Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results

Abstract

Let (U(t))t≥0 be a C0-semigroup of bounded linear operators on a Banach space X. In this paper, we establish that if, for some t0>0, U(t0) is a Fredholm (resp., semi-Fredholm) operator, then (U(t))t≥0 is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that (U(t))t≥0 can be imbedded in a C0-group on X. Also we study semigroups which are near the identity in the sense that there exists t0>0 such that U(t0)−I∈𝒥(X), where 𝒥(X) is an arbitrary closed two-sided ideal contained in the set of Fredholm perturbations. We close this paper by discussing the case where 𝒥(X) is replaced by some subsets of the set of polynomially compact perturbations.

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IJMMS 2003:22, 1421–1431
PII. S0161171203011839
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
REMARKS ON EMBEDDABLE SEMIGROUPS IN GROUPS
AND A GENERALIZATION OF SOME
CUTHBERT’S RESULTS
KHALID LATRACH and ABDELKADER DEHICI
Received 5 February 2001 and in revised form 27 July 2001
Let (U(t))t≥0be a C0-semigroup of bounded linear operators on a Banach space X.
In this paper, we establish that if, for some t0> 0, U(t0) is a Fredholm (resp., semi-
Fredholm) operator, then (U(t))t≥0 is a Fredholm (resp., semi-Fredholm) semi-
group. Moreover, we give a necessary and sufficient condition guaranteeing that
(U(t))t≥0can be embedded in a C0-group on X. Also we study semigroups which
are near the identity in the sense that there exists t0> 0 such that U(t0)−I ∈ ?(X),
where ?(X) is an arbitrary closed two-sided ideal contained in the set of Fredholm
perturbations. We close this paper by discussing the case where ?(X) is replaced
by some subsets of the set of polynomially compact perturbations.
2000 Mathematics Subject Classification: 47A53, 47A55, 47D03.
1. Introduction. Let X be a Banach space over the complex field and let
?(X) denote the Banach algebra of bounded linear operators on X. The subset
of all compact operators of ?(X) is designated by ?(X). For A ∈ ?(X), we let
σ(A), ρ(A), R(λ,A), N(A), and R(A) denote the spectrum, the resolvent set,
the resolvent operator, the null space, and the range of A, respectively. The
nullity of A, α(A), is defined as the dimension N(A) and the deficiency of A,
β(A), is defined as the codimension of R(A) in X.
Write
Φ+(X) =?A ∈ ?(X) : α(A) < ∞, R(A) is closed in X?,
Φ−(X) =?A ∈ ?(X) : β(A) < ∞?then R(A) is closed in X??.
(1.1)
By Φ±(X) := Φ+(X)∪Φ−(X) we denote the set of semi-Fredholm operators in
?(X), while Φ(X) := Φ+(X)∩Φ−(X) is the set of Fredholm operators in ?(X).
If A ∈ Φ±(X), the number i(A) = α(A)−β(A), a finite or infinite integer is the
index of A. Let X∗denotes the dual space of X and A∗the dual operator of A.
Let (U(t))t≥0be a C0-semigroup of bounded linear operators on X. We say
that (U(t))t≥0 is a Fredholm (resp., semi-Fredholm) semigroup if U(t) is in
Φ(X) (resp., Φ±(X)) for all t > 0.

