Some spectral domain in approximate point-spectrum-preserving maps on B ( X )

Let X be an inﬁnite-dimensional complex Banach space, B ( X ) the algebra of all bounded linear operators on X . Denote the spectral domain by σ γ ( T ) = { λ ∈ σ a ( T ) : T that is semi-Fredholm and asc ( T – λ I ) < ∞} . In this paper, we characterize the structure of additive surjective maps ϕ : B ( X ) → B ( X ) with σ γ ( ϕ ( T )) = σ γ ( T ) for all T ∈ B ( X ).


Introduction
The study of preserver problems has a long history and has established many remarkable results in past decades.Preserver problems aim to characterize those linear or nonlinear maps on operator algebras preserving certain properties, subsets or relations ( [1-6, 9, 12, 15, 16, 22]).One of the most famous problems in this area is Kaplansky's problem [12] asking whether every surjective unital invertibility preserving linear map between two semisimple Banach algebras is a Jordan homomorphism.This problem was solved in some special cases of semisimple Banach algebras ( [10,13,23]).Then, Aupetit in [2] solved it for von Neumann algebras.It is known that a spectrum is a very fundamental and key concept in operator theory.Some results about linear or additive maps preserving the spectrum as well as certain parts of the spectrum have been established by many authors ([2, 9, 20, 21]).Recently, many authors are interested in nonadditive preserver problems related to spectral domains of operators.For example, Hajighasemi and Hejazian in [11] characterized the nonlinear surjective maps on B(X ) preserving the semi-Fredholm domain, the Fredholm domain, and the Weyl domain respectively.In [4], Bouramdane and Ech-Chérif El Kettani investigated the form of maps preserving some spectral domains of the skew product of operators.As is known, certain parts of a spectrum of operators are introduced to analyze the structure of operators, such as various spectra in the Weyl-type theorem.The Weyl-type theorem can reflect the connections between several spectra, which has been studied for more than one hundred years.There have been numerous significant results in terms of this.Note that some spectral domains play a key role in the research of the Weyl-type theorem and its perturbation [8,[17][18][19].Moreover, these spectral domains are at most countable and very "small" subsets of a spectrum in general, such as normal eigenvalues, the semi-Fredholm domain in a spectrum, and so on.Thus, how may these spectral domains influence the structure of automorphisms on the algebra of all bounded linear operators on a Banach space?In [20], the authors characterized additive surjective maps ϕ on B(X ) that preserve the semi-Fredholm domain in a spectrum, and showed that such a map is an automorphism or an antiautomorphism on B(X ).In [7], Cao discussed the linear surjective maps preserving upper semi-Weyl operators, and showed that their induced maps on the Calkin algebra are Jordan automorphisms.In this paper, we combine the approximate point spectrum with a semi-Fredholm domain of operators, and consider an additive map that preserves the intersection of a semi-Fredholm domain with finite ascent and approximate point spectrum.How does the spectral domain influence the structure of automorphisms on the algebra of all bounded linear operators on a Banach (or Hilbert) space?
Throughout this paper let X be a complex infinite-dimensional Banach space and B(X ) the algebra of all bounded linear operators on X .For T ∈ B(X ), we denote by X * , T * , N (T), and R(T) the dual space of X , the conjugate operator, the null space, and the range of T, respectively.Let dimN (T) be the dimension of N (T), and codimR(T) be the codimension of R(T).
Recall that a bounded operator T is said to be bounded below if it is injective and has closed range.For an operator T, the ascent of T is defined by If the infimum does not exist, then the asc(T) is defined as ∞.It is known that for some p ≥ 0.
Denote the spectrum, the point spectrum, and the approximate point spectrum of T, respectively, by In [24,25], Cao considered property (R), which is a variant of Weyl's theorem.Also, there is a spectral domain that plays an important role in the study of property (R).Now, we define the spectral domain by Moreover, since asc(T -λI) < ∞, we have that there exists > 0 such that T -λI is bounded below for all λ with 0 < |λλ 0 | < .Then, λ ∈ isoσ a (T), which induces that Thus, the spectral domain σ γ (T) is at most countable and is a very "small" subset of a spectrum in general.
In this paper, we will characterize additive surjective maps ϕ on B(X ) preserving the spectral domain σ γ (T) in both directions.The main result is the following Theorem: , then one of the following assertions holds: (1) there is an invertible operator A ∈ B(X ) such that ϕ(T) = ATA -1 for all T ∈ B(X ); (2) there is a bounded invertible linear operator C : In this case, X must be a reflexive space.

