EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 4, 2013, 413-427 ISSN 1307-5543 – www.ejpam.com Natural Generalized Inverse and Core of an Element in Semigroups, Rings and Banach and Operator Algebras Xavier Mary Université Paris-Ouest Nanterre-La Défense, Laboratoire Modal’X Abstract. Using the recent notion of inverse along an element in a semigroup, and the natural partial order on idempotents, we study bicommuting generalized inverses and define a new inverse called natural inverse, that generalizes the Drazin inverse in a semigroup, but also the Koliha-Drazin inverse in a ring. In this setting we get a core decomposition similar to the nilpotent, Kato or Mbekhta decom- positions. In Banach and Operator algebras, we show that the study of the spectrum is not sufficient, and use ideas from local spectral theory to study this new inverse. 2010 Mathematics Subject Classifications: 15A09, 47A05, 47A11 Key Words and Phrases: generalized inverses, Koliha-Drazin inverse 1. Introduction In this paper, S, R and A denote respectively a semigroup, a unital ring and a uni- tal Banach algebra. In particular, R and A with only their multiplication structure will be considered as semigroups. For any semigroup S, S1 denotes the monoid generated by S (R1 = R,A 1 =A ). E(S) denotes the set of idempotents. For any subset A of S, A′ = {x ∈ S, (∀a ∈ A) xa = ax} denotes the commutant (or centralizer) of A. We say a is (von Neumann) regular in S if a ∈ aSa. A particular solution to axa = a is called an associate, or inner inverse, of a. A solution to xax = a is called a weak (or outer) inverse. Finally, an element that satisfies axa = a and xax = x is called an inverse (or reflexive inverse, or relative inverse) of a and is denoted by a′. The set of all associates of a is denoted by A(a), and the set of weak inverses of a by W (a). A commuting inverse, if it exists, is unique and denoted by a#. It is usually called the group inverse of a. A classical reference for generalized inverses is [2]. Email address: xavier.mary@u-paris10.fr http://www.ejpam.com 413 c© 2013 EJPAM All rights reserved. X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 414 We will make use of the Green’s preorders and relations in a semigroup [9]. For elements a and b of S, Green’s preorders ≤L , ≤R and ≤H are defined by a ≤L b⇐⇒ S1a ⊂ S1 b⇐⇒∃x ∈ S1, a = x b; a ≤R b⇐⇒ aS1 ⊂ bS1⇐⇒∃x ∈ S1, a = bx; a ≤H b⇐⇒ {a ≤L b and a ≤R b}. If ≤K is one of these preorders, then aK b ⇐⇒ {a ≤K b and b ≤K a}, and Ka = {b ∈ S, bK a} denotes the K -class of a. We recall the following characterization of group invertibility in terms of Green’s relation H (see [9, 22]): Lemma 1. a# exists if and only if aH a2 if and only ifHa is a group. The study of generalized inverses has been conducted in many different mathematical ar- eas, from semigroup theory to operator theory, and applied to various domains such as Markov chains or differential equations. In these studies, it may be useful to consider commuting (or bicommuting) inverses. Since the existence of a commuting inner inverse is a very strong property, it is common to look at outer commuting inverses, following the seminal work of M. Drazin [4], who introduced the Drazin inverse in the context of semigroups and rings. Later, this inverse has been generalized in the setting of operators by Koliha [16] using spectral properties and functional calculus. This generalized Drazin inverse (also called Koliha-Drazin inverse) finds many applications, in particular to singular differential equations. In [18] the author introduced a special outer inverse, called inverse along an element in the context of semigroups. The aim of this article is to use this new inverse to study bicommut- ing generalized inverses. Then, using the natural partial order on idempotents, we will define a new inverse called natural inverse, that generalizes the Drazin inverse in a semigroup, but also the Koliha-Drazin inverse in a ring. In this setting, this provides a decomposition of an el- ement similar to the