




































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

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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.

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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.

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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)}.

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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),

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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.

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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].

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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).

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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.

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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

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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.

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	Introduction
	Preliminaries
	Main Results
	Illustrative Examples
	Example 3.4 (Damped wave equation)

	Conditions for Coincidence
	Spectral-Bound Table
	Stability Hierarchy Diagram

	Conclusion

