© Hindawi Publishing Corp. REMARKS ON EMBEDDABLE SEMIGROUPS IN GROUPS AND A GENERALIZATION OF SOME CUTHBERT’S RESULTS

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., semiFredholm) 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 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) .F orA ∈ (X) ,w e 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


Introduction.
Let X be a Banach space over the complex field and let ᏸ(X) denote the Banach algebra of bounded linear operators on X.The subset of all compact operators of ᏸ(X) is designated by (X).For A ∈ ᏸ(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 By Φ ± (X) := Φ + (X) ∪ Φ − (X) we denote the set of semi-Fredholm operators in ᏸ(X), while Φ(X) := Φ + (X) ∩ Φ − (X) is the set of Fredholm operators in ᏸ(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.
In [7,Theorem 16.3.6],it is proved that a C 0 -semigroup of bounded linear operators (U (t)) t≥0 can be embedded in a C 0 -group if and only if there exists t 0 > 0 such that 0 ∈ ρ(U(t 0 )).The main goal of Section 2 is to give a generalization of this result to Fredholm semigroup.Our approach consists in relaxing the requirement there exists t 0 > 0 such that 0 ∈ ρ(U(t 0 )) and replacing it by the weaker one there exists t 0 > 0 such that U(t 0 ) ∈ Φ(X).In fact, we prove under this hypothesis that (U (t)) t≥0 is a Fredholm semigroup, that is, U(t) ∈ Φ(X) for all t ≥ 0. In particular, we show that if there exists t 0 > 0 such that U(t 0 ) ∈ Φ ± (X), then (U (t)) t≥0 is 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 C 0 -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 t 0 > 0, U(t 0 )−I ∈ (X) where (X) is an arbitrary closed twosided ideal of ᏸ(X) contained in the ideal of Fredholm perturbations Ᏺ(X).In the last section, some generalizations of the results obtained in Section 3 to polynomially compact perturbations are also given.

2.
Embeddable C 0 -semigroups in C 0 -groups.Let X be a Banach space and let (U (t)) t≥0 be a C 0 -semigroup of bounded linear operators on X. Theorem 2.1.A C 0 -semigroup (U (t)) t≥0 can be embedded in a C 0 -group on X if and only if there exists t 0 > 0 such that U(t 0 ) ∈ Φ(X).
To prove Theorem 2.1, the following proposition is required.Proposition 2.2.Let t 0 > 0 and let (U (t)) t≥0 be a C 0 -semigroup on X.

Proof of Proposition 2.2. (i)
We first show that U(t 0 ) is injective.Since α(U(t 0 )) < ∞, then 0 is an eigenvalue with finite multiplicity of U(t 0 ).Let x = 0 be an eigenvector associated to 0. Putting t 1 = t 0 /2, then U(t 0 )x = U(t 1 )U (t 1 )x = 0, hence 0 is an eigenvalue of U(t 1 ).Proceeding by induction, we define a sequence (t n ) n∈N with t n → 0 as n → ∞ such that 0 is an eigenvalue of U(t n ), ∀n ∈ N.For n ≥ 0, we define the sets Clearly, the inclusion N(U(s)) ⊆ N(U(t)), for s ≤ t, and the compactness of Λ 0 imply that (Λ n ) n is a decreasing sequence (in the sense of the inclusion) of nonempty compact subsets of X.Thus Since t n → 0 as n → ∞, (2.2) contradicts the strong continuity of (U(t)) t≥0 .This shows that N(U(t 0 )) = {0}, that is, α(U(t 0 )) = 0. Let 0 ≤ t ≤ t 0 .The inclusion N(U(t)) ⊆ N(U(t 0 )) implies that α(U(t)) = 0. Assume now that t > t 0 and x ∈ N(U(t)), then there exists an integer n such that nt 0 > t and therefore U(nt 0 )x = U(nt 0 − t)U(t)x = 0. Hence, we have x = 0 and consequently N(U(t)) = {0} for all t > t 0 which ends the proof of (i).