nilpotent, Kato or Mbekhta decompositions [21]. In the first sections we introduce the main notions (inverse along an element, natural generalized inverse) entirely in the semigroup setting. We then study further properties of the natural inverse in rings, making the link with quasipolar (generalized Drazin invertible) elements [10, 11, 16, 17]. In the last sections, a particular attention is given to Banach and operators algebras. The main result is that this inverse relies on finer properties than spectral properties only. Local spectral theory[12] is then an interesting tool. 2. Inverse Along an Element 2.1. Definition and First Properties The inverse along an element was introduced in [18], and in [19], it was interpreted as a kind of inverse modulo H . We recall the definition and properties of this inverse. Note that in this article, this new inverse is denoted by a−d instead of a‖d , extending the case d = 1. X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 415 Definition 1. Given a, d in S, we say a is invertible along d if there exists b ∈ S such that bad = d = dab and b ≤H d. If such an element exists then it is unique and is denoted by a−d . Another characterization is the following: Lemma 2. a is invertible along d if and only if there exists b ∈ S such that bab = b and bH d, and in this case a−d = b. Theorem 1. Let a, d ∈ S. Then the following are equivalent: i) a−d exists. ii) d ≤R da and (da)# exists. iii) d ≤L ad and (ad)# exists. iv) dadH d. v) d ≤H dad. In this case, b = d(ad)] = (da)]d. For another look at this inverse, we also refer to [5], where M. Drazin independently defined an new outer inverse that is actually similar to the inverse along an element. 2.2. Commutativity and Idempotents A remarkable feature of the inverse along an element is the following [18, Theorem 10]. Theorem 2. Let a, d ∈ S and pose A= (a, d). If a is invertible along d, then a−d ∈ A′′. As a direct corollary, we get: Corollary 1. Let a, d ∈ S, dadH d and pose b = a−d . If ad = da, then ab = ba and bd = d b. We define the following sets: i) Σ0(a) = {e ∈ E(S), eaeH e}; ii) Σ1(a) = {a}′ ∩Σ0(a); iii) Σ2(a) = {a}′′ ∩Σ0(a). (If S is commutative, or the idempotents are central, then the three sets are equal. We then simply denote it Σ(a).) Lemma 3. Let e ∈ E(S) and a ∈ S such that ae = ea. Then e ∈ Σ0(a) ⇐⇒ e ≤H a. X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 416 Proof. Assume e ∈ Σ0(a). Then e ≤H eae = ea = ae ≤H a. Conversely, if e ≤H a and ae = ea, then e ≤R a⇒ e = ee ≤R ea ≤R e that is eRea. But ea = ae, hence eRea⇒ eRae⇒ e = eeReae. By symmetry, we get eH eae. Combining the previous lemmas and theorems we get: Theorem 3. τa : W (a) −→ E(S) x 7−→ ax • is one to one from W (a)∩ {a}′ onto Σ1(a); • is one to one from W (a)∩ {a}′′ onto Σ2(a). Its reciprocal τ−1 a associates e to b = a−e. Proof. Let b, c ∈ W (a) ∩ {a}′. Then ab = ac ⇒ b = bab = bac. But also ba = ca by commutativity and bac = cac = c. Finally b = c. Obviously, ab = ba = e is an idempotent commuting with a. Conversely, if e ∈ Σ1(a), then e ≤R a ⇒ e = ee ≤R ea = ae ≤R e. Also e ≤L a⇒ e = ee ≤R ae = ea ≤L e. It follows that ea = eaH e, a is invertible along e. Pose b = a−e. Then b ∈ {a, e}′′ hence ab = ba and ab = abe = bae = e. For the second statement, we have only to prove that τa maps W (a) ∩ {a}′′ onto Σ2(a), but this follows from Theorem 2. As a consequence, looking for commuting or bicommuting outer inverses can be handled through idempotents. Recall that any set of idempotents may be partially ordered by e ≤ f ⇐⇒ e f = f e = e, the natural partial order, and if this set is commutative, then this partial order is compatible with multiplication. We then have two partial orders on E(S), the natural partial order and the H preorder (that reduces to a partial order for idempotents since a H -class contains at most one idempotent [9]). Actually, they coincide for idempotents. If e ≤ f , then e = e f = f e and e ≤H f and conversely, if e = f x = y f then f e = f f x = f x = e = y f = y f f = e f . It is interesting