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LINEAR PRESERVER PROBLEMS: generalizedinverse

Mostafa Mbekhta

Université Lille 1, France

Banach Algebras 2011,Waterloo August 3-10, 2011

Mostafa Mbekhta LINEAR PRESERVER PROBLEMS: generalized inverse

I. IntroductionLinear preserver problems is an active research area in Matrix,Operator Theory and Banach Algebras. These problems involvecertain linear transformations on spaces of matrices, operators orBanach algebras... that will be designates by A in what follows.First, we start by listing the type of problems that are related to LinearPreserved Problems topic.

Problem I

Problem I. Let F be a (scalar-valued, vector-valued, or set-valued)given function on A. We would like to characterize those lineartransformations φ on A which satisty

F(φ(x)) = F(x) (x ∈ A).

Examples : (i) scalar-valued : A = Mn(C) and F(x) = det(x)

Theorem [G. Frobenius 1897]A linear map φ fromMn(C) intoMn(C) preserves the determinant(i.e. det(φ(x)) = det(x)) if and only if it takes one of the followingforms

φ(x) = axb or φ(x) = axtrb, for all x ∈Mn(C)

where a, b ∈Mn(C) are non-singular matrices such thatdet(ab) = 1, and xtr stands for the transpose of x.

Problem I

Problem I. Let F be a (scalar-valued, vector-valued, or set-valued)given function on A. We would like to characterize those lineartransformations φ on A which satisty

F(φ(x)) = F(x) (x ∈ A).

Examples : (i) scalar-valued : A = Mn(C) and F(x) = det(x)

Theorem [G. Frobenius 1897]A linear map φ fromMn(C) intoMn(C) preserves the determinant(i.e. det(φ(x)) = det(x)) if and only if it takes one of the followingforms

φ(x) = axb or φ(x) = axtrb, for all x ∈Mn(C)

where a, b ∈Mn(C) are non-singular matrices such thatdet(ab) = 1, and xtr stands for the transpose of x.

Problem I

(ii) A = C(K) and F(X) = ||X|| (scalar-valued).Let K be a compact metric space and C(K) the Banach space ofcontinuos real valued functions defined on K, with the supremunnorm.

Theorem [Banach 1932]Let φ : C(K)→ C(K) be a surjective linear map. Thenφ is an isometry (i.e. ||φ(f )||∞ = ||f ||∞) if and only if

φ(f (t)) = h(t)f (ϕ(t)), t ∈ K

where |h(t)| = 1 and ϕ is a homeomorphism of K onto itself.

Problem I

Kadison (1951) described the surjective linear isometries ofC∗-algebras.

Theorem [Kadison]Let A and B be C∗-algebras and φ : A → B be a surjective linearmap. Thenφ is an isometry if and only if there is unitary u ∈ B and aC∗-isomorphism ϕ : A → B such that φ = uϕ.

Problem I

(iii) vector-valued : A C∗-algebra and F : A → A, F(X) = |X|, theabsolute value of X (i.e. |X| = (X∗X)1/2).

TheoremLet A be a C∗-algebra and φ : A → A be a surjective linear map.Then |φ(x)| = |x| for all x ∈ A if and only if there exists an unitaryelement u ∈ A such that φ(x) = ux for all x ∈ A.

(iv) set-valued : A Banach algebra and F(X) = σ(X).

Problem II

Problem II. Let S be a given subset of A. We would like tocharacterize those linear map φ on A wich satisfy :

X ∈ S =⇒ φ(X) ∈ S (X ∈ A)

or satisfyX ∈ S ⇐⇒ φ(X) ∈ S (X ∈ A).

Examples :(i) S the set of idempotent, nilpotent, Fredholm, semi-Fredholmoperators.(ii) S = A−1 = {X ∈ A; X invertible}(iii) S = {X ∈ A;∃Y ∈ A; XYX = X}, the set of all the elements of Ahaving a generalized inverse.The two previous examples, where the first one is related toKaplansky problem, will be developed in detail in this presentation,later on.