(ii) To prove this item, we will proceed by duality.Let (U * (t)) t≥0 be 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 * (t 0 )) < ∞.Let x * be an element of N(U * (t 0 )).Arguing as above, we construct a sequence (t n ) n∈N with t n → 0 as n → ∞ such that 0 is an eigenvalue of U * (t n ), for all n ∈ N a decreasing sequence of nonempty compact subsets of X * .We infer that Using the fact that (U * (t)) t≥0 is continuous in the weak * topology at t = 0, we conclude that Combining (2.4) and (2.5), we obtain x * ,x = 0 for all x ∈ X.This shows that x * = 0 and therefore α(U * (t 0 )) = 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 be the dual operator of U(t 0 ).Obviously, U * (t 0 ) ∈ Φ − (X) and consequently β(U * (t 0 )) < ∞.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] . It follows from the first part of the statement (ii) that β(U (t)) < ∞ for all t ≥ 0. Again using Kato's lemma [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(t 0 ) ∈ Φ(X), then α(U(t 0 )) < ∞ and β(U(t 0 )) < ∞.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. 3.An extension of some results by Cuthbert.Throughout this section X denotes a Banach space and (U (t)) t≥0 designates a strongly continuous semigroup with infinitesimal generator A.

Proof of Theorem
As mentioned in the introduction, this section is motivated by Cuthbert's work [2] dealing with C 0 -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 ᏸ(X).To this purpose, we introduce the concept of Fredholm perturbations (see [1,4,12]).Definition 3.1.We say that an operator The sets of Fredholm, upper semi-Fredholm, and lower semi-Fredholm perturbations are denoted by Ᏺ(X), Ᏺ + (X), and Ᏺ − (X), respectively.These sets of operators were introduced and investigated in [4] (see also [12]).In particular, it is proved that Ᏺ + (X) and Ᏺ(X) are closed two-sided ideals of ᏸ(X) while Ᏺ − (X) is a closed subset of ᏸ(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 Ᏺ(X).
In the following, (X) denotes an arbitrary nonzero closed two-sided ideal of ᏸ(X) satisfying (X) ⊆ Ᏺ(X). (3.1) (1) It is worth noticing that, in general, the structure ideal of ᏸ(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 operators.We can quote, for example, compact operators, weakly compact operators, strictly singular operators, strictly cosingular operators, upper semi-Fredholm perturbations, and Fredholm perturbations.In general, we have where (X) and C(X) denote, respectively, the ideals of ᏸ(X) consisting of strictly singular and strictly cosingular operators on X.The inclusion (X) ⊆ Ᏺ + (X) is due to Kato (cf.[8]) while C(X) ⊆ Ᏺ − (X) was proved by Vladimirskiȋ [13].
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, c 0 and l p (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 ᏸ(X).This, together with (3.2), implies that We denote by ᏻ the set It should be noted that for a given C 0 -semigroup, the set ᏻ can be empty.
Observe that the relation implies that It follows from these relations that ᏻ is the intersection of an additive subgroup of real number with the positive real line.Therefore, ᏻ may be in one of the following forms: (ii) ᏻ = {nx, for some x > 0; and n = 1, 2,...}; (iii) ᏻ is a dense subset of ]0, ∞[ with empty interior.
The following examples taken from [2] show that all the three types of sets may occur, the above classification of ᏻ-sets is not empty; and sets of type (ii) can arise from semigroups having bounded or unbounded infinitesimal generators.
In the next theorem, we derive some relationships between the type of ᏻ-sets and the structure of the semigroup.In particular, we show that ᏻ has the first form if and only if A is a Fredholm perturbation.If ᏻ takes the third form, then A is necessarily unbounded.This result extends [2, Theorem 2] to large classes of operators which contain properly the set of compact operators.
Proof of Theorem 3.5.(i)⇒(ii).The first step in the proof of this implication consists in showing that (i) implies that A is bounded.The proof of this implication is similar to that of [2, Theorem 2].Details are omitted.