to notice that even in the noncommutative case, invertibilty along an idempotent e can be expressed as invertibity in the local submonoid eSe (ring theorists use the word “corner ring”). Lemma 4. Let a ∈ S, e ∈ E(S). Then e ∈ Σ0(a) (a−e exists) if and only if eae is invertible in the local submonoid eSe. In this case a−e = (ea)#e = e(ae)# = (eae)# = (eae)−1. Proof. Assume a−e exists. Then a−eH e hence a−e = ea−e = a−ee = ea−ee ∈ eSe. It also satisfies a−eae = e = eaa−e hence a−e(eae) = e = (eae)a−e and eae is invertible in the monoid eSe (with unit e). Conversely, assume eae is invertible in eSe with inverse b ∈ eSe. Then b ≤H e and bae = b(eae) = e = (eae)b = eab and b is the inverse of a along e. Finally, note that Σ2(a) is a commutative band (commutative semigroup of idempotents, semilattice with e ∧ f = e f = f e). X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 417 Proposition 1. Σ2(a) is a commutative subsemigroup of S. Proof. If e, f ∈ Σ2(a), then e f = f e ≤H e ≤H a. We have to show that e f is an idempo- tent. e f e f = e f f e = e f e = ee f = e f and e f is an idempotent. 3. The Natural Generalized Inverse in a Semigroup 3.1. Definition and First Properties Definition 2. Let S be a semigroup, a ∈ S. i) Let j = 0, 1,2. The element a is j−maximally invertible if the set Σ j(a) admits maximal elements for the natural partial order. Elements b = a−e where e is maximal are then called j−maximal generalized inverses of a. ii) If there exists a greatest element M ∈ Σ j(a), then we say that a is j−naturally invertible, and b = a−M is called the j−natural (generalized) inverse of a. iii) Finally, if a is 2−naturally invertible, the element aM = aba is called the core of a. We will mainly deal with the 2−natural inverse in the sequel, and we will also refer to it as the natural inverse. As noted before, if S is commutative or the idempotents central then the three notions coincide. Recall that a semilattice is distributive if e ∧ f ≤ x implies the existence of e′, f ′ such that e ≤ e′, f ≤ f ′ and x = e′ ∧ f ′. Proposition 2. If the semilattice Σ2(a) is distributive, then any 2−maximally invertible element is naturally invertible. Proof. let e be a maximal element of Σ2(a), f ∈ Σ2(a). Then e f = f e ≤ e and exists e′, f ′ such that e ≤ e′, f ≤ f ′ and e = e′ f ′. By maximality, e′ = e and we get e = e f ′ = f ′e. It follows that e ≤ f ′ hence e = f ′ and f ≤ e. e is the greatest element in Σ2(a). The natural inverse generalizes the Drazin inverse [4]. Theorem 4. Assume a is Drazin invertible with inverse aD. Then a is 1 and 2−naturally invert- ible with inverse a−M = aD. Proof. Let a be Drazin invertible with index n and inverse aD. Then e = aaD = aDa ∈ Σ2(a)⊆ Σ1(a), and aDan+1 = an+1aD = an. Let f ∈ Σ1(a). Then a− f satisfies a− f a f = f = aa− f f = f aa− f = f a− f a. It follows that f = f � a− f �n+1 an+1. Then f e = f � a− f �n+1 an+1aDa = f . Also f e = e f (e = aaD = aDa ∈ Σ2(a)) hence f ≤ e. X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 418 3.2. Examples Many Maximal inverses Let S be the semigroup generated by three elements e, f , a subject to the conditions e = e2 = ea = ae, f = f 2 = f a = a f and e f = f e. Then S is commutative, a is maximally invertible but not naturally invertible, with two maximal inverses a−e = e and a− f = f . We consider now a simple variant of the previous example. Let S′ be the semigroup generated by three elements e, f , a subject to the conditions e = e2 = ea = ae, f = f 2 = f a = a f and e f = e, f e = f . Then Σ1(a) = {e, f } but Σ2(a) is empty since e and f do not commute. Right Hereditary Semigroups with Central Idempotents [6] In this example we notably show that elements of a right hereditary semigroup with central idempotents are naturally invertible, and describe the set Σ(a). Let S be a right principal projective (p.p.) semigroup with central idempotents as defined in [6]. Then E(S) is a semilattice (for the natural partial order). For any e ∈ E(S), define Ye = {x ∈ S, xe = x and xs = x t ⇒ es = et} (that is the