Stability in Semigroups of Bounded Linear Operators: Bridging Algebraic and Analytic Notions 1Otobong J. Tom∗, and 2Otobong G. Udoaka. 1 Fed. Uni. Tech., Ikot Abasi & Akwa Ibom State Uni., Ikot Akpaden. 2Akwa Ibom State University, Ikot Akpaden. 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 Udo�a (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 notions�strong, asymptotic, exponential, and uniform�thereby 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 signi�cant 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. Keywords: Semigroup theory; KW-stability; analytic stability; spectral bound; Green's relations; evolution equations. 1 Introduction The concept of stability is central to mathematics, capturing how systems behave under iteration, evolution, or perturbation. Within algebraic semigroup theory, stability was formally introduced by Koch and Wallace (1956) [1], who de�ned a semigroup S to be stable if aS ⊆ abS =⇒ aS = abS, Sa ⊆ Sab =⇒ Sa = Sab. This condition enforces the collapse of Green's relations�fundamental equivalence rela- tions describing semigroup structure�so that D = J = L = R. Stability in this sense IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 10 has deep structural implications, simplifying semigroup decompositions and embedding properties [4, 20, 21, 22, 23, 24, 25, 26]. In operator theory and functional analysis, a di�erent approach to stability has devel- oped through the study of C0-semigroups, which arise naturally in solving the abstract Cauchy problem du dt = uAu, (0) = u0, where A is the generator [2, 6, 11, 12, 13, 15]. Here, stability is measured analytically in terms of operator norms and spectral conditions. Classical notions include asymptotic stability, strong stability, exponential stability, and uniform stability, each re�ecting dif- ferent aspects of long-time behaviour. These notions are closely tied to the spectrum of the generator and results such as the Gearhart�Prüss theorem [5, 7, 8, 9, 10, 17, 32]. Although both traditions revolve around the idea of stability, they have historically evolved in relative isolation: the algebraic approach is structural and norm-free, while the analytic approach is spectral and dynamical. A recent advance has begun to bridge this divide. Tom, Udoaka, and Udo�a (2025) [3] introduced KW-stability into the setting of semigroups of bounded linear operators, proving that every strongly continuous (C0) semigroup on a Banach space is stable in the sense of Koch�Wallace. This recognition provides a new link between classical semigroup stability theory and operator semigroup analysis, placing algebraic stability at the foundation of analytic operator theory. Their work suggests further directions for stability research, including the study of unbounded operator semigroups, hypersemigroups, and semigroups arising in stochastic analysis [7, 9, 16, 32]. In what follows, we continue this line of investigation by examining how KW-stability interacts with analytic stability notions. In particular, we identify the spectral conditions under which algebraic and analytic stability coincide and illustrate this interplay with canonical examples such as translation, shift, heat, and damped wave semigroups. 2 Preliminaries De�nition 2.1 (Normed linear space). A normed linear space is a pair (X, ∥ · ∥) where X is a vector space over the �eld R or C, and ∥ · ∥ : X → [0,∞) is a function, called a norm, satisfying the following properties for all x, y ∈ X and all scalars α: 1. Positivity: ∥x∥ ≥ 0, and ∥x∥ = 0 if and only if x = 0. 