Problem II

Problem II. Let S be a given subset of A. We would like tocharacterize those linear map φ on A wich satisfy :

X ∈ S =⇒ φ(X) ∈ S (X ∈ A)

or satisfyX ∈ S ⇐⇒ φ(X) ∈ S (X ∈ A).

Examples :(i) S the set of idempotent, nilpotent, Fredholm, semi-Fredholmoperators.(ii) S = A−1 = {X ∈ A; X invertible}(iii) S = {X ∈ A;∃Y ∈ A; XYX = X}, the set of all the elements of Ahaving a generalized inverse.The two previous examples, where the first one is related toKaplansky problem, will be developed in detail in this presentation,later on.

Problem III

Problem III. LetR be a relation on A. We would like to Characterizethose linear transformations φ on A which satisfy

X R Y =⇒ φ(X) R φ(Y), (X,Y ∈ A)

or satisfy

X R Y ⇐⇒ φ(X) R φ(Y), (X,Y ∈ A).

Examples :(i)R could be commutativity(ii)R could be similarity(iii) “zero product preserving mappings” i.e.XY = 0 ⇐⇒ φ(X)φ(Y) = 0.

Problem III

Problem III. LetR be a relation on A. We would like to Characterizethose linear transformations φ on A which satisfy

X R Y =⇒ φ(X) R φ(Y), (X,Y ∈ A)

or satisfy

X R Y ⇐⇒ φ(X) R φ(Y), (X,Y ∈ A).

Examples :(i)R could be commutativity(ii)R could be similarity(iii) “zero product preserving mappings” i.e.XY = 0 ⇐⇒ φ(X)φ(Y) = 0.

Problem IV

Problem IV. Given a map F : A → A. We would like to characterizethose linear transformations φ on A which satisfy

F(φ(X)) = φ(F(X)) (X ∈ A).

Examples :(i) A = Mn(C) and F(X) = X†

(X† stands for the Moore-Penrose inverses of X).

(ii) F(X) = |X|, the absolute value of X (i.e. |X| = (X∗X)1/2).

(iii) F(X) = Xn, n being a fixed integer not less than 2. When n = 2,φ is a Jordan homomorphism.

Problem IV

Problem IV. Given a map F : A → A. We would like to characterizethose linear transformations φ on A which satisfy

F(φ(X)) = φ(F(X)) (X ∈ A).

Examples :(i) A = Mn(C) and F(X) = X†

(X† stands for the Moore-Penrose inverses of X).

(ii) F(X) = |X|, the absolute value of X (i.e. |X| = (X∗X)1/2).

(iii) F(X) = Xn, n being a fixed integer not less than 2. When n = 2,φ is a Jordan homomorphism.

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,

or equivalently, φ(a2) = φ(a)2 for all a in A.We say that φ is unital if φ(1) = 1.Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,or equivalently, φ(a2) = φ(a)2 for all a in A.

We say that φ is unital if φ(1) = 1.Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Jordan homomorphism

A linear map φ : A → B between two unital complex Banachalgebras is called Jordan homomorphism ifφ(ab + ba) = φ(a)φ(b) + φ(b)φ(a) for all a, b in A,or equivalently, φ(a2) = φ(a)2 for all a in A.We say that φ is unital if φ(1) = 1.Remarks(1) φ is homomorphism⇒ φ is Jordan homomorphism.(2) φ is anti-homomorphism (i.e. φ(ab) = φ(b)φ(a))⇒ φ is Jordanhomomorphism.(3) The converse is true in some cases :for instance if φ is onto and if B is prime, that is aBb = {0} ⇒ a = 0or b = 0.(4) It is well-known that a unital Jordan homomorphism preservesinvertibility,(i.e. x ∈ A−1 ⇒ φ(x) ∈ B−1).