L ∗−class of e for the extended Green’s relation L ∗ [7]). Then Ye is a cancellative monoid (with unit e) and the structure theorem of Fountain says that S is the semilattice of these disjoints monoids. If S is right semi-hereditary then it is right p.p. and incomparable principal right ideals are disjoints[3]. It follows notably that E(S) is a chain (any two idempotents are comparable), and maximal invertibility implies natural invertibility. Let now a ∈ S. If a is regular, then a is group invertible hence naturally invertible. We assume in the sequel that a is not regular. By centrality of the idempotents, Σ0(a) = Σ1(a) = Σ2(a) = {e ∈ E(S), e ≤H a}. Let a0 ∈ E(S) be the idempotent such that a ∈ Ya0 . Since aa0 = a0a = a, any e ≤H a satisfies e ≤ a0 for the H order hence the natural partial order, and since a is not regular, e < a0. Conversely, let e < a0 and assume S is semi-hereditary. From ae = ea ∈ eS ∩ aS, eS and aS are comparable, and from e < a0 we get eS ⊂ aS (ae ∈ Ye disjoint from Ya0 hence ae 6= a). It follows that e ≤R a and in particular ea = aeRe is regular. Since for regular elements, the appartenance in Ye is simply Green’s relation L , we get that ae = eaH e and e ≤H a. Finally, we have proved that Σ(a) is the chain of idempotents {e ∈ E(S), e < a0}. If we finally assume that S is right hereditary (right ideals are projective), then Doro- feeva [3] showed that S satifisies the maximum condition for principal right ideals. As a consequence, the chain of idempotents {e ∈ E(S), e < a0} has a greatest element M and a is naturally invertible with inverse a−M . X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 419 4. The Ring Case 4.1. Invertibility Along an Element in a Ring In this section, R denotes a unital ring. In particular, it is a semigroup and the previous notations and results apply. In [19], invertibilty along an element is characterized in terms of existence of units. Theorem 5. Let d be a regular element of a unital ring R, d ′ ∈ A(d). Then the following are equivalent: i) a−d exists. ii) u= da+ 1− dd ′ is a unit. iii) v = ad + 1− d ′d is a unit. In this case, a−d = u−1d = dv−1. Note that in the particular case of invertibility along an idempotent e, this reduces to: Corollary 2. Let e ∈ E(R) be a idempotent element of a unital ring R. Then the following are equivalent: i) a−e exists. ii) u= ea+ 1− e is a unit. iii) v = ae+ 1− e is a unit. In this case, a−e = u−1e = ev−1. Corollary 3. If ae = ea, then e ≤H a if and only if u= 1+ ae− e is a unit. Remark that a sufficient condition for this to happen is the following: Lemma 5. If ae = ea and a+ 1− e is a unit, then e ≤H a. Proof. let u= a+ 1− e. Then ue = ae = ea hence e = u−1ea = au−1e. 4.2. Natural Inverse in a Ring Let R be a unital ring, and let a ∈ R. Then the semilattice Σ2(a) is actually a distributive lattice (with e∨ f = e+ f − e f ) hence a is 2−maximally invertible if and only if it is naturally invertible. We derive new criterion for the natural inverse to exists. X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 420 Theorem 6. Let a ∈ R. Then the following are equivalent: i) a is naturally invertible with inverse a−M ; ii) There exists b ∈ {a}′′, bab = b and Σ2(a− aba) = {0}; iii) a = x + y with x ∈ {a}′′, x# exists, x y = 0 and Σ2(y) = {0}. In this case, a−M = b = x#. Proof. 1)⇒ 2) Assume a is naturally invertible with inverse b = a−M . Then M = ab = ba. Let e ∈ Σ2(a− aba). The ca = ac⇒ cb = bc⇒ c(a− aba) = (a− aba)c⇒ ec = ce. Hence e ∈ {a}′′. But also ∃t, s ∈ R, e = a(1− ba)t = s(1− ab)a and e ≤H a. Finally, e ∈ Σ2(a) hence e ≤ M , eM = Me = e. Computation give e = eM = s(1− ab)aba = 0. 2)⇒ 3) Let b ∈ {a}′′, bab = b and Σ(a− aba) = {0}. Then x = aba and y = a− aba satisfy the required relations (x# = b). 