2. Homogeneity (absolute scalability): ∥αx∥ = |α| ∥x∥. 3. Triangle inequality: ∥x+ y∥ ≤ ∥x∥+ ∥y∥. De�nition 2.2 (Banach space). A Banach space is a vector space X over the �eld R or C together with a norm ∥ · ∥ : X → [0,∞) such that (X, ∥ · ∥) is complete; that is, every Cauchy sequence {xn} in X converges to some x ∈ X with respect to the norm ∥ · ∥. Formally, for every sequence {xn} in X, if lim m,n→∞ ∥xn − xm∥ = 0, then there exists x ∈ X such that lim n→∞ ∥xn − x∥ = 0. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 11 De�nition 2.3 (Bounded linear operator). Let X and Y be normed linear spaces. A mapping T : X → Y is called a linear operator if T (αx+ βy) = αT (x) + βT (y), ∀ x, y ∈ X, α, β ∈ R or C. The operator T is said to be bounded if there exists a constant M > 0 such that ∥T (x)∥Y ≤ M∥x∥X , ∀ x ∈ X. Equivalently, T is bounded if and only if it is continuous at 0 (and hence continuous everywhere). De�nition 2.4 (Operator Semigroup). Let X be a Banach space and B(X) the algebra of all bounded linear operators on X. A family {T (t)}t≥0 ⊆ B(X) is called a strongly continuous semigroup (C0-semigroup) if: 1. T (0) = I (the identity operator), 2. T (t+ s) = T (t)T (s) for all t, s ≥ 0, 3. For every x ∈ X, limt→0+ T (t)x = x. For more about this, the reader is referred to [3]. De�nition 2.5 (Koch and Wallace Stability [1]). A semigroup S is called stable if for all a, b ∈ S: � (Right stability): aS ⊆ abS =⇒ aS = abS, � (Left stability): Sa ⊆ Sab =⇒ Sa = Sab. This de�nition was also given by East using Green's relation in [3, 33]. Equivalently, if a ≤J ab, then aRab, and if a ≤J ba, then aLba. 3 Main Results Proposition 3.1 (KW-Stability of C0−Semigroups [3]). Every C0-semigroup of bounded linear operators is stable in the sense of Koch�Wallace. Proof. Let S = {T (t) : t ≥ 0}. Pick a = T (t), b = T (s) with t, s ≥ 0. By the semigroup law, ab = T (t)T (s) = T (t+ s) = T (s+ t) = T (s)T (t) = ba, so S is commutative. Suppose aS ⊆ abS. For x = T (u) ∈ S, (ab)x = T (t+ s)T (u) = T (t+ s+ u). By commutativity, (ab)x = aT (s+u) ∈ aS. Thus (ab)S ⊆ aS. Combined with aS ⊆ abS, we get aS = abS. Similarly, if Sa ⊆ Sab, then for x = T (u) ∈ S, x(ab) = T (u)T (t+ s) = T (u+ t+ s) = T (u+ t)T (s) = (xa)b ∈ Sa, so Sab ⊆ Sa, hence Sa = Sab. Therefore S satis�es KW-stability. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 12 Theorem 3.2 (Equivalence of Green's Relations [3]). For a C0-semigroup of bounded linear operators, D = J = L = R. Proof. Since S is commutative, left and right ideals coincide: Sa = aS, so L = R. For any a ∈ S, SaS = {xay : x, y ∈ S}. But commutativity gives xay = (xy)a ∈ Sa, so SaS ⊆ Sa. Conversely, for xa ∈ Sa, write xa = (x)a · I ∈ SaS. Thus Sa = SaS, so J = L. Finally, D = L ◦ R and L = R imply D = L = R = J . 4 Illustrative Examples Example 4.1 (Translation Semigroup). Let X = C0(R), the Banach space of continuous functions on R vanishing at in�nity, equipped with the supremum norm ∥f∥∞ = sup x∈R |f(x)|. De�ne a family of operators {T (t)}t≥0 by (T (t)f)(x) = f(x+ t), f ∈ X, t ≥ 0, x ∈ R. For Strong continuity. We verify that {T (t)}t≥0 is a strongly continuous semigroup. For �xed f ∈ X, ∥T (t)f − f∥∞ = sup x∈R |f(x+ t)− f(x)|. Since f is uniformly continuous on R (as every f ∈ C0(R) is uniformly continuous), the right-hand side tends to zero as t → 0. Hence, lim t→0+ ∥T (t)f − f∥∞ = 0, so {T (t)}t≥0 is a C0-semigroup. For In�nitesimal generator. Let A denote the generator of {T (t)}t≥0. By de�ni- tion, Af = lim t→0+ T (t)f − f t f, ∈ D(A), where the domain consists of those f ∈ X for which the above limit exists in X. A direct computation shows that T (t)f(x)− f(x) t = f(x+ t)− f(x) t . Thus, the limit exists precisely when f is continuously di�erentiable with derivative van- ishing at in�nity. Therefore, Af = f ′, D(A) = {f ∈ C0(R) : f ′ ∈ C0(R)}. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 13 For KW-stability. By Proposition 3.1, every C0-semigroup on a Banach space is stable in the sense of Koch�Wallace. Explicitly, for f ∈ X and t, s ≥ 0, T (t+ s)f = T (t)T (s)f, and the KW-condition T (t)X ⊆ T (t+ s)X ⇒ T (t)X = T (t+ s)X is satis�ed. Thus the translation semigroup is KW-stable. For Analytic behavior. We compute the operator norm: ∥T (t)∥ sup= ∥f∥∞=1 ∥T (t)f∥∞. But for any f ∈ X, ∥T (t)f∥∞ = sup x∈R |f(x+ t)| = sup y∈R |f(y)| = ∥f∥∞. Hence, ∥T (t)∥ = 1 for all t ≥ 0. Therefore, there is no decay as t → ∞, and the semigroup fails to be analytically stable (in the sense of uniform exponential stability). Graphical interpretation. The operator T (t) acts as a horizontal shift of the function graph. For example, if f(x) = e−x2 is a bell-shaped curve centered at the origin, then T (1)f(x) = f(x+1) is the same curve shifted left by one unit. Importantly, the height of the curve is unchanged, so ∥T (t)f∥∞ = ∥f∥∞ for all t ≥ 0. This shows why the semigroup is KW-stable (algebraically the orbits are preserved) but not analytically stable (no decay in norm). x f(x) f(x) = e−x2 T (1)f(x) = f(x+ 1) 0−1 Figure 1: Translation semigroup illustrated on f(x) = e−x2 . The original function (blue) and its translated version (red dashed) have identical amplitude but shifted position, explaining why KW-stability holds while analytic stability fails. Example 4.2 (Right Shift on ℓ2). Let X = ℓ2(N) with norm ∥x∥2 = ∑ k≥1 |xk|2. De�ne T (n) for n ∈ N by T (n)(x1, x2, x3, . . . ) = (0, . . . , 0︸ ︷︷ ︸ n , x1, x2, x3, . . . ). For Semigroup property. For m,n ∈ N, T (m)T (n) = T (m+ n), IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 14 so {T (n)}n∈N is a (discrete) semigroup. For KW-stability. The orbit structure is preserved under shifts: T (m)T (n) = T (m+ n) and the KW-condition T (n)X ⊆ T (n+m)X ⇒ T (n)X = T (n+m)X holds. For Isometry / norm. For any x ∈ ℓ2, ∥T (n)x∥2 = ∑ k≥1 |(T (n)x)k|2 = ∑ k≥1 |xk|2 = ∥x∥2, so ∥T (n)∥ = 1 for all n: each T (n) is an isometry. For Analytic behaviour. Since there is no decay (∥T (n)∥ = 1 always), the semi- group is not analytically (exponentially) stable. Interpretation. The right-shift moves entries to the right while preserving total ℓ2 energy. Algebraic stability (KW) holds but analytic decay does not. x1 x2 x3 · · · 0 x1 x2 x3 · · · Original After T (1) Figure 2: Schematic of the right shift T (1) on ℓ2: each component moves one box to the right and a 0 is inserted at the left. Example 4.3 (Heat Semigroup). Let X = L2(Rn), the Hilbert space of square-integrable functions on Rn. De�ne, for t > 0, (T (t)f)(x) = (Gt ∗ f)(x), Gt(x) = (4πt)−n/2e− |x|2 4t . Here Gt is the Gaussian heat kernel, representing the fundamental solution of the heat equation. For Strong continuity. For each f ∈ L2(Rn), the convolution T (t)f = Gt ∗ f de�nes a continuous function of t. As t → 0+, Gt tends to the Dirac delta distribution δ, so T (t)f → f in L2, ensuring lim t→0+ ∥T (t)f − f∥2 = 0. Hence {T (t)}t≥0 is a strongly continuous semigroup (a C0-semigroup). For The generator. We recall that T (t) solves the Cauchy problem for the heat equation: ∂u ∂t = ∆ uu, (0, x) = f(x). Thus, the generator is the Laplacian operator Af = ∆f, D(A) = H2(Rn) = {f ∈ L2(Rn) : ∆f ∈ L2(Rn)}. This follows by di�erentiating T (t)f at t = 0 under the Fourier transform. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 15 For KW-stability. By Proposition 3.1, every C0-semigroup is KW-stable in the algebraic sense. In particular, for the heat semigroup, {T (t)f : t ≥ 0} = {T (s+ t)f : t ≥ 0}, for all s ≥ 0, re�ecting the invariance of reachable states. For Analytic stability. The Fourier transform of Gt is given by Ĝt(ξ) = e−t|ξ|2 , so in the Fourier domain, T (t)f has the multiplier e−t|ξ|2, which decays exponentially in t for each ξ ̸= 0. Since the spectrum of ∆ is σ(∆) = (−∞, 0], the spectral bound is strictly negative. Therefore, there exists ω > 0 such that ∥T (t)f∥2 ≤ e−ωt∥f∥2, ∀t ≥ 0. Hence the semigroup is not only contractive but also exponentially stable. Interpretation. In this example, algebraic stability (KW-stability) and analytic stabil- ity (exponential decay of norms) coincide. Unlike the translation and shift semigroups, which preserve norm without decay, the heat semigroup smooths and dissipates initial data over time. Physically, this corresponds to the di�usion of heat: local peaks �atten, energy spreads out, and the system relaxes exponentially fast. −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 0 0.2 0.4 x G t( x ) t = 0.5 t = 1 t = 2 Figure 3: Gaussian heat kernel Gt(x) at di�erent times t = 0.5 (blue), t = 1 (red), and t = 2 (green). Graphical interpretation (Figure 3). The curves show the Gaussian kernel Gt(x) for di�erent times: � At t = 0.5 (blue): the kernel is tall and narrow, concentrated near x = 0. Heat is still localized. � At t = 1 (red): the peak is lower but wider, showing partial di�usion and �attening of the initial concentration. � At t = 2 (green): the kernel is very �at and spread out, indicating that heat has dissipated signi�cantly. Thus, as t → ∞, the peak decays while the width grows like √ t. This illustrates exponen- tial stability in the semigroup sense and di�usion in the physical sense. For detailed expositions of heat semigroups and their stability properties, see [16, 18, 19, 27, 28, 29, 30, 31]. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 16 4.1 Example 3.4 (Damped wave equation) We consider the damped wave equation on the bounded interval (0, L) with homogeneous Dirichlet boundary conditions: utt(x, t) + αut(x, t)− uxx(x, t) = 0, x ∈ (0, L), t > 0, u(0, t) = u(L, t) = 0 t, ≥ 0, u(x, 0) = u0(x) u, t(x, 0) = v0(x) x, ∈ (0, L), (1) where α ∈ R is the (constant) damping coe�cient. State space and energy. Set X = H1 0 (0, L)× L2(0, L), with state variable U(t) = (u(·, t), ut(·, t))⊤. We equip X with the energy inner product 〈 (u1, v1), (u2, v2) 〉 X := ∫ L 0 u′ 1(x)u ′ 2(x) dx+ ∫ L 0 v1(x)v2(x) dx, and corresponding norm ∥(u, v)∥2X = ∥u′∥2L2(0,L) + ∥v∥2L2(0,L). The physical energy associated to a solution of (1) is E(t) = 1 2 ( ∥ut(·, t)∥2L2 + ∥ux(·, t)∥2L2 ) . First-order formulation and generator. Write (1) as a �rst-order system U ′(t) = AU(t) by setting U = (u, v)⊤ with v = ut. De�ne A ( u v ) = ( v uxx − αv ) , D(A) = ( H2(0, L) ∩H1 0 (0, L) ) ×H1 0 (0, L). Equivalently, in matrix form, A = ( 0 I ∂xx −αI ) , D(A) = (H2 ∩H1 0 )×H1 0 . Generation of a C0-semigroup (sketch). The operator ∂xx with Dirichlet bound- ary conditions is self-adjoint and has compact inverse on L2(0, L). Standard