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KHALID LATRACH AND ABDELKADER DEHICI
In [7, Theorem 16.3.6], it is proved that a C0-semigroup of bounded linear
operators (U(t))t≥0can be embedded in a C0-group if and only if there exists
t0> 0 such that 0 ∈ ρ(U(t0)). The main goal of Section 2 is to give a gen-
eralization of this result to Fredholm semigroup. Our approach consists in
relaxing the requirement there exists t0> 0 such that 0 ∈ ρ(U(t0)) and replac-
ing it by the weaker one there exists t0> 0 such that U(t0) ∈ Φ(X). In fact,
we prove under this hypothesis that (U(t))t≥0is a Fredholm semigroup, that
is, U(t) ∈ Φ(X) for all t ≥ 0. In particular, we show that if there exists t0> 0
such that U(t0) ∈ Φ±(X), then (U(t))t≥0is a semi-Fredholm semigroup, that
is, U(t) ∈ Φ±(X) for all t ≥ 0.
In Section 3, we extend some results owing to Cuthbert [2] which deal with
C0-semigroups having the property of being near the identity, in the sense that,
for some value of t, U(t)−I ∈ ?(X). We show that Cuthbert’s results remain
valid if, for some t0> 0, U(t0)−I ∈ ?(X) where ?(X) is an arbitrary closed two-
sided ideal of ?(X) contained in the ideal of Fredholm perturbations ?(X). In
the last section, some generalizations of the results obtained in Section 3 to
polynomially compact perturbations are also given.
2. Embeddable C0-semigroups in C0-groups. Let X be a Banach space and
let (U(t))t≥0be a C0-semigroup of bounded linear operators on X.
Theorem 2.1. A C0-semigroup (U(t))t≥0can be embedded in a C0-group on
X if and only if there exists t0> 0 such that U(t0) ∈ Φ(X).
To prove Theorem 2.1, the following proposition is required.
Proposition 2.2. Let t0> 0 and let (U(t))t≥0be a C0-semigroup on X.
(i) If U(t0) ∈ Φ+(X), then U(t) ∈ Φ+(X) and α(U(t)) = 0 for all t ≥ 0.
(ii) If U(t0) ∈ Φ−(X), then U(t) ∈ Φ−(X) and β(U(t)) = 0 for all t ≥ 0.
(iii) If U(t0) ∈ Φ(X), then U(t) ∈ Φ(X) and i(U(t)) = 0 for all t ≥ 0.
Obviously, Proposition 2.2 shows that if, for some t0> 0, U(t0) ∈ Φ±(X),
then (U(t))t≥0is a semi-Fredholm semigroup. In the case where U(t0) ∈ Φ(X),
(U(t))t≥0is a Fredholm semigroup and i(U(t)) = 0 for all t ≥ 0.
Proof of Proposition 2.2. (i) We first show that U(t0) is injective. Since
α(U(t0)) < ∞, then 0 is an eigenvalue with finite multiplicity of U(t0). Let
x ?= 0 be an eigenvector associated to 0. Putting t1= t0/2, then U(t0)x =
U(t1)U(t1)x = 0, hence 0 is an eigenvalue of U(t1). Proceeding by induction,
we define a sequence (tn)n∈Nwith tn→ 0 as n → ∞ such that 0 is an eigenvalue
of U(tn), ∀n ∈ N. For n ≥ 0, we define the sets
Λn= N?U?tn
Clearly, the inclusion N(U(s)) ⊆ N(U(t)), for s ≤ t, and the compactness of
Λ0imply that (Λn)nis a decreasing sequence (in the sense of the inclusion) of
????x ∈ X : ?x? = 1?.
(2.1)