3)⇒ 1) Finally, let a = x+ y with x ∈ {a}′′, x# exists, x y = 0 and Σ(y) = {0}. By properties of the group inverse, x# ∈ {x}′′⇒ x x# ∈ {a}′′. Pose M = x x#. Since x = x x# x = x x#a = ax x#, M ≤H a and M ∈ Σ2(a). Let e ∈ Σ2(a). Then e = a−eae, and e is in the bicommutant of y = a− x . Then e− eM = e(1− x x#) = ea−ea(1− x x#) = ea−e(x + y)(1− x x#) = ea−e y and e− eM ∈ Σ2(y). By hypothesis, e− eM = 0 and e ≤ M , M is the greatest element of Σ2(a) and a is naturally invertible with inverse a−x x# . The unique decomposition a = x + y = aM + (a − aM) = aba + (a − aba) as in the previous theorem will be called the natural core decomposition of a. 4.3. Link with the Koliha-Drazin Inverse We recall the following definitions of quasinilpotency and quasipolarity in unital rings due to R. Harte [10]. Definition 3. An element q of a unital ring R is quasinilpotent if ∀x ∈ {q}′, 1+ xq ∈ R−1, and quasi-quasinilpotent if ∀x ∈ {q}′′, 1+ xq ∈ R−1 Note that quasi-quasinilpotent elements need not be quasinilpotent in general. The two notions however coincide for Banach algebras. Definition 4. An element a of a unital ring R is quasipolar (resp. quasi-quasipolar) if there exists a idempotent (called spectral idempotent) p in {a}′′ such that ap is quasinilpotent (resp. quasi-quasinilpotent) and a+ p ∈ R−1. X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 421 It was remarked in [17] that the last condition can be replaced by the following one 1 − p ≤H a. This is the content of lemma 5. It was proved by J. Koliha and P. Patricio (Theorem 4.2 in [17]) that quasipolar elements are exactly the generalized Drazin invertible elements (also called Koliha-Drazin invertible elements): Definition 5. An element a of a unital ring R is generalized Drazin invertible (resp. quasi- generalized Drazin invertible) if there exists b in {a}′′ such that bab = b and a2 b − a is quasinilpotent (resp. quasi-quasinilpotent). Theorem 7. An element a of a unital ring R is generalized Drazin invertible (resp. quasi- generalized Drazin invertible) if and only if it is quasipolar (resp. quasi-quasipolar). In this case b = (a+ p)−1(1− p). Next theorem proves that the natural inverse generalizes not only the Drazin inverse, but also the Koliha-Drazin inverse in a ring: Theorem 8. Let R be a unital ring, and a ∈ R be quasi-quasipolar with spectral idempotent p and quasi-Koliha-Drazin inverse b. Then a is naturally invertible, M = 1− p is the greatest element of Σ2(a) and the quasi-generalized Drazin inverse b is equal to a−M , the natural generalized inverse of a. Proof. Assume a is quasi-quasipolar in the ring sense. Then exists spectral idempotent p in {a}′′ such that ap is quasi-quasinilpotent and a+ p ∈ R−1. By lemma 5, M = 1− p ≤H a, hence it is in Σ2(a). Let f ∈ Σ2(a). Then exists x ∈ S, f = xa. By quasi-quasinilpotency, (1− f p) = (1− xap) ∈ R−1. But by commutativity of {a}′′ and the fact that f , p ∈ E(S), we have (1− f p)(1+ f p) = 1− f p. By invertibility, 1+ f p = 1 hence f p = 0. It follows that f M = f (1 − p) = f − f p = f and f ≤ M for the natural partial order. M is the greatest element of Σ2(a). Now the generalized Drazin inverse of a b = (a+ p)−1(1− p) is obviously inH(1−p) and is an outer inverse of a by definition. By unicity, it is a−M . If we require the element a to be quasipolar instead of quasi-quasipolar with Koliha-Drazin inverse b, then the idempotent M = 1− p is actually the greatest element Σ1(a) and b = a−M is also the 1−natural generalized inverse of a. 5. The Banach Algebra Case In this section,A denotes a unital Banach algebra. For any a ∈A , we denote its spectrum by σ(a) and its spectral radius by r(a). Recall that in a Banach algebra, an element is quasinilpotent if its spectrum reduces to 0, or equivalently if its spectral radius is 0, and quasipolar if 0 an isolated point of the spectrum. It is known [10] that these notions coincide with their ring counterpart, and also with the quasi-quasi notion (for instance, σ(a) = {0C} if and only if a is quasinilpotent in the ring sense if and only if a is quasi-quasinilpotent in the ring sense). Corollary 4. Let a ∈ A . If 0 is an isolated point of the spectrum of a, then a is naturally invertible. X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 422 Proof. If 0 is an isolated point of the spectrum of a, then a is