Kaplansky’s problem

Kaplansky’s problem :Let φ be a unital surjective linear map between two semi-simpleBanach algebras A and B which preserves invertibility (i.e.,φ(x) ∈ B−1 whenever x ∈ A−1). Is it true that φ is a Jordanhomomorphism ?

This problem has been first solved in the finite-dimensional case:

Theorem [J. Dieudonné, 1943]Let φ : Mn(C)→ Mn(C) be a surjective, unital linear map. Ifσ(φ(x)) ⊆ σ(x) for all x ∈ Mn(C), then there is an invertible elementa ∈ Mn(C) such that φ takes one of the following forms:

φ(x) = axa−1 or φ(x) = axtra−1.

In the commutative case the well-known Gleason-Kahane-Zelazkotheorem provides an affirmative answer.

Theorem [Gleason-Kahan-Zelazko, 1968]Let A be a Banach algebra and B a semi-simple commutative Banachalgebra.If φ : A → B is a linear map such that σ(φ(x)) ⊆ σ(x) for all x ∈ A,then φ is an homomorphism.

In the non-commutative case, the best known results so far are due toAupetit and Sourour.

Theorem [Sourour, 1996]Let X,Y be two Banach spaces and let φ : B(X)→ B(Y) a linear,bijective and unital map. Then the following properties areequivalent:(i) σ(φ(T)) ⊆ σ(T) for all T ∈ B(X);(ii) φ is an isomorphism or an anti-isomorphism;(iii) either there is A : Y → X invertible such that φ(T) = A−1TA forall T ∈ B(X);or, there is B : Y → X∗ invertible such that φ(T) = B−1T∗B for allT ∈ B(X), and in this case X and Y are reflexive.

We will say that a C∗-algebra A is of real rank zero if the set formedby all the real linear combinations of (orthogonal) projections is densein the set of self-adjoint elements of A.It is well known that this property is satisfied by every von Neumannalgebra, and in particular by the C∗-algebra B(H) of all boundedlinear operators on a Hilbert space H, and by the Calkin algebraC(H) = B(H)/K(H) .

Theorem [B. Aupetit, 2000]

Let A be a C∗-algebra of real rank zero and B a semi-simple Banachalgebra.If φ : A → B is a surjective linear map such that σ(φ(x)) = σ(x) forall x ∈ A, then φ is a Jordan isomorphism.

RemarkThe Kaplansky problem is still open when A,B are C∗-algebras.

Remark Many other linear preserver problems, like the problem ofcharacterizing linear maps preserving idempotents or nilpotents orcommutativity,... that were first solved for matrix algebras, have beenrecently extended to the infinite-dimensional case ( B. Aupetit;Brešar; M. Mathien, L. Molnar; L. Rodman; Šemrl; A.R. Sourour;...).

Generalized inverse

II. Generalized inverse preservers mapsDefinition An element b ∈ A is called a generalized inverse of a ∈ Aif b satisfies the following two identities

aba = a and bab = b.

Let A∧ denote the set of all the elements of A having a generalizedinverse.

Theorem [Kaplansky, 1948]Let A be a Banach algebra:- If A = A∧ then A is finite dimensional.- Furthermore, if A is semi-simple, thenA = A∧ ⇐⇒ dim(A) <∞.

Generalized inverse

II. Generalized inverse preservers mapsDefinition An element b ∈ A is called a generalized inverse of a ∈ Aif b satisfies the following two identities

aba = a and bab = b.

Let A∧ denote the set of all the elements of A having a generalizedinverse.

Theorem [Kaplansky, 1948]Let A be a Banach algebra:- If A = A∧ then A is finite dimensional.- Furthermore, if A is semi-simple, thenA = A∧ ⇐⇒ dim(A) <∞.

Generalized inverse

Definition We say that φ : A → B preserves generalized invertibilityin both directions if x ∈ A∧ ⇐⇒ φ(x) ∈ B∧.

