić for insightful comments and meticulous proofreading which improved the manuscript. The first author would also like to thank Marko for professional guidance. Special thanks belongs to Jochen Glück for his valuable remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors acknowledge financial support from the Slovenian Research Agency, Grant No. P1-0222.
Rights and permissions
About this article
Cite this article
Kaplin, M., Fijavž, M.K. Generation of relatively uniformly continuous semigroups on vector lattices. Anal Math 46, 293–322 (2020). https://doi.org/10.1007/s10476-020-0025-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10476-020-0025-y