quasi-quasipolar in the Banach sense, hence it is quasi-quasipolar in the ring sense. We then apply Theorem 8. We now investigate the link between Σi(a), i = 1,2 and σ(a). Theorem 9. LetA be a unital Banach algebra, a ∈A . Then i) σ(a) = {0C} ⇒ Σ1(a) = {0}. ii) Σ2(a) = {0} ⇒ σ(a) is connected and contains 0C. Proof. i) If the spectrum of a reduces to 0, the its spectral radius is equal to 0. Let e ∈ Σ1(a). Then e = aa−e = a−ea. We get ||e|| 1 n = ||en|| 1 n = ||an(a−e)n|| 1 n ≤ ||an|| 1 n ||(a−e)|| → 0 and ||e||= 0. ii) If a is invertible, then 1 ∈ Σ2(a). Hence assume σ(a) contains 0 but is not connected. Then σ(a) = C0∪C1 with 0 ∈ C0 and C0, C1 disjoint and open and closed in σ(a). Then the holomorphic calculus for f (z) = 1 z on U open set containing C1 and 0 outside U , such that U contains an open neighbourhood of C0, defines an element x = f (a) of {a}′′ such that ax = xa = e is idempotent and non zero, and Σ2(a) does not reduce to {0}. Now, we consider three different (commutative) Banach algebras to show that we cannot do better in the theorem, nor define natural invertibility in terms of the spectrum. • Consider the Banach algebra A = C0([0,1]) of continuous functions on [0, 1], and let a(t) = t. Then σ(a) = [0, 1] and Σ(a) = {0}. a is naturally invertible with b = 0. • Consider the Banach algebraA = C0([0,1]∪ [2,3]) of continuous functions on [0, 1]∪[2, 3], and let a(t) = t, 0≤ t ≤ 1 and a(t) = t−1, 2≤ t ≤ 3. Then σ(a) = [0, 2] and Σ(a) = {1[2,3]}. a is naturally invertible with b(t) = 0, 0 ≤ t ≤ 1 and b(t) = 1 t−1 , 2≤ t ≤ 3. • Consider now the Banach algebra A = L∞([0, 1]) of essentially bounded measurable functions on [0, 1], and let a(t) = t. Then σ(a) = [0, 1] and Σ(a) = {1A, ∃0< c ≤ 1,λ(A∩ [0, c]) = 0}. This set admits no maximal element, hence a(t) = t is not naturally invertible. X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 423 It appears that natural invertibility is strongly linked with the nature of the structure space (or spectrum) of the whole commutative Banach algebra B = {a}′′, independently of the nature of the spectrum of the element a. Obviously, if the spectrum of {a}′′ is not connected, then Shilov’s idempotent theorem gives the existence of a nontrivial idempotent. This idempotent needs not to be in Σ(a). Next theorem uses the generalized spectral theory of Hile and Pfaffenberger [14, 15] and the associated functional calculus to construct elements in Σ j(a), j = 1,2. The construction is similar to the case of a disconnected spectrum, but instead of using σ(a) (that can be connected), we use the generalized spectrum of Hile and Pfaffenberger. If a, q ∈ A , then the spectrum of a relative to q, or q−spectrum of a σq(a), is the set of points z such that a− z.1− z̄q is not invertible in A. Theorem 10. Let a, q ∈ A , with σ(a) connected set that contains 0. Assume σ(q) ∩ T = ;, where T is the unit circle, and σq(a) is not connected. Then i) if q ∈ {a}′, Σ1(a) is not empty; ii) if q ∈ {a}′′, Σ2(a) is not empty. Proof. This is a consequence of Theorem 12 in [14]. Indeed, since a is not invertible, 0 is in the q spectrum of a. Since σq(a) is not connected, we can find a closed rectifiable curve Γ in the q resolvent such that 0 is in its exterior and its interior contains elements of σq(a) (a component of σq(a) that does not contains 0). Choosing z = 0 in equation 4.3 gives an idempotent p ≤R a. The rest follows from commutation properties. 6. Operators Finally, we apply the previous results to the operator algebra A = B(X ) of bounded operators on a Banach space X . 