Remark Observe that, every n× n complex matrix has a generalizedinverse, and therefore, every map on a matrix algebra preservesgeneralized invertibility in both directions. So, we have here anexample of a linear preserver problem which makes sense only in theinfinite-dimensional case.Let H be an infinite-dimensional separable complex Hilbert space andB(H) the algebra of all bounded linear operators on H andK(H) ⊂ B(H) be the closed ideal of all compact operators.We denote the Calkin algebra B(H)/K(H) by C(H). Letπ : B(H)→ C(H) be the quotient map.

Generalized inverse

Definition We say that φ : A → B preserves generalized invertibilityin both directions if x ∈ A∧ ⇐⇒ φ(x) ∈ B∧.Remark Observe that, every n× n complex matrix has a generalizedinverse, and therefore, every map on a matrix algebra preservesgeneralized invertibility in both directions. So, we have here anexample of a linear preserver problem which makes sense only in theinfinite-dimensional case.

Let H be an infinite-dimensional separable complex Hilbert space andB(H) the algebra of all bounded linear operators on H andK(H) ⊂ B(H) be the closed ideal of all compact operators.We denote the Calkin algebra B(H)/K(H) by C(H). Letπ : B(H)→ C(H) be the quotient map.

Generalized inverse

Definition We say that φ : A → B preserves generalized invertibilityin both directions if x ∈ A∧ ⇐⇒ φ(x) ∈ B∧.Remark Observe that, every n× n complex matrix has a generalizedinverse, and therefore, every map on a matrix algebra preservesgeneralized invertibility in both directions. So, we have here anexample of a linear preserver problem which makes sense only in theinfinite-dimensional case.Let H be an infinite-dimensional separable complex Hilbert space andB(H) the algebra of all bounded linear operators on H andK(H) ⊂ B(H) be the closed ideal of all compact operators.We denote the Calkin algebra B(H)/K(H) by C(H). Letπ : B(H)→ C(H) be the quotient map.

Generalized inverse

Theorem [M. Mbekhta, L. Rodman and P. Šemrl, 2006]Let H be an infinite-dimensional separable Hilbert space and letφ : B(H)→ B(H) be a bijective continuous unital linear mappreserving generalized invertibility in both directions. Then

φ(K(H)) = K(H),

and the induced map ϕ : C(H)→ C(H), (i.e. ϕ ◦ π = π ◦ φ), is eitheran automorphism, or an anti-automorphism.

Generalized inverse

Let F(H) ⊂ B(H) the ideal of all finite rank operators,We say that φ : B(H)→ B(H) is surjective up to finite rank (resp.compact) operators if for every A ∈ B(H) there exists B ∈ B(H) suchthat A− φ(B) ∈ F(H) (resp. K(H)).

Theorem [M.Mbekhta and P. Šemrl, 2009]Let H be an infinite-dimensional separable Hilbert space andφ : B(H)→ B(H) a surjective up to finite rank operators linear map.If φ preserves generalized invertibility in both directions, then

φ(K(H)) ⊆ K(H)

and the induced map ϕ : C(H)→ C(H) is either an automorphism,or an anti-automorphism multiplied by an invertible elementa ∈ C(H).

Generalized inverse

Let F(H) ⊂ B(H) the ideal of all finite rank operators,We say that φ : B(H)→ B(H) is surjective up to finite rank (resp.compact) operators if for every A ∈ B(H) there exists B ∈ B(H) suchthat A− φ(B) ∈ F(H) (resp. K(H)).

Theorem [M.Mbekhta and P. Šemrl, 2009]Let H be an infinite-dimensional separable Hilbert space andφ : B(H)→ B(H) a surjective up to finite rank operators linear map.If φ preserves generalized invertibility in both directions, then

φ(K(H)) ⊆ K(H)

Semi-Fredholm

III. Semi-Fredholm preserver mapsWe recall that an operator A ∈ B(H) is said to be Fredholm if its rangeis closed and both its kernel and cokernel are finite-dimensional,and is semi-Fredholm if its range is closed and its kernel or itscokernel is finite-dimensional.We denote by SF(H) ⊂ B(H) the subset of all semi-Fredholmoperators.