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REMARKS ON EMBEDDABLE SEMIGROUPS IN GROUPS ...
nonempty compact subsets of X. Thus?∞
??U?tn
Since tn→ 0 as n → ∞, (2.2) contradicts the strong continuity of (U(t))t≥0.
This shows that N(U(t0)) = {0}, that is, α(U(t0)) = 0.
Let 0 ≤ t ≤ t0. The inclusion N(U(t)) ⊆ N(U(t0)) implies that α(U(t)) = 0.
Assume now that t > t0and x ∈ N(U(t)), then there exists an integer n such
that nt0> t and therefore U(nt0)x = U(nt0−t)U(t)x = 0. Hence, we have
x = 0 and consequently N(U(t)) = {0} for all t > t0 which ends the proof
of (i).
(ii) To prove this item, we will proceed by duality. Let (U∗(t))t≥0be the dual
semigroup of (U(t))t≥0. Since β(U(t)) = α(U∗(t)), then it suffices to show that
α(U∗(t)) = 0 for all t ≥ 0. By hypothesis, we have α(U∗(t0)) < ∞. Let x∗be
an element of N(U∗(t0)). Arguing as above, we construct a sequence (tn)n∈N
with tn→ 0 as n → ∞ such that 0 is an eigenvalue of U∗(tn), for all n ∈ N a
decreasing sequence
1423
n=0Λn?= ∅. If x ∈?∞
∀n ≥ 1.
n=0Λn, then
?x−x??= ?x? = 1(2.2)
Σn= N?U∗?tn
????x∗∈ X∗:??x∗??= 1?
(2.3)
of nonempty compact subsets of X∗. We infer that
?∞
???U∗?tn
Using the fact that (U∗(t))t≥0is continuous in the weak∗topology at t = 0, we
conclude that
?∞
n=0Σn?= ∅. Let x∗∈
n=0Σn, then for all n ∈ N
?x∗−x∗,x???=???x∗,x???
∀x ∈ X.
(2.4)
lim
t→0
???U∗?tn
?x∗−x∗,x???= 0
∀x ∈ X.
(2.5)
Combining (2.4) and (2.5), we obtain ?x∗,x? = 0 for all x ∈ X. This shows
that x∗= 0 and therefore α(U∗(t0)) = 0. Arguing as above, we show that
α(U∗(t)) = 0 for all t ≥ 0.
(iii) This follows from (i) and (ii).
To complete the proof of (i) it suffices to show that R(U(t)) is closed in X
for all t ≥ 0. Assume that U(t0) ∈ Φ+(X), then α(U(t0)) < ∞ and β(U(t0)) = ∞
(if β(U(t0)) < ∞ the proof is contained in (ii) see below). Let U∗(to) be the dual
operator of U(t0). Obviously, U∗(t0) ∈ Φ−(X) and consequently β(U∗(t0)) <
∞. Hence β(U∗(t)) < ∞ for all t ≥ 0. Now applying Kato’s lemma [8, Lemma
332] we infer that R(U∗(t)) is closed in X∗for all t ≥ 0. This together with the
closed graph theorem of Banach [15, page 205] implies that R(U(t)) is closed
in X for all t ≥ 0.
Assume now that U(t0) ∈ Φ−(X), then β(U(t0)) < ∞ and α(U(t0)) = ∞ (if
α(U(t0)) < ∞ the proof is contained in (i)). It follows from the first part of
the statement (ii) that β(U(t)) < ∞ for all t ≥ 0. Again using Kato’s lemma