Next, since A is bounded, then U(t) is uniformly continuous for t ≥ 0 (see [7]).Hence, for all ε > 0 there exists δ > 0 such that U(t)− I < ε for t < δ. (3.8) Accordingly, for any t < δ, we have Hence, for ε small enough, t 0 U(s)ds is invertible for all t < δ.Moreover, using the identity together with the fact that A and U(t) commute, we infer that Since (X) is an ideal, we infer that A ∈ (X).
(ii)⇒(i).Assume that A ∈ (X).Using again identity (3.10) and the ideal structure of (X) we see that U(t)− I ∈ (X) for all t ≥ 0.

R(λ, A) U(t)− I = λR(λ, A)
The next result asserts that if the ᏻ-set is in the form (iii), then the infinitesimal generator of (U (t)) t≥0 is necessarily unbounded.It generalizes [2, Theorem 3].Proposition 3.6.Assume that condition (3.1) holds true.If ᏻ is a dense subset of ]0, ∞[ with no interior points, then A is unbounded.
Proof.Assume, for contradiction, that A is bounded.Then, proceeding as in the proof of the implication (i)⇒(ii) in Theorem 3.5 we see that if t < δ and t ∈ ᏻ, then A ∈ (X).So, by Theorem 3.5, we get ᏻ = ]0, ∞[.This contradicts the hypothesis.Remark 3.7.(1) Notice that if (X) is a nonzero closed two-sided ideal of ᏸ(X) satisfying (3.1), then it follows from [4, Proposition 4, page 70] that Ᏺ 0 (X) ⊆ (X) ⊆ Ᏺ(X), (3.15) where Ᏺ 0 (X) stands for the ideal of finite rank operators on X.This shows that Ᏺ 0 (X) is the minimal ideal (in the sense of the inclusion) in ᏸ(X) for which the results of this section are valid.Evidently, if X has the approximation property, then we have Ᏺ 0 (X) = (X).
(2) Even though the description of the ideal structure of ᏸ(X) is a complex task, there exist some Banach spaces X for which ᏸ(X) has only one proper nonzero closed two-sided ideal.The first result in this direction was established by Calkin (cf.[4]).He proved that if X is a separable Hilbert space, then (X) is the unique proper nonzero closed two-sided ideal of ᏸ(X).An extension of this result was obtained by Gohberg et al. [4].They proved the same result for X = l p , 1 ≤ p < ∞, and X = c 0 .In [6], Herman establishes the same result for a large class of Banach spaces, namely Banach spaces which have perfectly homogeneous block bases and satisfy (+) (for the definition and the meaning of the symbol (+) we refer to [6]).(Evidently, the spaces l p , 1 ≤ p < ∞, and c 0 belong to this class.)Thus, if X has perfectly homogeneous block bases which satisfy (+), then (3.16) Consequently, for this class of spaces the results of this section use the ideal of compact operators and coincide with those obtained in [2].Hence, for such spaces the Cuthbert results are optimal.
4. Further extensions.Let X be a Banach space.An operator R ∈ ᏸ(X) is called a Riesz operator if λ − R ∈ Φ(X) for all scalars λ ≠ 0. Let (X) denote the class of all Riesz operators.For further discussions concerning this family of operators, we refer to [1,12] and the references therein.For our purpose, we recall that Riesz operators satisfy the Riesz-Schauder theory of compact operators, (X) is not an ideal of ᏸ(X) [1], and Ᏺ(X) is the largest ideal contained in (X) [12].Hence the sets (X), (X), C(X), Ᏺ + (X), and Ᏺ − (X) are also contained in (X).
We say that an operator F ∈ ᏸ(X) is polynomially compact (see [3]) if there is a nonzero complex polynomial p(z) such that the operator p(F ) is compact.We designate by P (X) the set of polynomially compact operators on X.Let F ∈ P (X), the nonzero polynomial p(z) of least degree and leading coefficient 1 such that p(F ) is compact will be called the minimal polynomial of F .We denote by Ξ(X) the subset of P (X) defined by We first prove the following lemma which is required in the sequel.