results for second-order hyperbolic operators with bounded damping (see [7, 9, 10, 32]) imply that A with domain above is the generator of a C0-semigroup {T (t)}t≥0 on X. In particular: - For α ≥ 0 the semigroup is contractive with respect to a suitable equivalent energy norm (damping is nonnegative). - For α < 0 the operator has a component that can generate growth (negative damping gives energy injection). We therefore treat {T (t)} as the evolution operator for (1). IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 17 Modal decomposition and spectrum. Let {φn}n≥1 denote the Dirichlet Laplacian eigenfunctions φn(x) = sin (nπx L ) , −φ′′ n = ω2 nφn, ωn = nπ L , n ∈ N. Expand the solution as u(x, t) = ∑ n≥1 qn(t)φn(x). Each modal coe�cient satis�es the scalar ODE q′′n(t) + αq′n(t) + ω2 nqn(t) = 0. The characteristic equation is λ2 + αλ+ ω2 n = 0 with roots λ± n = −α± √ α2 − 4ω2 n 2 . Hence ℜ(λ± n ) ≤ −α 2 for every n ≥ 1, and the spectral bound of A satis�es s(A) := sup{ℜλ : λ ∈ σ(A)} = −α 2 . (Here we used that the full spectrum of A consists of these modal eigenvalues due to compactness of the spatial resolvent and separation of variables.) Energy identity and exponential decay for α > 0. Multiply (1) by ut and integrate over (0, L) to obtain the standard energy balance: d dt E(t) = −α ∫ L 0 |ut(x, t)|2 dx ≤ 0. Thus energy is nonincreasing. To obtain exponential decay we combine this dissipation with a coercivity (Poincaré) inequality: for u ∈ H1 0 (0, L), ∥u∥L2 ≤ 1 ω1 ∥u′∥L2 , ω1 = π L . Using the energy E(t) and the modal spectral gap one can show (standard multiplier or resolvent estimates; see [27, 28, 29, 30, 31]) that there exist constants M ≥ 1 and γ > 0 (depending on α and L) such that ∥T (t)∥L(X) ≤ Me−γt, t ≥ 0. A simple modal estimate gives a concrete lower bound γ ≥ α/2 in the case of uniform damping (constant α); more careful resolvent estimates may yield the optimal rate γ = α/2 when the Poincaré constant is accounted for. Non-decay when α = 0. If α = 0 equation (1) reduces to the undamped wave equa- tion. Modal eigenvalues are purely imaginary λ± n = ±iωn, so s(A) = 0. Energy is conserved (dE/dt = 0) and no decay of the norm occurs in general (solutions persist as undamped oscillations). Thus analytic stability fails when α = 0. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 18 Instability for α < 0. If α < 0 the modal real parts satisfy ℜ(λ± n ) ≥ −α/2 > 0 (note sign), and high modes may exhibit growth; hence the semigroup is not stable and solutions typically grow exponentially [6, 7, 11, 19]. Remarks. � The exponential decay argument above uses that damping is uniform (constant α > 0) and the spatial domain is bounded so the Laplacian has compact resolvent. For localized damping (e.g. α(x) ≥ 0 supported only on a subregion) exponential decay may fail or require geometric control/observability conditions (see Bardos� Lebeau�Rauch-type results). � The modal description also explains why s(A) = −α/2 in this uniform case: the real parts of all modal eigenvalues are bounded above by −α/2. Thus the spectral criterion s(A) < 0 is equivalent to α > 0 here. Conclusion for Example 3.4. With the state space X = H1 0 (0, L) × L2(0, L) and generator A de�ned above, the semigroup {T (t)} satis�es: � KW-stability for all α ∈ R (algebraic property of the one-parameter family). � Exponential (analytic) stability if and only if α > 0 (spectral bound negative). � Conservation of energy (no decay) when α = 0. � Instability when α < 0 [6, 11, 12, 13, 24]. For more about PDE: see [8, 14, 18, 19] for generation results, spectral mapping, and standard energy/multiplier proofs; for localized damping and geometric control see the survey by Lebeau and Rauch and the literature cited therein. 