Semi-Fredholm

The above theorem allows us to establish the following result which isof independent interest.

We say that a map φ : B(H)→ B(H) preserves semi-Fredholmoperators in both directions if for every A ∈ B(H) the operator φ(A)is semi-Fredholm if and only if A is.

TheoremLet H be an infinite-dimensional separable Hilbert space andφ : B(H)→ B(H) a surjective up to compact operators linear map. Ifφ preserves semi-Fredholm operators in both directions, then

φ(K(H)) ⊆ K(H)

Semi-Fredholm

For a Fredholm operator T ∈ B(H) we define the index of T by

ind(T) = dim Ker T − codim Im T ∈ Z.

TheoremUnder the same hypothesis and notation as in the above theorem, thefollowing statements hold :(i) φ preserves Fredholm operators in both directions;(ii) there is an n ∈ Z such that either

ind(φ(T)) = n + ind(T)

for every Fredholm operator T, or

ind(φ(T)) = n− ind(T)

for every Fredholm operator T.

Semi-Fredholm

LemmaLet A ∈ B(H). Then the following are equivalent:(i) A is semi-Fredholm,(ii) for every B ∈ B(H) there exists δ > 0 such that A + λB ∈ B(H)∧

for every complex λ with |λ| < δ.

CorollaryLet H be an infinite-dimensional separable Hilbert space andφ : B(H)→ B(H) a surjective up to finite rank op linear map.Then φ preserves generalized invertibility in both directions implies φpreserves semi-Fredholm operators in both directions.

Semi-Fredholm

LemmaLet A ∈ B(H). Then the following are equivalent:(i) A is semi-Fredholm,(ii) for every B ∈ B(H) there exists δ > 0 such that A + λB ∈ B(H)∧

for every complex λ with |λ| < δ.

CorollaryLet H be an infinite-dimensional separable Hilbert space andφ : B(H)→ B(H) a surjective up to finite rank op linear map.Then φ preserves generalized invertibility in both directions implies φpreserves semi-Fredholm operators in both directions.

strongly preserving generalized inverses

IV. Additive maps strongly preserving generalized inverses

We denote by A−1 the set of invertible elements of A.We shall say that an additive map φ : A → B strongly preservesinvertibility if φ(x−1) = φ(x)−1 for every x ∈ A−1.Similarly, we shall say that φ strongly preserves generalizedinvertibility if φ(y) is a generalized inverse of φ(x) whenever y is ageneralized inverse of x.

Remark. One easily checks that a Jordan homomorphism stronglypreserves invertibility (resp. generalized inverses).

The motivation for this problem is Hua’s theorem (1949) which statesthat

every unital additive map φ between two fields such thatφ(x−1) = φ(x)−1 is an isomorphism or an anti-isomorphism.

Theorem [N.Boudi and M.Mbekhta, 2010 ]Let A and B be unital Banach algebras and let φ : A → B be anadditive map. Then φ strongly preserves invertibility if and only ifφ(1)φ is a unital Jordan homomorphism and φ(1) commutes with therange of φ.

The motivation for this problem is Hua’s theorem (1949) which statesthat

every unital additive map φ between two fields such thatφ(x−1) = φ(x)−1 is an isomorphism or an anti-isomorphism.

Theorem [N.Boudi and M.Mbekhta, 2010 ]Let A and B be unital Banach algebras and let φ : A → B be anadditive map. Then φ strongly preserves invertibility if and only ifφ(1)φ is a unital Jordan homomorphism and φ(1) commutes with therange of φ.

For the special case of the complex matrix algebra A =Mn(C), wederive the following corollary that provides a more explicit form oflinear maps strongly preserving invertibility.

