Stability in Semigroups of Bounded Linear Operators: Bridging Algebraic and Analytic Notions

Otobong Johnson Tom; Otobong G. Udoaka

Otobong Johnson Tom Federal University of Technology, Ikot Abasi
Otobong G. Udoaka Akwa Ibom State University, Ikot Akpaden

Keywords: Semigroup theory, KW-stability, analytic stability;, spectral bound, Green's relations, evolution equations

Abstract

Stability is a cornerstone in the theory of semigroups, shaping the study of evo-
lution equations, operator theory, and algebraic structures. Yet, algebraic and ana-
lytic perspectives on stability have traditionally developed in isolation. This paper
builds a novel bridge between the two. Tom, Udoaka and Udoa (2025) established
that every strongly continuous (C0) semigroup of bounded linear operators is stable
in the sense of Koch and Wallace (KW), a universal algebraic property that forces
Green's relations to collapse (D = J = L = R). This recognition is new in operator
semigroup theory, where stability has typically been studied only in analytic terms.
We further provide precise spectral conditions under which KW-stability aligns with
analytic stability notionsstrong, asymptotic, exponential, and uniformthereby
unifying algebraic semigroup stability with spectral/operator-theoretic stability. Il-
lustrative examples, including the translation, right shift, heat, and damped wave
semigroups, demonstrate the stability gap and the exact conditions under which
the two approaches coincide. The study is signicant because it supplies a uni-
versal structural property of operator semigroups, a spectral criterion for analytic
decay, and practical insights for evolution equations, control design, and numerical
discretization.

