INVITATION TO LINEAR PRESERVER PROBLEMS, PART IIand the induced map ’: C(H) !C(H), (i.e. ’ ˇ=...

INVITATION TO LINEAR PRESERVERPROBLEMS, PART II

Mostafa Mbekhta

Université Lille 1

R.T.O.T.A. 3-5 May 2018Memphis

Mostafa Mbekhta INVITATION TO LINEAR PRESERVER PROBLEMS, PART II

Generalized inverse

I. 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) <∞.

In particular, for the special case of the complex matrix algebraA =Mn(C), we have A = A∧.

Generalized inverse

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

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 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.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 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.

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 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.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 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.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 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.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.

Can we relax the assumptions of this theorem ? The followingtheorem, answers this question in the affirmative.

Generalized inverse

φ(K(H)) = K(H),

Generalized inverse

φ(K(H)) = K(H),

Semi-Fredholm

II. 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 and let F(H) ⊂ B(H) the ideal of all finite rank operators,

Definition We say that φ : B(H)→ B(H) is surjective up to finiterank (resp. compact) operators if for every A ∈ B(H) there existsB ∈ B(H) such that A− φ(B) ∈ F(H) (resp. K(H)).

Semi-Fredholm

II. 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 and let F(H) ⊂ B(H) the ideal of all finite rank operators,

Semi-Fredholm

II. 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 and let F(H) ⊂ B(H) the ideal of all finite rank operators,

Semi-Fredholm

II. 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 and let F(H) ⊂ B(H) the ideal of all finite rank operators,

Semi-Fredholm

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, thenφ(K(H)) ⊆ K(H) and the induced map ϕ : C(H)→ C(H) is eitheran automorphism, or an anti-automorphism multiplied by aninvertible element a ∈ C(H).

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.For a Fredholm operator T ∈ B(H) we define the index of T byind(T) = dim Ker T − codim Im T ∈ Z.

Semi-Fredholm

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, thenφ(K(H)) ⊆ K(H) and the induced map ϕ : C(H)→ C(H) is eitheran automorphism, or an anti-automorphism multiplied by aninvertible element a ∈ C(H).

Semi-Fredholm

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, thenφ(K(H)) ⊆ K(H)

and the induced map ϕ : C(H)→ C(H) is eitheran automorphism, or an anti-automorphism multiplied by aninvertible element a ∈ C(H).

Semi-Fredholm

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, thenφ(K(H)) ⊆ K(H) and the induced map ϕ : C(H)→ C(H) is eitheran automorphism, or an anti-automorphism multiplied by aninvertible element a ∈ C(H).

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.For a Fredholm operator T ∈ B(H) we define the index of T byind(T) = dim Ker T − codim Im T ∈ Z.

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.

For a Fredholm operator T ∈ B(H) we define the index of T byind(T) = dim Ker T − codim Im T ∈ Z.

Semi-Fredholm

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, thenφ(K(H)) ⊆ K(H) and the induced map ϕ : C(H)→ C(H) is eitheran automorphism, or an anti-automorphism multiplied by aninvertible element a ∈ C(H).

For the behavior of the index, we have the following result

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 n0 ∈ Z such that either ind(φ(T)) = n0 + ind(T) forevery Fredholm operator T, or ind(φ(T)) = n0 − ind(T) for everyFredholm operator T.

Semi-Fredholm

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, thenφ(K(H)) ⊆ K(H) and the induced map ϕ : C(H)→ C(H) is eitheran automorphism, or an anti-automorphism multiplied by aninvertible element a ∈ C(H).

Semi-Fredholm

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, thenφ(K(H)) ⊆ K(H)

and the induced map ϕ : C(H)→ C(H) is eitheran automorphism, or an anti-automorphism multiplied by aninvertible element a ∈ C(H).

Semi-Fredholm

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, thenφ(K(H)) ⊆ K(H) and the induced map ϕ : C(H)→ C(H) is eitheran automorphism, or an anti-automorphism multiplied by aninvertible element a ∈ C(H).

Semi-Fredholm

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 n0 ∈ Z such that either ind(φ(T)) = n0 + ind(T) forevery Fredholm operator T, or ind(φ(T)) = n0 − ind(T) for everyFredholm operator T.