Corollary

Let φ :Mn(C)→Mn(C), be a linear map. Then the followingconditions are equivalent:(1) φ preserves invertibility ;(2) there is a λ ∈ {−1, 1} such that φ takes one of the followingforms:

φ(x) = λaxa−1 or φ(x) = λaxtra−1,

for some invertible element a ∈Mn(C).

Theorem [N.Boudi and M.Mbekhta]Let A and B be unital complex Banach algebras and let φ : A → Bbe an additive map such that 1 ∈ Im(φ) or φ(1) is invertible. Then thefollowing conditions are equivalent:(i) φ strongly preserves generalized invertibility;(ii) φ(1)φ is a unital Jordan homomorphism and φ(1) commutes withthe range of φ.

For the special case of linear maps over the complex matrix algebraA =Mn(C), we deduce the following corollary that gives a moreexplicit form of the linear maps strongly preserving generalizedinvertibility.

Corollary

Let φ :Mn(C)→Mn(C), be a linear map. Then φ stronglypreserves generalized inverses if and only if either φ = 0 or there isλ ∈ {−1, 1} such that φ takes one of the following forms:

N. Boudi and M.Mbekhta Additive maps preserving stronglygeneralized inverses, J. Operator Theory 64 (2010), 117-130

Corollary

Moore-Penrose inverses

V. Moore-Penrose inverses preservers mapsIn the context C∗-algebras, it is well known that every generalizedinvertible element a has a unique generalized inverse b for which aband ba are projections, such an element b is called the Moore-Penroseinverse of a and denoted by a†.In other words, a† is the unique element of A that satisfies:

aa†a = a, a†aa† = a†, (aa†)∗ = aa†, (a†a)∗ = a†a.

Let A† denotes the set of all elements of A having a Moore-Penroseinverse.We will say that a linear map φ : A → B preserves stronglyMoore-Penrose invertibility if

φ(x†) = φ(x)†, ∀x ∈ A†.

A and B will denote a C∗-algebras, and we will say that a linear mapφ : A → B is C∗-Jordan homomorphism if it is a Jordanhomomorphism which preserves the adjoint operation, i.e.φ(x∗) = φ(x)∗ for all x in A.The C∗-homomorphism and C∗-anti-homomorphism are analogouslydefined.

TheoremLet A be a C∗-algebra of real rank zero and B a prime C∗-algebra.Let φ : A → B be a surjective, unital linear map. Then the followingconditions are equivalent:1) φ(x†) = φ(x)† for all x ∈ A†;2) φ is either a C∗-homomorphism or a C∗-anti-homomorphism.

As an application of the above theorem in the context of theC∗-algebra B(H) of bounded linear operator on complex separableHilbert space, we derive the following result which characterizes thesurjective unital additive maps from B(H) onto itself that stronglypreserves Moore-Penrose inverses.Denote by B†(H) the set of the operators on H that possess aMoore-Penrose inverse.

Corollary

Let φ : B(H)→ B(H) be a surjective unital additive map. Then thefollowing conditions are equivalent:(1) φ(T†) = φ(T)† for all T ∈ B†(H);(2) there is a unitary operator U in B(H) such that φ takes one of thefollowing forms

φ(T) = UTU∗ or φ(T) = UT trU∗ for all T,

where T tr is the transpose of T with respect to an arbitrary but fixedorthonormal basis of H.Mostafa Mbekhta LINEAR PRESERVER PROBLEMS: generalized inverse

In connection with Theorem, we conclude by the following conjecture

conjecture

Let A and B be C∗-algebras. Let φ : A → B be a surjective linearmap. Then the following conditions are equivalent:1) φ(x†) = φ(x)† for all x ∈ A†;2) φ is a C∗-Jordan homomorphism.

ascent and descent

VI. Additive preservers of the ascent, descent and related subsetsThe ascent a(T) and descent d(T) of T ∈ L(X) are defined bya(T) = inf{n ≥ 0 : ker(Tn) = ker(Tn+1)}d(T) = inf{n ≥ 0 : R(Tn) = R(Tn+1)}, where the infimum over theempty set is taken to be infinite.

An operator T ∈ L(X) is said to have a Drazin inverse, or to beDrazin invertible, if there exists S ∈ L(X) and a non-negative integern such that

Tn+1S = Tn, STS = S and TS = ST. (1)

Note that if T possesses a Drazin inverse, then it is unique and thesmallest non-negative integer n in (1) is called the index of T and isdenoted by i(T). It is well known that T is Drazin invertibe if and onlyif it has finite ascent and descent, and in this case a(T) = d(T) = i(T).

ascent and descent

VI. Additive preservers of the ascent, descent and related subsetsThe ascent a(T) and descent d(T) of T ∈ L(X) are defined bya(T) = inf{n ≥ 0 : ker(Tn) = ker(Tn+1)}d(T) = inf{n ≥ 0 : R(Tn) = R(Tn+1)}, where the infimum over theempty set is taken to be infinite.An operator T ∈ L(X) is said to have a Drazin inverse, or to beDrazin invertible, if there exists S ∈ L(X) and a non-negative integern such that

Tn+1S = Tn, STS = S and TS = ST. (1)

Note that if T possesses a Drazin inverse, then it is unique and thesmallest non-negative integer n in (1) is called the index of T and isdenoted by i(T). It is well known that T is Drazin invertibe if and onlyif it has finite ascent and descent, and in this case a(T) = d(T) = i(T).

Recall also that an operator T ∈ L(X) is called upper (resp. lower)semi-Fredholm if R(T) is closed and dim N(T) (resp. codimR(T)) isfinite. The set of such operators is denoted by F+(X) (resp. F−(X)).The Fredholm and semi-Fredholm subsets are defined byF(X) := F+(X) ∩ F−(X) and F±(X) := F+(X) ∪ F−(X),respectively.

Let us introduce the following subsets :

(i) A(X) := {T ∈ L(X) : a(T) <∞} the set of finite ascentoperators,

(ii) D(X) := {T ∈ L(X) : d(T) <∞} the set of finite descentoperators,

(iii) Dr(X) := A(X) ∩ D(X) the set of Drazin invertible operators,

(iv) B+(X) := F+(X) ∩ A(X) the set of upper semi-Browderoperators,

(v) B−(X) := F−(X) ∩ D(X) the set of lower semi-Browderoperators,

(vi) B±(X) := B+(X) ∪ B−(X) the set of semi-Browder operators,

(vii) B(X) := B+(X) ∩ B−(X) the set of Browder operators.

Let S denotes one of the subsets (i)-(vii). A surjective additive mapsΦ : L(X)→ L(Y) is said to preserve S in the both direction if T ∈ Sif and only if Φ(T) ∈ S.

Theorem [Mbekhta, Muller, Oudghiri]

Let Φ : L(H)→ L(K) be a surjective additive continuous map. Thenthe following assertions are equivalent :

(i) Φ preserves the finiteness-of-ascent.

(ii) Φ preserves the finiteness-of-descent.

(iii) Φ preserves in both direction B+.

(iv) Φ preserves in both direction B−.

(v) there exists an invertible bounded linear, or conjugate linear,operator A : H → K and a non-zero complex number c such thatΦ(S) = cASA−1 for all S ∈ L(H).

(i) Φ preserves the Drazin invertibility.

(ii) Φ preserves in both direction Semi-Browder operators.

(iii) Φ preserves in both direction B.

(iv) there exists an invertible bounded linear, or conjugate linear,operator A : H → K and a non-zero complex number c such thateither Φ(S) = cASA−1 for all S ∈ L(H), or Φ(S) = cAS∗A−1 forall S ∈ L(H).

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