5 Conditions for Coincidence Theorem 5.1 (Coincidence of Stability). Let {T (t)} be a C0-semigroup with generator A. KW-stability and analytic stability yield the same conclusion i� s(A) < 0 and σ(T (t)) \ {0} = etσ(A). Proof. If s(A) < 0, then r(T (t)) = ets(A) < 1 for t > 0. By spectral mapping and Gearhart�Prüss theorem, exponential stability follows. Conversely, if s(A) ≥ 0, analytic decay fails although KW-stability holds (Examples 4.1, 4.2). Thus both s(A) < 0 and spectral mapping are necessary and su�cient. 5.1 Spectral-Bound Table Spectral bound s(A) KW-stabilityAnalytic conclusionSpectral mapping s(A) < 0 Always trueExponentially stableHolds s(A) = 0 Always trueNeutral (no decay)Holds s(A) > 0 Always trueUnstable (growth)Holds IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 19 Stability Hierarchy Diagram5.2 Exponential Stability Uniform Stability Strong Stability Asymptotic Stability Koch�Wallace Stability (Always holds) Figure 4: Hierarchy of analytic stability notions with Koch�Wallace stability in parallel. 6 Conclusion This work has revealed a fundamental connection between algebraic and analytic stability in semigroups of bounded linear operators. We showed that every C0−semigroup is stable in the sense of Koch�Wallace, a universal algebraic property that enforces the collapse of Green's relations. At the same time, we identi�ed precise spectral conditions under which this algebraic stability coincides with analytic stability in the form of decay properties such as strong, asymptotic, and exponential stability. The examples of the translation semigroup, right shift, heat semigroup, and damped wave semigroup illustrate the subtle boundary between structural invariance and spectral decay, giving rise to what may be described as a stability gap. This recognition clari�es why operator semigroups can exhibit robust algebraic structure while displaying very di�erent analytic behavior depending on their spectral placement. Beyond its theoretical interest, the study o�ers insight into the analysis of evolution equations, the design of stable control systems, and the assessment of numerical schemes where stability properties are decisive. By showing that KW-stability is always present while analytic stability is conditional, we provide a uni�ed framework that advances semi- group theory and strengthens its applications in mathematics, physics, and engineering. References [1] S.B. Koch and A.D. Wallace, �Stability in semigroups,� Duke Math. J., vol. 23, pp. 193�202, 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. Udo�a KW�Stability 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. 9�14 2025 www.iiardjournals.org [4] J. East and P. 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Mui, Spectral properties of locally eventually positive operator semigroups, Semi- group Forum, 106(2), pp. 460�480, 2023. [33] J. East and P. M. Higgins, �Green's relations and stability for subsemigroups,� Semi- group Forum , vol. 101, no 1, pp. 77�86, 2020. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 22 Introduction Preliminaries Main Results Illustrative Examples Example 3.4 (Damped wave equation) Conditions for Coincidence Spectral-Bound Table Stability Hierarchy Diagram Conclusion