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KHALID LATRACH AND ABDELKADER DEHICI
[8, Lemma 332] we see that R(U(t)) is closed in X for all t ≥ 0 which completes
the proof of (ii).
Now if U(t0) ∈ Φ(X), then α(U(t0)) < ∞ and β(U(t0)) < ∞. It follows from
the discussion above that R(U(t)) is closed in X for all t ≥ 0. This ends the
proof of Proposition 2.2.
Proof of Theorem 2.1.
2.2 and [7, Theorem 16.3.6].
The proof follows immediately from Proposition
3. An extension of some results by Cuthbert. Throughout this section X
denotes a Banach space and (U(t))t≥0designates a strongly continuous semi-
group with infinitesimal generator A.
As mentioned in the introduction, this section is motivated by Cuthbert’s
work [2] dealing with C0-semigroups which have the property of being near the
identity, in the sense that, for some positive value of t > 0, U(t)−I ∈ ?(X). We
discuss the possibility of extending Cuthbert’s results to other operator ideals
of ?(X). To this purpose, we introduce the concept of Fredholm perturbations
(see [1, 4, 12]).
Definition 3.1. We say that an operator F ∈ ?(X) is a Fredholm pertur-
bation if A+F ∈ Φ(X) whenever A ∈ Φ(X). The operator F is called an up-
per (resp., lower) semi-Fredholm perturbation if F +A ∈ Φ+(X) (resp., F +A ∈
Φ−(X)) whenever A ∈ Φ+(X) (resp., A ∈ Φ−(X)).
The sets of Fredholm, upper semi-Fredholm, and lower semi-Fredholm per-
turbations are denoted by ?(X), ?+(X), and ?−(X), respectively. These sets of
operators were introduced and investigated in [4] (see also [12]). In particular,
it is proved that ?+(X) and ?(X) are closed two-sided ideals of ?(X) while
?−(X) is a closed subset of ?(X).
Our main objective here is to show that Cuthbert’s results remain valid if
we replace ?(X) by any closed two-sided ideal contained in ?(X).
In the following, ?(X) denotes an arbitrary nonzero closed two-sided ideal of
?(X) satisfying
?(X) ⊆ ?(X).
(3.1)
Remark 3.2. (1) It is worth noticing that, in general, the structure ideal of
?(X) is extremely complicated. Most of the results on ideal structure deal with
the well-known closed ideals which have arisen from applied work with opera-
tors.Wecanquote,forexample,compactoperators,weaklycompactoperators,
strictly singular operators, strictly cosingular operators, upper semi-Fredholm
perturbations, and Fredholm perturbations. In general, we have
?(X) ⊆ ?(X) ⊆ ?+(X) ⊆ ?(X),
?(X) ⊆ C?(X) ⊆ ?−(X) ⊆ ?(X),
(3.2)

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REMARKS ON EMBEDDABLE SEMIGROUPS IN GROUPS ...
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where ?(X) and C?(X) denote, respectively, the ideals of ?(X) consisting of
strictly singular and strictly cosingular operators on X. The inclusion ?(X) ⊆
?+(X) is due to Kato (cf. [8]) while C?(X) ⊆ ?−(X) was proved by Vladimirski˘ ı
[13].
(2) If X is isomorphic to an Lpspace with 1 ≤ p ≤ ∞ or to C(Ξ) where Ξ is a
compact Hausdorff space, then we have
?(X) ⊆ ?(X) = ?+(X) = C?(X) = ?−(X) = ?(X)
(3.3)
(cf. [9, equations (2.9) and (2.10)]).
A Banach space X is said to be an h-space if each closed infinite-dimensional
subspace of X contains a complemented subspace isomorphic to X [14]. Any
Banach space isomorphic to an h-space; c, c0and lp(1 ≤ p < ∞) are h-spaces.
In [14, Theorem 6.2], Whitley proved that, if X is an h-space, then ?(X) is the
greatest proper ideal of ?(X). This, together with (3.2), implies that
?(X) ⊆ ?+(X) = ?(X) = ?(X),
?(X) ⊆ ?−(X) ⊆ ?(X) = ?(X).
(3.4)
We denote by ? the set
? =?t > 0 such that U(t)−I ∈ ?(X)?.
(3.5)
It should be noted that for a given C0-semigroup, the set ? can be empty.
Remark 3.3. Note that, under assumption (3.1), if ? ?= ∅, then the C0-
semigroup (U(t))t≥0can be embedded in a C0-group on X. (It suffices to write
U(t0) = I +[U(t0)−I] for some to∈ ? and to apply Theorem 2.1.) This state-
ment improves [2, Theorem 1].
Observe that the relation
?U(t)−I??U(s)−I?=?U(t+s)−I?−?U(s)−I?−?U(t)−I?,
implies that
(3.6)
s ∈ ?, t ∈ ? ?⇒ s+t ∈ ?,s ∈ ?, t ∉ ? ?⇒ s+t ∉ ?.
(3.7)
It follows from these relations that? is the intersection of an additive subgroup
of real number with the positive real line. Therefore, ? may be in one of the
following forms:
(i) ? = ]0,∞[;
(ii) ? = {nx, for some x > 0; and n = 1,2,...};
(iii) ? is a dense subset of ]0,∞[ with empty interior.