Semi-Fredholm

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 n0 ∈ Z such that either ind(φ(T)) = n0 + ind(T) forevery Fredholm operator T, or ind(φ(T)) = n0 − ind(T) for everyFredholm operator T.

Semi-Fredholm

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 n0 ∈ Z such that either ind(φ(T)) = n0 + ind(T) forevery Fredholm operator T, or ind(φ(T)) = n0 − ind(T) for everyFredholm operator T.

Semi-Fredholm

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 n0 ∈ Z such that either ind(φ(T)) = n0 + ind(T) forevery Fredholm operator T, or

ind(φ(T)) = n0 − ind(T) for everyFredholm 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)∧

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)∧

Semi-Fredholm

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

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

For T ∈ B(H), the essential spectrum, σe(T), of T , is defined as thespectrum of π(T) in the Calkin algebra C(H), i.e. σe(T) = σ(π(T)).Obviously,

σe(T) = {λ ∈ C; T − λI is not Fredholm }.

We say that a linear map φ : B(H)→ B(H) preserves the essentialspectrum if σe(φ(T)) = σe(T) for all T ∈ B(H).

Theorem[M.M. 2007]Let H be an infinite-dimensional Hilbert space and letφ : B(H)→ B(H) be a linear map. Assume that φ is surjective up tocompact operators. Then the following are equivalent:

(i) φ preserves the essential spectrum;

(ii) φ preserves the set of Fredholm operators in both directions andφ(I) = I − K where K ∈ K(H);

(iii) φ(K(H)) ⊆ K(H) and the induced mapϕ : C(H)→ C(H), ϕ ◦ π = π ◦ φ, is either an automorphism, oran anti-automorphism.

ConjectureLet H be an infinite-dimensional Hilbert space and letφ : B(H)→ B(H) be a linear map. Assume that φ is surjective up tocompact operators. Then the following conditions are equivalent:

(I) φ preserves the essential spectrum;

(II) there exists ψ : B(H)→ B(H) either an automorphism or ananti-automorphism and there exists a linear mapχ : B(H)→ K(H) such that φ(T) = ψ(T) +χ(T), T ∈ B(H);

(III) either

(i) φ(T) = ATA−1 + χ(T) for every T ∈ B(H) where A is aninvertible operator in B(H) and χ : B(H)→ K(H) linear, or

(ii) φ(T) = BT trB−1 + χ(T) for every T ∈ B(H) where B is aninvertible operator in B(H) and χ : B(H)→ K(H) linear.

Notice that the implications (II) ⇐⇒ (III) =⇒ (I) hold. Therefore, itremains to prove that (I) =⇒ (II) or (I) =⇒ (III).

(III) either

(III) either(i) φ(T) = ATA−1 + χ(T) for every T ∈ B(H) where A is an

invertible operator in B(H) and χ : B(H)→ K(H) linear, or

(III) either(i) φ(T) = ATA−1 + χ(T) for every T ∈ B(H) where A is an

invertible operator in B(H) and χ : B(H)→ K(H) linear, or(ii) φ(T) = BT trB−1 + χ(T) for every T ∈ B(H) where B is an

invertible operator in B(H) and χ : B(H)→ K(H) linear.

strongly preserving generalized inverses

III. 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 which states that

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

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.

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 which states that

Theorem [N.Boudi and M.M. 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.

Corollary

Let φ :Mn(C)→Mn(C), be a linear map. Then the followingconditions are equivalent:(1) φ preserves invertibility ;(2) φ strongly preserves invertibility;(3) 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.M. 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.

Corollary

Theorem [N.Boudi and M.M. 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.

Corollary

For generalized invertibility, we have

Theorem [N.Boudi and M.M. 2010]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 linear maps over the complex matrix algebra A =Mn(C),

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:φ(x) = λaxa−1 or φ(x) = λaxtra−1,for some invertible element a ∈Mn(C).

Theorem [N.Boudi and M.M. 2010]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.

Corollary

Theorem [N.Boudi and M.M. 2010]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.

Corollary

Theorem [N.Boudi and M.M. 2010]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.

Corollary

Theorem [N.Boudi and M.M. 2010]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.