Preliminaries
Let z ∈ X and f ∈ X * , we denote by z ⊗ f the bounded linear rank-one operator if both z and f are nonzero.The rank-one operator z ⊗ f is defined by (z ⊗ f )x = f (x)z for all x ∈ X .For a subset M of X , {M} denotes the closed subspace spanned by M. We first establish some useful results that are needed for the proof of our main Theorem.
The next Proposition gives a criterion of two operators being equal by rank-one operators and the spectral domain σ γ (•).
Proof For any nonzero vector x ∈ X , let G = {f ∈ X * |f (x) = 1}.We choose a scalar α ∈ C such that α > A + B .For any f ∈ G, we define an operator where A e is the essential norm of A. We derive that α ∈ σ γ (A -F f ), and so α ∈ σ γ (B -F f ).It follows that α ∈ σ p (B -F f ).Then, there exists a nonzero vector y f ∈ X such that (B -F f )y f = αy f .We can obtain Putting y = (B -αI) -1 (A -αI)x, it follows from (B -F f )y f = αy f that (B -F f )y = αy for any f ∈ G.We claim that x and y are linearly dependent.Indeed, if x and y are linearly independent, then there exists some f ∈ G such that f (y) = 0.This implies that By = αy.This is a contradiction with the fact α > A + B .It follows that (B -F f )x = αx.Therefore, we obtain Ax = Bx.From the arbitrariness of x, we have A = B.

Proof of main result
In the following, we will give the proof of the main theorem and show the result in four steps.
Then, by Proposition 2.1, we have that dimR(ϕ(T)) ≥ 2. Since ϕ is bijective and ϕ -1 has the same property as ϕ, it follows that ϕ preserves the set of operators of rank one in both directions.
For the operator y ⊗ h and z ⊗ g, since ϕ is surjective, there are two rank-one operators y 0 ⊗ h 0 and z 0 ⊗ g 0 such that Then, x ⊗ f + y 0 ⊗ h 0 and x ⊗ f + z 0 ⊗ g 0 are two rank-one operators.Thus, for any nonzero λ ∈ C, λx ⊗ f + y 0 ⊗ h 0 and λx ⊗ f + z 0 ⊗ g 0 are also rank-one operators.Fix a nonzero complex number λ, we let ϕ(λx ⊗ f ) = λy λ ⊗ g λ , where y λ ⊗ g λ is a rank-one idempotent.Then, ϕ(λx ⊗ f + y 0 ⊗ h 0 ) = λy λ ⊗ g λ + y ⊗ h is also rank one.We obtain that y λ and y are linearly dependent or the same is true for g λ and h.We claim that y λ and y are linearly dependent.Indeed, suppose on the contrary, we can obtain g λ and h are linearly dependent.
Therefore, ϕ preserves idempotents of rank one and their linear spans in both directions.
From the main result of [16] it gives that (1) There is an invertible operator A ∈ B(X ) such that ϕ(T) = AFA -1 for all finite-rank operators F ∈ B(X ), or (2) There is a bounded invertible linear operator C : X * → X such that ϕ(T) = CF * C -1 for all finite-rank operators F ∈ B(X ).In this case X must be a reflexive space.
Step 4. ϕ takes the desired from.Assume that (1) holds.Let T ∈ B(X ) and for any rank-one operator F, we have Then, we obtain that T = A -1 ϕ(T)A by the Proposition 2.3.Consequently, ϕ(T) = ATA -1 for all T ∈ B(X ).If (2) holds, then we similarly have that ϕ(T) = CT * C -1 for all T ∈ B(X ).
Thus, we obtain that w, z are nonzero vectors such thatw ∈ N (T + K) and z ∈ N (T -K).Thus, we have that the operators T -K and T + K are Fredholm with N (T -K) = {0} and N (T + K) = {0}.
22 A : R → R, there exists a complex number μ ∈ C such that the operators μI + T 22 A, μI + 2T 22 A, and μI -2T 22 A are invertible.It follows that the operators μA -1 + T 22 , μA -1 + 2T 22 and μA -1 -2T 22 are invertible.With respect to the decomposition X = L ⊕ K = M ⊕ R, we now define an operator S ∈ B(X ) by