6.1. Local Spectral Theory In the operator case, we can improve somehow the results of the previous section. Let X be a Banach space and T ∈ B(X ). T (X ), or R(T ) denotes its range, N(T ) its kernel. We use ideas from local spectral theory [1, 12, 20, 21] and define the following sets: Definition 6. • The hyperrange of T is the linear space T∞(X ) = ⋂ n∈N T n(X ); • The hyperkernel of T is the linear space N∞(T ) = ⋃ n∈N N(T n); • The quasinilpotent part (or transfinite kernel) of T is the linear space H0(T ) = {x ∈ X , ||T n x || 1 n → 0}; • The algebraic core C(T ) of T is the largest subspace such that T (M) = M; X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 424 • The analytic core (or transfinite range) K(T ) of T consists of all vectors x0 ∈ X for which there exist a sequence xn ∈ X such that T xn = xn−1 and exists c > 0, ||xn|| ≤ cn||x0||. The algebraic core can also be defined as follows: C(T ) consists of all vectors x0 ∈ X for which there exist a sequence xn ∈ X such that T xn = xn−1. We then have the following inclusions: K(T )⊂ C(T )⊂ T∞(X ), N∞(T )⊂ H0(T ). In [12], it is proved that for a bounded operator T , the analytic core corresponds to the holomorphic range {limz→0(T − zI) f (z), f ∈ Holo(0, X )}, and that the intersection if the analytic core with N(T ) is the holomorphic kernel of T {g(0), (T − zI)g(z) = 0, g ∈ Holo(0, X )}. We have the following relations: Proposition 3. Let P ∈ Σ1(T ). Then P(X )⊂ K(T ) and H0(T )⊂ N(P). Proof. Let P ∈ Σ1(T ). Then P = T T−P P = T T−P = T−P T . Let x0 ∈ P(X ), and for all n> 0, pose xn = (T−P)n x0. Then T xn = T (T−P)n x0 = P(T−P)n−1 x0 = (T −P)n−1P x0 = (T −P)n−1 x0 = xn−1. Also ||xn|| ≤ ||T−P ||n||x0||, hence x0 ∈ K(T ). Let now x ∈ H0(T ). Then P(x) = T−P T (x) = (T−P)nT n(x) forall n> 0 and ||P(x)|| 1 n ≤ ||(T−P)n|| 1 n ||T n(x)|| 1 n ≤ ||(T−P)||||T n(x)|| 1 n → 0 and P(x) = 0. Corollary 5. K(T ) = {0} ⇒ Σ1(T ) = {0}; H0(T ) = X ⇒ Σ1(T ) = {0}. Obviously, the existence of a greatest element in Σ2(T ) is guaranteed by a decomposition of the form X = H0(T )⊕K(T ), with both subspaces closed (choose P the associated projection on K(T )). But such a decomposition occurs only for quasipolar elements: Theorem 11. [21, Theorem 1.6] Let T ∈ B(X ). Then 0 is an isolated point of the spectrum if and only if H0(T ), K(T ) are closed and X = H0(T )⊕ K(T ). Theorem 12. Assume K(T ) is closed and complemented with complement N hyperinvariant, and N(T )∩ K(T ) = {0}. Then T is naturally invertible with greatest idempotent the projection on K(T ) parallel to N. X. Mary / Eur. J. Pure Appl. Math, 6 (2013), 413-427 425 Proof. Let X = K(T ) ⊕ N and M the idempotent of the theorem. First, we must prove that M ∈ Σ2(T ). Since K(T ) and N are hyperinvariant, we only have to prove that M ≤H T . Consider T|K(T ) : K(T ) → K(T ) the restriction of T to K(T ). T|K(T ) is well defined since T (K(T ))⊂ K(T ), and surjective since T (K(T )) = K(T ). But from the hypothesis N(T )⊂ N it is also injective, hence invertible and exists S bounded operator, TS = ST = M . Let now P be and idempotent in Σ2(T ). Then P(X )⊂ K(T ) from Proposition 3, hence P(X ) ⊂ M(X ). It follows that PM P = P and by commutation (Σ2(T ) is a commutative semigroup), PM = M P = PM P = P and M is the greatest element of Σ2(T ). By the results of Harte [12], N(T )∩K(T ) is the holomorphic kernel of T , and it reduces to 0 precisely when T has the single valued extension property (SVEP) at 0 (Theorem 9 p. 180). We get the following corollary. Corollary 6. Let T be a bounded operator on X with the SVEP at 0. If K(T ) is closed and hyperinvariantly complemented, then T is naturally invertible. As a final result, we investigate the range of the core of a naturally invertible element: Proposition 4. Let T be naturally invertible with natural inverse B, greatest idempotent M = T B = BT and core T M = T BT. Then Kν(T ) = T M(X ) is a closed, hyperinvari- ant, complemented (with hyperinvariant complement) subspace of the analytic core K(T ), and T Kν(T ) = Kν(T ). Proof. By commutation, T M(X ) ⊂ M(X ). But also M(X ) = M2(X ) = BT M(X ) ⊂ T M(X ) and the two subspaces are equal. The other properties follow. 6.2. Miscellanous In this last section we give examples and results relative to natural invertiblity. The Shift Operator Let S be the shift operator on l2(N). Then S is not quasinilpotent, but its hyperrange reduces to 0. As a consequence, Σ1(S) = {0}. The spectrum of S is the unit disk. 6.2.1. Strongly Irreducible Operators In 1972, F. Gilfeather [8] introduced the concept of strongly irreducible operator. A bounded linear operator T is said to be strongly irreducible, if there exists no non-trivial idempotent p in the commutant of T . This concept actually coincides with the concept of Banach irreducible operator (a bounded linear operator T is said to be Banach irreducible, if T can not be written as a direct sum of two bounded linear operators). It is clear that strongly irreducible operators satisfy Σ1(T ) = {0}. Also, the following spectral result is due to Herrero and Jiang [13]: Theorem 13. σ(T ) is connected if and only if T is in the norm closure of strongly irreducible operators. REFERENCES 426 6.2.2. Rosenblum’s Corollary, Commutant and Bicommutant Let T = � X 0 0 Y � be the a decomposition of T with X invertible and M = � X X−1 0 0 0 � the greatest element of Σ1(T ). If σ(X ) ∩ σ(Y ) = {0}, then by Rosenblum’s corollary (see [23]), � X 0 0 0 � , � 0 0 0 Y � ∈ {T}′′ and T = � X 0 0 0 � + � 0 0 0 Y � is the natural core decomposition of T , with M = � X X−1 0 0 0 � the greatest element of Σ2(T ). This is the case for instance when Y is quasinilpotent. References [1] P. Aiena. Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, Boston, UUS. 2004 [2] A. B. Israel and T.N.E. Greville. Generalized Inverses, Theory and Applications, 2nd Edition, Springer 2003. [3] M. P. Dorofeeva. Hereditary and semi-hereditary monoids, Semigroup Forum 4, 301–311. 1972. [4] M.P. Drazin. Pseudo-Inverses in Associative Rings and Semigroups, American Mathematical Monthly 65, no. 7, 506–514. 1958 [5] M.P. Drazin. A class of outer generalized inverses, Linear Algebra and its Applications 436, no. 7, 1909–1923 2012. [6] J. Fountain. Right PP monoids with central idempotents, Semigroup Forum 13, no. 3, 229–237. 1977. [7] J. Fountain. Abundant semigroups, Proceedings of the London Mathematical Society 3, no. 1, 103–129. 1982. [8] F. Gilfeather. Strong reducibility of operators, Indiana University Mathematics Journal 22, 393–397. 1972. [9] J.A. Green. On the structure of semigroups, Annals of Mathematics 54, no. 1, 163–172. 1951. [10] R. Harte. On quasinilpotents in rings, Panamerican Mathematical Journal 1, 10–16. 1991. [11] D. Kitson and R. Harte. On Browder tuples, Acta Scientiarum Mathematicarum 75, no. 3–4, 665–677. 2009. REFERENCES 427 [12] R. Harte. On local spectral theory, Recent advances in operator theory and applications 175–183, Operator Theory: Advances and Applications, 187, Birkhäuser, Basel, 2009. [13] D.A. Herrero and C.L. Jiang. Limits of strongly irreducible operators, and the Riesz decom- position theorem, Michigan Mathematical Journal 37, no. 2, 283–291. 1990. [14] G. N. Hile and W. E. Pfaffenberger. Generalized spectral theory in complex Banach alge- bras, Canadian Journal of Mathematics 37, no. 6, 1211–1236. 1985. [15] G. N. Hile and W. E. Pfaffenberger. Idempotents in complex Banach algebras, Canadian Journal of Mathematics 39 no. 3, 625–630. 1987. [16] J.J. Koliha. A generalized Drazin inverse, Glasgow Mathematical Journal 38, no. 3, 367– 381. 1996. [17] J.J. Koliha and P. Patricio. Elements of rings with equal spectral idempotents, Journal of the Australian Mathematical Society 72, 137–152. 2002. [18] X. Mary. On generalized inverses and Green’s relations, Linear Algebra and its Applications 434, no. 8, 1836–1844. 2011. [19] X. Mary and P. Patricio. Generalized invertibilty moduloH in semigroups and rings, Linear Multilinear Algebra 61, no. 8, 1130–1135. 2013. [20] M. Gonzalez, M. Mbekhta, and M. Oudghiri. On the isolated points of the surjective spec- trum of a bounded operator, Proceedings of the American Mathematical Society 136, no. 10, 3521–3528. 2008. [21] M. Mbekhta. Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux, Glasgow Mathematical Journal 29, no. 2, 159–175. 1987. [22] D.D. Miller and A.H. Clifford. RegularD-Classes in Semigroups, Transactions of the Amer- ican Mathematical Society 82, no. 1, 270–280. 1956 [23] H. Radjavi and P. Rosenthal. Invariants Subspaces, Springer-Verlag, Berlin, 1973.