Corollary

Moore-Penrose inverses

IV. 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†.We will say that a linear map φ : A → B is C∗-Jordanhomomorphism if it is a Jordan homomorphism which preserves theadjoint operation, i.e. φ(x∗) = φ(x)∗ for all x in A.The C∗-homomorphism and C∗-anti-homomorphism are analogouslydefined.

IV. 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:

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†.We will say that a linear map φ : A → B is C∗-Jordanhomomorphism if it is a Jordan homomorphism which preserves theadjoint operation, i.e. φ(x∗) = φ(x)∗ for all x in A.The C∗-homomorphism and C∗-anti-homomorphism are analogouslydefined.

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†.

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

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†.We will say that a linear map φ : A → B is C∗-Jordanhomomorphism if it is a Jordan homomorphism which preserves theadjoint operation, i.e. φ(x∗) = φ(x)∗ for all x in A.

The C∗-homomorphism and C∗-anti-homomorphism are analogouslydefined.

Theorem[M.M.]Let 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.

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

Theorem[M.M.]Let 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.

Corollary

Theorem[M.M.]Let 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.

Corollary

Theorem[M.M.]Let 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.

Corollary

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 or a C∗-anti-homomorphism..

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 or a C∗-anti-homomorphism..

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 or a C∗-anti-homomorphism..

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 or a C∗-anti-homomorphism..

ascent, descent

V. ascent and descent preserver mapsThe 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, descent

V. ascent and descent preserver mapsThe 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 [M.M, V.Muller, M.Oudghiri 2014]

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(T) = cATA−1 for all T ∈ L(H).

Theorem [M.M, Muller, Oudghiri 2014]

(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(T) = cATA−1 for all T ∈ L(H), or Φ(T) = cAT∗A−1

for all T ∈ L(H).

Group inverse

DefinitionT ∈ L(X) is said to be group invertible if there exists S ∈ L(X) suchthat

TST = T, STS = S and ST = TS.

In this case, S is unique and is denoted by T#.

The set of group invertible operators is denoted by G.

Inv(L(X)) ⊆ G, in this case T# = T−1 for T invertible.

{P ∈ L(X) : P2 = P} ⊆ G, in this case P# = P.

T ∈ G ⇔ T is Drazin invertible with i(T) ≤ 1.

T ∈ G ⇔ T∗ ∈ G(X∗). Furthermore, (T∗)# = (T#)∗.

T ∈ G ⇔ ∃P; P2 = P and T + P is invertible.

Group inverse

Additive preservers of group invertible operators

Theorem [M.M-Oudghiri, 2017]Let X be a real or complex infinite-dimensional Banach space, and letΦ : L(X)→ L(X) be a surjective additive map. The followingassertions are equivalent.

(i) Φ preserves G in both directions;

(ii) there exist a non-zero α ∈ C and a bounded invertible linear, orconjugate linear, operator A between suitable spaces such that

Φ(T) = αATA−1 for all T ∈ L(X)

orΦ(T) = αAT∗A−1 for all T ∈ L(X).

QuestionIt would be interesting to know if the main theorem holds true forsurjective additive maps Φ on L(X) preserving group invertibility inone direction : T ∈ G ⇒ Φ(T) ∈ G.

For each positive integer n, let Dn(X) denote the set of all Drazininvertible operators in L(X) of index n.

Theorem [M.M-Oudghiri-Souilah, 2017]Let X be a real or complex infinite-dimensional Banach space, and letΦ : L(X)→ L(X) be a surjective additive map. Then φ preservesDn(X) if and only ifthere exist a scalar α 6= 0 and a bounded invertible linear, orconjugate linear, operator A such that either

Φ(T) = αATA−1 or Φ(T) = αAT∗A−1.

For T ∈ L(X), we denote by Pn(T) the set of all the poles of order nof its resolvent.

Corollary

Let Φ : L(X)→ L(X) be an additive surjective map, and let n ≥ 2 bean integer.Then Pn(Φ(T)) = Pn(T) for all T ∈ L(X) if and only ifone of the following assertions holds :

1 There is a bijective continuous mapping A : X → X, either linearor conjugate linear, such that

Φ(T) = ATA−1 for all T ∈ L(X).

2 X is reflexive and here is a bijective continuous mappingB : X∗ → X, either linear or conjugate linear, such that

Φ(T) = BT∗B−1 for all T ∈ L(X).

Corollary

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