References

[1] S.B. Koch and A.D. Wallace, Stability in semigroups, Duke Math. J., vol. 23, pp.
193202, 1956.
[2] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Math-
ematical Society, 1957.
[3] O. J. Tom, O. G. Udouaka, and E. S. Udoa KWStability in Semigroup of
Bounded Linear Operators, International Journal of Applied Science and Mathe-
matical Theory E- ISSN 2489-009X P-ISSN 2695-1908, Vol. 11 No. 7, pp. 914 2025
www.iiardjournals.org
[4] J. East and P. Higgins, Stability in semigroups: Green's relations and beyond,
Semigroup Forum, 101:125, 2020.
11
[5] A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in
Mathematics 10, A K Peters, 2005.
[6] J. Glück and A. Mironchenko, Stability criteria for positive semigroups on or-
dered Banach spaces, J. Evol. Equ, vol. 25, No. 12, 1424-3199/25/010001-49, 2024.
https://doi.org/10.1007/s00028-024-01044-8.
[7] J. Mui, Spectral properties of locally eventually positive operator semigroups, Semi-
group Forum, vol. 106, No. 2, pp. 460480, 2023.
[8] R. C. Penney, Self-dual cones in Hilbert space, J. Funct. Anal., vol. 21, pp. 305315,
1976.
[9] H. H. Schaefer, Halbgeordnete lokalkonvexe Vektorräume, III. Math. Ann., 141, pp.
113142, 1960.
[10] H. H. Schaefer, Invariant ideals of positive operators in C(X), I, Ill. J. Math., 11,
pp. 703715, 1967.
[11] H. Vogt, Stability of uniformly eventually positive C0−semigroups on Lp−spaces,
Proc. Am. Math. Soc., vol. 150 No. 8, pp. 35133515, 2022.
[12] L. Weis, The stability of positive semigroups on Lp spaces, Proc. Am. Math. Soc.,
vol. 123, No. 10, pp. 30893094, 1995.
[13] L. Weis, A short proof for the stability theorem for positive semigroups on Lp(μ).
Proc. Am. Math. Soc., vol. 126, No. 11, pp. 32533256, 1998.
[14] A. W. Wickstead, Compact subsets of partially ordered Banach spaces, Math. Ann.,
212, pp. 271284, 1975.
[15] P. P. Zabreko and S. V. Smickih, A theorem of M. G. Kren and M. A. Rutman,
Funktsional. Anal. i Prilozhen, vol. 13, No. 3, pp. 8182, 1979.
[16] B. Simon, The bound state of weakly coupled Schrödinger operators in one and two
dimensions, Ann. Phys., 97, pp. 279288, 1976.
[17] Josep Martinez and José M. Mazon. C0−semigroups norm continuous at innity,
Semigroup Forum, vol. 52, No. 2, pp. 213224, 1996.
[18] Desheng Li and Mo Jia, A dynamical approach to the Perron-Frobenius theory and
generalized Krein-Rutman type theorems, J. Math. Anal. Appl., vol. 496(2):Paper
No. 124828, 22, 2021.
[19] Matthias Keller, Daniel Lenz, Hendrik Vogt, and Radosª aw Wojciechowski, Note on
basic features of large time behaviour of heat kernels, J. Reine Angew. Math., 708,
pp. 73-95, 2015.
[20] O. G. Udoaka. Rank of some semigroups, International Journal of Applied Science
and Mathematical Theory, vol. 9 no. 3, pp. 90-100, 2023.
12
[21] M. N. John, and O. G. Udoaka, Algebraic and Topological Analysis of En-
veloping Semigroups in Transformation Groups: Proximal Equivalence and Ho-
momorphic Image, IJO - International Journal of Mathematics, vol. 6, no.
12, pp. 923, 2023. http://ijojournals.com/index.php/m/article/view/769, DOI;
https://doi.org/10.5281/zenodo.10443958.
[22] R. U. Ndubuisi, O. G. Udoaka, K P Shum, and R B Abubakar, On Homomorphisms
(Good Homomorphisms) Between Completely J◦- Simple Semigroups, Canadian
Journal of Pure and Applied Sciences, vol. 13, no. 2, pp. 4793-4797, 2019.
[23] R. U. Ndubisi and O. G. Udoaka, A Structure Theorem for Left Restriction Semi-
groups of Type F, International Journal of Semigroup Theory Appl., vol. 2, 2018.
[24] O. J. Tom, and O. G. Udoaka, Semigroup Approach for the Solution of Boundary
Layer Euation with Sinc Function Term, IJO - International Journal of Mathematics,
vol. 8, issue 4, pp. 2237. https://ijojournals.com/index.php/m/article/view/1057.
[25] A.H. Cliord and G.B. Preston, The Algebraic Theory of Semigroups, Vols. I & II,
AMS Mathematical Surveys, 1961/1967.
[26] J.A. Green, On the structure of semigroups, Annals of Mathematics, vol. 54, pp.
163172, 1951.
[27] D. Daners, J. Glück, and J. B. Kennedy, Eventually positive semigroups of linear
operators, J. Math. Anal. Appl., 433(2), pp. 15611593, 2016.
[28] M. D. Donsker and S. R. Srinivasa Varadhan, On a variational formula for the
principal eigenvalue for operators with maximum principle, Proc. Natl. Acad. Sci.
USA, 72, pp. 780783, 1975.
[29] M. D. Donsker and S. R. Srinivasa Varadhan. On the principal eigenvalue of second-
order elliptic dierential operators, Commun. Pure Appl. Math., 29, pp. 595621,
1976.
[30] S. Friedland, Characterizations of the spectral radius of positive operators, Linear
Algebra Appl., 134, pp. 93105, 1990.
[31] S. Friedland, The Collatz-Wielandt quotient for pairs of nonnegative operators, Appl.
Math., Praha, 65(5), pp. 557597, 2020.
[32] J. Mui, Spectral properties of locally eventually positive operator semigroups, Semi-
group Forum, 106(2), pp. 460480, 2023.
[33] J. East and P. M. Higgins, Green's relations and stability for subsemigroups, Semi-
group Forum , vol. 101, no 1, pp. 7786, 2020.