Next, making use of the spectral mapping theorem for the Browder essential spectrum [5, Theorem 4] The developments below are mainly suggested by the fact that, in general, the sets Ᏺ(X) and Ξ(X) do not coincide.Indeed, if is the minimal polynomial of F ∈ Ξ(X), then, by the structure theorem of Gilfeather [3, Theorem 1], the spectrum of F consists of countably many points with {λ 1 ,...,λ k } as only possible limit points and such that all but possibly {λ 1 ,...,λ k } are eigenvalues with finite-dimensional generalized eigenspaces.This, together with the fact that the operators belonging to Ᏺ(X) satisfy the Riesz-Schauder theory of compact operators (see above), implies that Ᏺ(X) = Ξ(X).Thus the next result improves Proposition 3.6.
then (U (t)) t≥0 can be embedded in a C 0 -group on X.
Due to some technical difficulties, we do not know whether or not Theorem 3.5 is valid for perturbations belonging to Ξ(X).So, we discuss this result for a subset of Ξ(X) consisting of power compact operators, that is, ᏼ(X) := F ∈ ᏸ(X) such that F n ∈ (X) for some integer n ≥ 1 . (4.5) Our principal motivation here rely on the fact that, for some classes of Banach spaces, we have Ᏺ(X) ⊆ ᏼ(X).In particular, if X is isomorphic to an L p space with 1 ≤ p ≤ ∞ or to C(Ω) where Ω is a metric compact Hausdorff space, then (X) = Ᏺ(X) (cf.(3.3)).Moreover, by [10, Theorem 1], we have (X)(X) ⊆ (X).These conclusions are also valid if X is an l p space with 1 ≤ p < ∞ and c 0 [6].Note also that if X has the Dunford-Pettis property (a Banach space X is said to have the Dunford-Pettis property if for every Banach space Y every weakly compact operator T : X → Y takes weakly compact sets in X into relatively norm compact sets of Y ), then ᐃ(X)ᐃ(X) ⊆ (X) where ᐃ(X) stands for the set of weakly compact operators.However, although the inclusion ᏼ(X) ⊆ (X) is valid for arbitrary Banach spaces (use the Ruston characterization of Riesz operators [1]), in general, we have ᏼ(X) ≠ Ᏺ(X).In the light of these observations, we project to extend Theorem 3.5 to semigroups (U (t)) t≥0 for which there exists t 0 > 0 such that U(t 0 ) − I ∈ ᏼ(X).Evidently, since ᏼ(X) ⊆ Ξ(X), Proposition 4.2 holds also true for power compact perturbations.More precisely, we have the following theorem.Theorem 4.3.Let (U (t)) t≥0 be a C 0 -semigroup on X with type ω and let A denote its infinitesimal generator.Define the set ᏻ by Then, the following items are equivalent: for some (in fact for all) λ > ω.
Proof.We try to imitate the procedure in the proof of Theorem 3.5.Let us first observe that if U(t) − I ∈ ᏼ(X), then there exists m ≥ 1 such that (U (t) − I) m ∈ (X).Using the spectral mapping theorem (see, e.g., [15, page 227]), one sees that that spectrum of U(t)−I is either finite or a countable set accumulating only at zero.Moreover, This means that, apart possibly from the point 1, σ (U(t)) = {e ηt : η ∈ P σ (A)} (P σ (A) stands for the point spectrum of A) and, for any ε > 0, the set {λ ∈ σ (U(t)) : |λ − 1| > ε} is finite for all t > 0. Then arguing as in the proof of [2, Theorem 2], we conclude that (i) implies that A ∈ ᏸ(X).Furthermore, similar arguments as in the proof of Theorem 3.5 [(i)⇒(ii)] imply that which leads to A ∈ ᏼ(X).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation