Linear Maps Preserving the Set of Semi-Weyl Operators

Abstract

Let H be an infinite-dimensional separable complex Hilbert space and $B\left(H\right)$ the algebra of all bounded linear operators on H. In this paper, we characterized the linear maps $\varphi :B\left(H\right)\to B\left(H\right)$, which are surjective up to compact operators preserving the set of left semi-Weyl operators in both directions. As an application, we proved that $\varphi$ preserves the essential approximate point spectrum if and only if the ideal of all compact operators is invariant under $\varphi$ and the induced map $\phi$ on the Calkin algebra is an automorphism. Moreover, we have $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ if both $\varphi \left(T\right)$ and T are Fredholm.

1. Introduction

Let H be an infinite-dimensional separable complex Hilbert space, $B\left(H\right)$ the algebra of all bounded linear operators on H, and $\mathcal{K}\left(H\right)\subseteq B\left(H\right)$ the closed ideal of all compact operators. For an operator $T\in B\left(H\right)$, we write $T^{*}$ for the conjugate operator of T, $N\left(T\right)$ for its kernel, and $R\left(T\right)$ for its range. The dimension, codimension, and index of T are denoted by $dimT, codimT$, and $indT$, respectively.

An operator $T\in B\left(H\right)$ is called upper semi-Fredholm if $R\left(T\right)$ is closed and $N\left(T\right)$ is finite- dimensional. If $R\left(T\right)$ is closed and finite-codimensional, $T\in B\left(H\right)$ is called a lower semi-Fredholm operator. We call $T\in B\left(H\right)$ Fredholm if $R\left(T\right)$ is closed and finite-codimensional and $N\left(T\right)$ is finite-dimensional. For a semi-Fredholm operator (upper semi-Fredholm operator or lower semi-Fredholm operator), let $n\left(T\right)=dimN\left(T\right)$ and $d\left(T\right)=dimH/R\left(T\right)=codimR\left(T\right)$. The index of a semi-Fredholm operator $T\in B\left(H\right)$ is given by $ind\left(T\right)=n\left(T\right)-d\left(T\right)$. The operator T is Weyl if it is Fredholm of index zero. $T\in B\left(H\right)$ is called left (right) semi-Weyl if T is upper (lower) semi-Fredholm with $ind\left(T\right)\le 0$ ($ind\left(T\right)\ge 0$). Let $S{F}_{+}^{-}\left(H\right)$ denote the set of all left semi-Weyl operators. For an operator $T\in B\left(H\right)$, the spectrum $\sigma \left(T\right)$, the essential spectrum ${\sigma }_e\left(T\right)$, the Weyl spectrum ${\sigma }_w\left(T\right)$, and the essential approximate point spectrum ${\sigma }_{ea}\left(T\right)$ of T are defined by $\sigma \left(T\right)=\{\lambda \in C:T-\lambda I\mathrm{is}\mathrm{not}\mathrm{invertible}\}$, ${\sigma }_e\left(T\right)=\{\lambda \in C:T-\lambda I\mathrm{is}\mathrm{not}\mathrm{Fredholm}\}$, ${\sigma }_w\left(T\right)=\{\lambda \in C:T-\lambda I\mathrm{is}\mathrm{not}\mathrm{Weyl}\}$, and ${\sigma }_{ea}\left(T\right)=\{\lambda \in C:T-\lambda I\mathrm{is}\mathrm{not}\mathrm{left}\mathrm{semi}-\mathrm{Weyl}\}$, respectively.

Let $\Phi \left(H\right)\subseteq B\left(H\right)$ be the set of all Fredholm operators. We denote the Calkin algebra $B\left(H\right)/\mathcal{K}\left(H\right)$ by $\mathcal{C}\left(H\right)$. Let $\pi :B\left(H\right)\to \mathcal{C}\left(H\right)$ be the quotient map. It is well known that $T\in \Phi \left(H\right)$ if and only if $\pi \left(T\right)$ is invertible in $\mathcal{C}\left(H\right)$.

A bijective linear map $\varphi :B\left(H\right)\to B\left(H\right)$ is called a Jordan isomorphism if $\varphi \left(A^2\right)={\left(\varphi \left(A\right)\right)}^2$ for every $A\in B\left(H\right)$ or, equivalently, $\varphi (AB+BA)=\varphi \left(A\right)\varphi \left(B\right)+\varphi \left(B\right)\varphi \left(A\right)$ for all A and B in $B\left(H\right)$. It is obvious that every isomorphism and every anti-isomorphism is a Jordan isomorphism. For further properties of Jordan homomorphisms, we refer the reader to [1,2].

In the last two decades, there has been considerable interest in the so-called linear preserver problems (see the survey articles [3,4,5]). The goal of studying linear preservers is to give structural characterizations of linear maps on algebras having some special properties such as leaving invariant a certain subset of the algebra or leaving invariant a certain function on the algebra. One of the most-famous problems in this direction is Kaplansky’s problem ([6]): Let $\varphi$ be a surjective linear map between two semi-simple Banach algebras $\mathcal{A}$ and $\mathcal{B}$. Suppose that $\sigma \left(\varphi \right(x\left)\right)=\sigma \left(x\right)$ for all $x\in \mathcal{A}$. Is it true that $\varphi$ is a Jordan isomorphism? This problem was first solved in the finite-dimensional case. Dieudonné ([7]) and Marcus and Purves ([8]) proved that every unital invertibility preserving linear map on a complex matrix algebra is either an inner automorphism or a linear anti-automorphism. This result was later extended to the algebra of all bounded linear operators on a Banach space by Sourour ([9]) and to von Neumann algebra by Aupetit ([10]). Many linear preserver problems have been of interest for infinite-dimensional cases. For the most-significant partial results relevant to our discussions, we refer the reader to [9,10,11]. New contributions to the study of the linear preserver problem have been recently made by Mbekhta in [12], Alizadeh and Shakeri in [13], Bueno, Furtado, and Sivakumar in [14], Buenoa, Furtadob, Klausmeierc, and Veltrid in [15], and Bendaoud, Bourhim and Sarih in [16].

In this article, we studied linear maps preserving left (right) semi-Weyl operators in both directions. We characterized the linear maps $\varphi :B\left(H\right)\to B\left(H\right)$, which are surjective up to compact operators preserving the set of semi-Weyl operators in both directions. As an application, we proved that $\varphi$ preserves the essential approximate point spectrum if and only if the ideal of all compact operators is invariant under $\varphi$, the induced map $\phi$ on the Calkin algebra is an automorphism, and $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ if both $\varphi \left(T\right)$ and T are Fredholm.

2. Linear Maps Preserving the Set of Left (Right) Semi-Weyl Fredholm Operators

We say that a linear map $\varphi$ preserves property X in both directions, which means that if T is in the domain, then T has property X if and only if $\varphi \left(T\right)$ has property X. Therefore, a linear map $\varphi :B\left(H\right)\to B\left(H\right)$ preserves the set of left semi-Weyl operators in both directions if $T\in S{F}_{+}^{-}\left(H\right)\iff \varphi \left(T\right)\in S{F}_{+}^{-}\left(H\right)$.

A linear map $\varphi :B\left(H\right)\to B\left(H\right)$ is said to be surjective up to compact operators if, for every $T\in B\left(H\right)$, there exists $T^{\prime }\in B\left(H\right)$ such that $T-\varphi \left(T^{\prime }\right)\in \mathcal{K}\left(H\right)$. It is clear that if $\varphi$ is surjective, then it is surjective up to compact operators.

In order to prove the theorem and the corollaries, we need some known results.

Lemma 1

(Theorem 4.2 in [5]). Let H be an infinite-dimensional separable Hilbert space and $\varphi :B\left(H\right)\to B\left(H\right)$ be a linear map surjective up to compact operators. Then, the following are equivalent:

(1) 

ϕ preserves upper semi-Fredholm operators in both directions;

(2) 

ϕ preserves lower semi-Fredholm operators in both directions;

(3) 

$\varphi \left(\mathcal{K}\right(H\left)\right)\subseteq \mathcal{K}\left(H\right)$, and the induced map $\phi :\mathcal{C}\left(H\right)\to \mathcal{C}\left(H\right),\phi \circ \pi =\pi \circ \varphi$ is an automorphism multiplied by an invertible element $a\in \mathcal{C}\left(H\right)$.

Lemma 2

(Theorem 2.1 in [12]). Let H be an infinite-dimensional separable Hilbert space and $\varphi :B\left(H\right)\to B\left(H\right)$ be a linear map surjective up to compact operators. Then, the following are equivalent:

 (1)

ϕ preserves the set of Fredholm operators in both directions;

 (2)

$\varphi \left(\mathcal{K}\right(H\left)\right)\subseteq \mathcal{K}\left(H\right)$, and the induced map $\phi :\mathcal{C}\left(H\right)\to \mathcal{C}\left(H\right),\phi \circ \pi =\pi \circ \varphi ,$ is the composition of either an automorphism or an anti-automorphism and left multiplication by an invertible element in $\mathcal{C}\left(H\right)$.

Lemma 3

(Theorem 4.8 in [3]). Let A be a factor, and let B be a primitive Banach algebra. For a surjective up to inessential elements linear map $\varphi :A\to B$, the following are equivalent:

 (1)

ϕ preserves Fredholm elements in both directions and $\varphi \left(I\right)$ is the Weyl element of B;

 (2)

ϕ preserves Weyl elements in both directions;

 (3)

Let $\mathcal{I}\left(A\right)$ and $\mathcal{I}\left(B\right)$ be the ideal of the inessential elements of A and B. Then, $\varphi \left(\mathcal{I}\right(A\left)\right)\subseteq \mathcal{I}\left(B\right)$, and the induced map $\phi :\mathcal{C}\left(A\right)\to \mathcal{C}\left(B\right)$ is either an isomorphism or an anti-isomorphism multiplied by an invertible element $a\in B$.

Lemma 4

(Theorem 3.1 in [4]). Let A be a unital C*-algebra of real rank zero and B a unital semi-simple complex Banach algebra. Let $\Delta (·)$ denote any one of the spectral functions $\sigma (·),{\sigma }_l(·),{\sigma }_r(·),{\sigma }_l(·)\cap {\sigma }_r(·),\partial \sigma (·)$, and $\eta \sigma (·)$. Suppose $\varphi :A\to B$ is a surjective linear map. If $\Delta \left(\varphi \right(T\left)\right)\subseteq \Delta \left(T\right)$ for every $T\in A$, then ϕ is a Jordan homomorphism. Furthermore, if B is prime, then ϕ is either a homomorphism or an anti-homomorphism.

Theorem 1. 

Let H be an infinite-dimensional Hilbert space, and let $\varphi :B\left(H\right)\to B\left(H\right)$ be a linear map preserving left (or right) semi-Weyl operators in both directions. Assume that ϕ is surjective up to compact operators and $\varphi \left(I\right)$ is Weyl, then $\varphi \left(\mathcal{K}\right(H\left)\right)\subseteq \mathcal{K}\left(H\right)$, and the induced map $\phi :\mathcal{C}\left(H\right)\to \mathcal{C}\left(H\right),\phi \circ \pi =\pi \circ \varphi$ is an automorphism multiplied by an invertible element $\left[B\right]\in \mathcal{C}\left(H\right)$.

Proof. 

Suppose that $\varphi :B\left(H\right)\to B\left(H\right)$ is a linear map preserving left semi-Weyl operators in both directions. Let $\varphi \left(I\right)=G+K_0$, where $G\in B\left(H\right)$ is invertible and $K_0\in \mathcal{K}\left(H\right)$. There exists $B_0\in B\left(H\right)$ such that $G{B}_0=B_{0}G=I$.

The linear map ${\varphi }_1:B\left(H\right)\to B\left(H\right)$ is defined by:

$${\varphi }_1\left(T\right)=B_0\varphi \left(T\right),\forall T\in B\left(H\right).$$

Then, ${\varphi }_1$ preserves the left semi-Weyl operators in both directions and ${\varphi }_1\left(I\right)=I+K_1$, where $K_1\in \mathcal{K}\left(H\right)$. Let us give some properties for the linear map ${\varphi }_1$: (i) ${\varphi }_1$ is surjective up to compact operators.

In fact, for any $T\in B\left(H\right)$, there exists $T^{\prime }\in B\left(H\right)$ and $K_2\in \mathcal{K}\left(H\right)$ such that $GT=\varphi \left(T^{\prime }\right)+K_2$. Then, $T=B_{0}GT=B_0\varphi \left(T^{\prime }\right)+K_3={\varphi }_1\left(T^{\prime }\right)+K_3$, where $K_3=B_0{K}_2\in \mathcal{K}\left(H\right)$.

(ii) For any $T\in B\left(H\right)$, ${\sigma }_{ea}\left(T\right)={\sigma }_{ea}\left({\varphi }_1\left(T\right)\right)$.

Since $T-\lambda I\in S{F}_{+}^{-}\left(H\right)$⇔${\varphi }_1(T-\lambda I)={\varphi }_1\left(T\right)-\lambda {\varphi }_1\left(I\right)={\varphi }_1\left(T\right)-\lambda I-\lambda K_1\in S{F}_{+}^{-}\left(H\right)$⇔${\varphi }_1\left(T\right)-\lambda I\in S{F}_{+}^{-}\left(H\right)$, it follows that ${\sigma }_{ea}\left(T\right)={\sigma }_{ea}\left({\varphi }_1\left(T\right)\right)$ for any $T\in B\left(H\right)$.

(iii) ${\varphi }_1$ preserves compact operators in both directions.

First, we claim that

$$\mathcal{K}\left(H\right)=\{K\in B\left(H\right):K+S{F}_{+}^{-}\left(H\right)\in S{F}_{+}^{-}\left(H\right)\}$$

$$=\{K\in B\left(H\right):{\sigma }_{ea}(T+K)={\sigma }_{ea}\left(T\right)\mathrm{for}\mathrm{all}T\in S{F}_{+}^{-}\left(H\right)\}.$$

From the stability properties of the index function, it is clear that $\mathcal{K}\left(H\right)\subseteq \{K\in B\left(H\right):K+S{F}_{+}^{-}\left(H\right)\in S{F}_{+}^{-}\left(H\right)\}$$=\{K\in B\left(H\right):{\sigma }_{ea}(T+K)={\sigma }_{ea}\left(T\right)\mathrm{for}\mathrm{all}T\in S{F}_{+}^{-}\left(H\right)\}.$

Let $\partial E$ and $\eta E$ denote the boundary and the polynomial convex hull of a compact subset E of ℂ, respectively. For any $T\in B\left(H\right)$, since

$$\partial {\sigma }_w\left(T\right)\subseteq \partial {\sigma }_e\left(T\right)\subseteq {\sigma }_e\left(T\right)\subseteq {\sigma }_w\left(T\right)\mathrm{and}\partial {\sigma }_w\left(T\right)\subseteq \partial {\sigma }_{ea}\left(T\right)\subseteq {\sigma }_{ea}\left(T\right)\subseteq {\sigma }_w\left(T\right),$$

it follows that $\eta {\sigma }_{ea}\left(T\right)=\eta {\sigma }_w\left(T\right)=\eta {\sigma }_e\left(T\right)$.

Now, let $K\in B\left(H\right)$ such that ${\sigma }_{ea}(T+K)={\sigma }_{ea}\left(T\right)$ for all $T\in B\left(H\right)$. Then, $\eta {\sigma }_e(T+K)=\eta {\sigma }_e\left(T\right)$ for all $T\in B\left(H\right)$. Taking into account the semisimplicity of $\mathcal{C}\left(H\right)$ and the spectral characterization of the radical, it is not difficult to prove that $\mathcal{K}\left(H\right)=\{K\in B\left(H\right):K+S{F}_{+}^{-}\left(H\right)\in S{F}_{+}^{-}\left(H\right)\}$$=\{K\in B\left(H\right):{\sigma }_{ea}(T+K)={\sigma }_{ea}\left(T\right) \mathrm{for}\mathrm{all}T\in S{F}_{+}^{-}\left(H\right)\}.$

Let $K\in \mathcal{K}\left(H\right)$, for any $T\in S{F}_{+}^{-}\left(H\right)$; since ${\varphi }_1$ preserves left semi-Weyl operators in both directions, there exists $T^{\prime }\in S{F}_{+}^{-}\left(H\right)$ and $K^{\prime }\in \mathcal{K}\left(H\right)$ for which $T={\varphi }_1\left(T^{\prime }\right)+K^{\prime }$. Hence, $T+{\varphi }_1\left(K\right)={\varphi }_1\left(T^{\prime }\right)+K^{\prime }+{\varphi }_1\left(K\right)={\varphi }_1(T^{\prime }+K)+K^{\prime }\in S{F}_{+}^{-}\left(H\right)$. Then, ${\varphi }_1\left(K\right)\in \mathcal{K}\left(H\right)$. For the converse, let ${\varphi }_1\left(K\right)\in \mathcal{K}\left(H\right)$, for any $T\in S{F}_{+}^{-}\left(H\right)$, ${\varphi }_1(T+K)={\varphi }_1\left(T\right)+{\varphi }_1\left(K\right)\in S{F}_{+}^{-}\left(H\right)$, then $T+K\in S{F}_{+}^{-}\left(H\right)$. It follows that $K\in \mathcal{K}\left(H\right)$. Now, we prove that ${\varphi }_1$ preserves compact operators in both directions.

(iv) $N\left({\varphi }_1\right)\subseteq \mathcal{K}\left(H\right)$, and consequently, $N\left(\varphi \right)\subseteq \mathcal{K}\left(H\right)$.

If $K\in N\left({\varphi }_1\right)$ and $T\in S{F}_{+}^{-}\left(H\right)$, then ${\varphi }_1(T+K)={\varphi }_1\left(T\right)\in S{F}_{+}^{-}\left(H\right)$. Thus, for all $T\in S{F}_{+}^{-}\left(H\right)$, $T+K\in S{F}_{+}^{-}\left(H\right)$. From the proof of (iii), we know that $K\in \mathcal{K}\left(H\right)$.

(v) Let ${\phi }_1:\mathcal{C}\left(H\right)\to \mathcal{C}\left(H\right)$ be an induced linear map such that ${\varphi }_1\circ \pi =\pi \circ {\varphi }_1$, then ${\phi }_1$ is an isomorphism or an anti-isomorphism.

From the fact that $\mathcal{K}\left(H\right)$ is invariant under ${\varphi }_1$, then ${\varphi }_1$ induces a linear map ${\phi }_1:\mathcal{C}\left(H\right)\to \mathcal{C}\left(H\right)$ such that ${\phi }_1\circ \pi =\pi \circ {\varphi }_1$. Clearly, ${\phi }_1$ is surjective, since ${\varphi }_1$ is surjective up to compact operators. We prove now that ${\phi }_1$ is injective. Since $N\left({\phi }_1\right)=\pi \left(N\left({\varphi }_1\right)\right)$ and $N\left({\varphi }_1\right)\subseteq \mathcal{K}\left(H\right)$, we can obtain that ${\phi }_1$ is injective.

From (ii), we know that, for any $T\in B\left(H\right)$, $\eta {\sigma }_{ea}\left(T\right)=\eta {\sigma }_{ea}\left({\varphi }_1\left(T\right)\right)$. Then, from (iii), $\eta {\sigma }_e\left(T\right)=\eta {\sigma }_e\left({\varphi }_1\left(T\right)\right)$. This shows that ${\varphi }_1$ is an $\eta {\sigma }_e$-preserving map. Thus, the induced mapping ${\phi }_1$ is an $\eta \sigma$-preserving map. By Lemma 4, ${\phi }_1$ is either an isomorphism or an anti-isomorphism.

(vi) ${\phi }_1$ is an isomorphism.

First, we will prove that ${\varphi }_1$ preserves upper semi-Fredholm operators in both directions. By Lemma 2, we know that ${\varphi }_1$ preserves Fredholm operators in both directions. Let $T\in B\left(H\right)$ be an upper semi-Fredholm; there are two cases to consider: $d\left(T\right)=\infty$ and $d\left(T\right)<\infty$. If $d\left(T\right)=\infty$, using the fact that ${\varphi }_1:B\left(H\right)\to B\left(H\right)$ is a linear map preserving left semi-Weyl operators in both directions, we know that ${\varphi }_1\left(T\right)$ is upper semi-Fredholm. If $d\left(T\right)<\infty$, then T is Fredholm; thus, ${\varphi }_1\left(T\right)$ is Fredholm since ${\varphi }_1$ preserves Fredholm operators in both directions. Using the same way, we can prove that T is upper semi-Fredholm if ${\varphi }_1\left(T\right)$ is upper semi-Fredholm. By Lemma 1, ${\phi }_1$ is an isomorphism.

From the definition of ${\varphi }_1$, we know that $\varphi$ preserves compact operators in both directions, and hence, $\mathcal{K}\left(H\right)$ is invariant under $\varphi$. Let $\varphi$ induce a linear map $\phi :\mathcal{C}\left(H\right)\to \mathcal{C}\left(H\right)$ such that $\phi \circ \pi =\pi \circ \varphi$. Then, $\phi ={\left[B\right]}^{-1}{\phi }_1$.

Similar to the above proof, the result is true if $\varphi$ is a linear map preserving right semi-Weyl operators in both directions. The proof is completed. □

Under the same hypothesis and notation as in Theorem 1, we obtain that ${\varphi }_1$ preserves the essential spectrum ([12], Theorem 3.2). Then, $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ or $ind\left(\varphi \right(T\left)\right)=-ind\left(T\right)$ for any $T\in \Phi \left(H\right)$. Since ${\varphi }_1$ preserves left semi-Weyl operators in both directions, it follows that $ind\left(\varphi \right(T\left)\right)·ind\left(T\right)\ge 0$ for any $T\in \Phi \left(H\right)$. Thus, $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ for any $T\in \Phi \left(H\right)$. Furthermore, we can prove that $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ for any upper (lower) semi-Fredholm operator $T\in B\left(H\right)$. By Lemma 1, Lemma 2, and Lemma 3, we can obtain:

Corollary 1.

Let $\varphi :B\left(H\right)\to B\left(H\right)$ be a linear map preserving left (right) semi-Weyl operators in both directions. Assume that ϕ is surjective up to compact operators and $\varphi \left(I\right)$ is Weyl, then:

 (1)

ϕ preserves Fredholm operators in both directions;

 (2)

ϕ preserves Weyl operators in both directions;

 (3)

ϕ preserves upper semi-Fredholm operators in both directions;

 (4)

ϕ preserves lower semi-Fredholm operators in both directions;

 (5)

ϕ preserves semi-Fredholm operators in both directions;

 (6)

For any $T\in \Phi \left(H\right)$, $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$;

 (7)

For any upper (lower) semi-Fredholm operator T, $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$.

Remark 1.

If $\varphi :B\left(H\right)\to B\left(H\right)$ is a linear map preserving Fredholm operators (or upper semi-Fredholm operators, or lower semi-Fredholm operators, or semi-Fredholm operators) in both directions, we cannot induce that ϕ is a linear map preserving left semi-Weyl operators in both directions. For example, let $A, B\in B\left({\ell }_2\right)$ be defined by:

$$A(x_1, x_2, x_3,\cdots )=(x_2, x_3, x_4,\cdots ),B(x_1, x_2, x_3,\cdots )=(0, 0, 0, x_1, x_2,\cdots ),$$

then there exists $A_1, B_1\in B\left({\ell }_2\right)$ such that $A{A}_1=B_{1}B=I$. Define $\varphi :B\left({\ell }_2\right)\to B\left({\ell }_2\right)$ as $\varphi \left(T\right)=ATB$, $T\in B\left({\ell }_2\right)$. We can see that ϕ is surjective and preserves Fredholm operators (upper semi-Fredholm operators, lower semi-Fredholm operators, semi-Fredholm operators) in both directions, but ϕ is not a linear map preserving left semi-Weyl operators in both directions.

From Remark 1, we have the question: If $\varphi :B\left(H\right)\to B\left(H\right)$ is a linear map preserving Fredholm operators (or upper semi-Fredholm operators, or lower semi-Fredholm operators, or semi-Fredholm operators) in both directions, when does ϕ preserve left semi-Weyl operators in both directions. To answer this question, let us begin by a Lemma (Lemma 2.4 in [5]).

Lemma 5.

Let $A\in B\left(H\right)$ be a lower (respectively upper) semi-Fredholm. If A is not Fredholm, then there exists a lower (respectively upper) semi-Fredholm operator B such that every non-trivial linear combination $\lambda A+\mu B$, $\lambda \ne 0$ or $\mu \ne 0$, is lower (respectively upper) semi-Fredholm, but not Fredholm.

Corollary 2.

Let $\varphi :B\left(H\right)\to B\left(H\right)$ be a linear map preserving left (right) semi-Weyl operators in both directions. Assume that ϕ is surjective up to compact operators, then $\varphi \left(I\right)$ is a Fredholm operator.

Proof. 

Denote $\varphi \left(I\right)=T$. We will prove that T is Fredholm. On the contrary, we assumed that this is not the case. Since I is a left semi-Weyl operator, T must be a left semi-Weyl operator. Then, by Lemma 5, there exists $S\in B\left(H\right)$ such that $\lambda T-S$ is upper semi-Fredholm, but not Fredholm, which means that $\lambda T-S$ is left semi-Weyl. We can further find $R\in B\left(H\right)$ such that $\varphi \left(R\right)=S+K$ for some $K\in \mathcal{K}\left(H\right)$. Any compact perturbation of a left semi-Weyl operator is a left semi-Weyl operator; thus, $\lambda T-\varphi \left(R\right)=\varphi (\lambda I-R)$ is left semi-Weyl for every $\lambda \in C$. As $\varphi :B\left(H\right)\to B\left(H\right)$ is a linear map preserving left semi-Weyl operators in both directions, we obtain that ${\sigma }_{ea}\left(R\right)=\varnothing$, a contradiction. □

Corollary 3.

Let linear map $\varphi :B\left(H\right)\to B\left(H\right)$ be surjective up to compact operators, then the following statements are equivalent:

 (1)

ϕ preserves left semi-Weyl operators in both directions, and $\varphi \left(I\right)$ is Weyl;

 (2)

ϕ preserves left semi-Weyl operators in both directions, and $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ if both $\varphi \left(T\right)$ and T are Fredholm;

 (3)

ϕ preserves right semi-Weyl operators in both directions, and $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ if both $\varphi \left(T\right)$ and T are Fredholm;

 (4)

ϕ preserves Fredholm operators in both directions, and $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ if both $\varphi \left(T\right)$ and T are semi-Fredholm;

 (5)

ϕ preserves upper semi-Fredholm operators in both directions, and $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ if both $\varphi \left(T\right)$ and T are upper semi-Fredholm;

 (6)

$\varphi \left(\mathcal{K}\right(H\left)\right)\subseteq \mathcal{K}\left(H\right)$; the induced map $\phi :\mathcal{C}\left(H\right)\to \mathcal{C}\left(H\right),\phi \circ \pi =\pi \circ \varphi$ is an automorphism multiplied by an invertible element $\left[B\right]\in \mathcal{C}\left(H\right)$, and $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ if both $\varphi \left(T\right)$ and T are Fredholm.

Proof. 

By the proof of Theorem 1 and Corollary 1, we only need to prove that $\left(6\right)\Rightarrow \left(1\right)$. By Lemma 1, we know that $\varphi$ preserves upper semi-Fredholm operators and Fredholm operators in both directions. Let $T\in S{F}_{+}^{-}\left(H\right)$, then $\varphi \left(T\right)$ is upper semi-Fredholm. If $d\left(T\right)=\infty$, then $d\left(\varphi \right(T\left)\right)=\infty$ because $\varphi$ preserves Fredholm operators in both directions, thus $\varphi \left(T\right)\in S{F}_{+}^{-}\left(H\right)$. If $d\left(T\right)<\infty$, then $\varphi \left(T\right)$ is Fredholm, and hence, $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)\le 0$, again $\varphi \left(T\right)\in S{F}_{+}^{-}\left(H\right)$. Using the same way, we can prove that $T\in S{F}_{+}^{-}\left(H\right)$ if $\varphi \left(T\right)\in S{F}_{+}^{-}\left(H\right)$. This proves that $\varphi$ preserves left semi-Weyl operators in both directions. Thus, $\varphi \left(I\right)$ is Fredholm. Since both $\varphi \left(I\right)$ and I are Fredholm, it follows that $ind\left(\varphi \right(I\left)\right)=ind\left(I\right)=0$. Then, $\varphi \left(I\right)$ is Weyl. □

Let $\varphi :B\left(H\right)\to B\left(H\right)$ be surjective up to compact operators. If $\varphi$ preserves left semi-Weyl operators in both directions and $\varphi \left(I\right)$ is Weyl, we cannot induce that $\varphi$ is ${\sigma }_{ea}$-preserving. For example, let $A_1,B_1\in B\left({\ell }_2\right)$ be defined by:

$$A_1(x_1,x_2,x_3,\cdots )=(x_2,x_3,x_4,\cdots ),B_1(x_1,x_2,x_3,\cdots )=(0,x_1,x_2,\cdots ),$$

and define $A=\left(\begin{array}{cc}{A}_1& 0\\ 0& I\end{array}\right)$ and $B=\left(\begin{array}{cc}I& 0\\ 0& B_1\end{array}\right)$. Let $\chi :B({\ell }_2\oplus {\ell }_2)\to \mathcal{K}({\ell }_2\oplus {\ell }_2)$ be a linear map, and consider the linear map $\varphi :B({\ell }_2\oplus {\ell }_2)\to B({\ell }_2\oplus {\ell }_2)$ defined by $\varphi \left(T\right)=ATB+\chi \left(T\right)$. Then, $\varphi$ is surjective up to compact operators and preserves the set of left semi-Weyl operators in both directions; also, $\varphi \left(I\right)$ is Weyl. According to the calculation, we obtain that ${\sigma }_{ea}\left(I\right)=\left\{1\right\}$, while ${\sigma }_{ea}\left(\varphi \left(T\right)\right)=\{\lambda \in C:|\lambda |=1\}$. This says that $\varphi$ is not ${\sigma }_{ea}$-preserving. There is a question: When does a map satisfying the hypothesis of Theorem 1 preserve the essential approximate point spectrum?

Corollary 4.

Let H be an infinite-dimensional Hilbert space, and let $\varphi :B\left(H\right)\to B\left(H\right)$ be a linear map. Assume that ϕ is surjective up to compact operators, then the following statements are equivalent:

 (1)

ϕ preserves left semi-Weyl operators in both directions and $I-\varphi \left(I\right)\in \mathcal{K}\left(H\right)$;

 (2)

ϕ preserves right semi-Weyl operators in both directions and $I-\varphi \left(I\right)\in \mathcal{K}\left(H\right)$;

 (3)

ϕ is ${\sigma }_{ea}$-preserving, i.e., ${\sigma }_{ea}\left(\varphi \left(T\right)\right)={\sigma }_{ea}\left(T\right)$ for all $T\in B\left(H\right)$;

 (4)

$\varphi \left(\mathcal{K}\right(H\left)\right)\subseteq \mathcal{K}\left(H\right)$; the induced map $\phi :\mathcal{C}\left(H\right)\to \mathcal{C}\left(H\right),\phi \circ \pi =\pi \circ \varphi$ is an automorphism, and $ind\left(\varphi \right(T\left)\right)=ind\left(T\right)$ if both $\varphi \left(T\right)$ and T are Fredholm.

Proof. 

In view of the preceding theorem and corollaries, we only need to prove the equivalence of (1) and (3). Suppose that $\varphi$ preserves the left semi-Weyl operators in both directions and $I-\varphi \left(I\right)\in \mathcal{K}\left(H\right)$. Let $\varphi \left(I\right)=I+K_0$, $K_0\in \mathcal{K}\left(H\right)$. Since $T-\lambda I\in S{F}_{+}^{-}\left(H\right)$⇔$\varphi (T-\lambda I)=\varphi \left(T\right)-\lambda \varphi \left(I\right)=\varphi \left(T\right)-\lambda I-\lambda K_0\in S{F}_{+}^{-}\left(H\right)$⇔$\varphi \left(T\right)-\lambda I\in S{F}_{+}^{-}\left(H\right)$, it follows that ${\sigma }_{ea}\left(T\right)={\sigma }_{ea}\left(\varphi \left(T\right)\right)$ for any $T\in B\left(H\right)$. For the converse, suppose that ${\sigma }_{ea}\left(\varphi \left(T\right)\right)={\sigma }_{ea}\left(T\right)$ for all $T\in B\left(H\right)$, then $\varphi$ preserves the left semi-Weyl operators in both directions. We need to prove that $I-\varphi \left(I\right)\in \mathcal{K}\left(H\right)$. Put $K=\varphi \left(I\right)-I$. Let $T\in B\left(H\right)$, $T^{\prime }\in B\left(H\right)$, and $K^{\prime }\in \mathcal{K}\left(H\right)$ for which $T=\varphi \left(T^{\prime }\right)+K^{\prime }$ ($\varphi$ is surjective up to compact operators). Then, ${\sigma }_{ea}\left(T\right)={\sigma }_{ea}(\varphi \left(T^{\prime }\right)+K^{\prime })={\sigma }_{ea}\left(\varphi \left(T^{\prime }\right)\right)={\sigma }_{ea}\left(T^{\prime }\right)$ and

$$\begin{array}{ccc}& & {\sigma }_{ea}(T+K)={\sigma }_{ea}(T+\varphi \left(I\right)-I)={\sigma }_{ea}(T+\varphi \left(I\right))-1\hfill \\ & =& {\sigma }_{ea}(\varphi \left(T^{\prime }\right)+\varphi \left(I\right)+K^{\prime })-1={\sigma }_{ea}\left(\varphi (T^{\prime }+I)\right)-1\hfill \\ & =& {\sigma }_{ea}(T^{\prime }+I)-1={\sigma }_{ea}\left(T^{\prime }\right)={\sigma }_{ea}\left(T\right),\hfill \end{array}$$

This gives ${\sigma }_{ea}(T+K)={\sigma }_{ea}\left(T\right)$ for all $T\in B\left(H\right)$. It follows from the proof of Theorem 1 that $K\in B\left(H\right)$ is compact. □

Let $SW\left(H\right)=\{T\in B(H):T$ be left semi-Weyl or right semi-Weyl}. Define the semi-Weyl spectrum ${\sigma }_{SW}\left(T\right)$ of an operator $T\in B\left(H\right)$ as ${\sigma }_{SW}\left(T\right)=\{\lambda \in C:T-\lambda I\notin SW\left(H\right)\}$. Similar to the proof of Theorem 1, we have that $\mathcal{K}\left(H\right)=\{K\in B(H):K+SW(H)\in SW(H\left)\right\}$$=\{K\in B\left(H\right):{\sigma }_{SW}(T+K)={\sigma }_{SW}\left(T\right)\mathrm{for}\mathrm{all}T\in SW\left(H\right)\}.$ We can prove the following:

Corollary 5.

Let H be an infinite-dimensional Hilbert space, and let $\varphi :B\left(H\right)\to B\left(H\right)$ be a linear map. Assume that ϕ is surjective up to compact operators, then the following statements are equivalent:

 (1)

ϕ preserves semi-Weyl operators in both directions, and $I-\varphi \left(I\right)\in \mathcal{K}\left(H\right)$;

 (2)

ϕ is ${\sigma }_{SW}$-preserving, i.e., ${\sigma }_{SW}\left(\varphi \left(T\right)\right)={\sigma }_{SW}\left(T\right)$ for all $T\in B\left(H\right)$;

 (3)

ϕ preserves semi-Fredholm operators in both directions, and $I-\varphi \left(I\right)\in \mathcal{K}\left(H\right)$;

 (4)

$\varphi \left(\mathcal{K}\right(H\left)\right)\subseteq \mathcal{K}\left(H\right)$; the induced map $\phi :\mathcal{C}\left(H\right)\to \mathcal{C}\left(H\right),\phi \circ \pi =\pi \circ \varphi$ is an automorphism or an anti-isomorphism.

We conclude this paper by a natural conjecture that we have been unable to answer:

Conjecture 1. 

Let H be an infinite-dimensional Hilbert space, and let $\varphi :B\left(H\right)\to B\left(H\right)$ be a linear map. Assume that $\varphi$ is surjective up to compact operators, then the following statements are equivalent:

 (1)

$\varphi$ preserves the essential approximate point spectrum;

 (2)

There exists $\psi :B\left(H\right)\to B\left(H\right)$ an automorphism and there exists $\chi :B\left(H\right)\to \mathcal{K}\left(H\right)$ a linear map such that $\varphi \left(T\right)=\psi \left(T\right)+\chi \left(T\right)$ for every $T\in B\left(H\right)$;

 (3)

$\varphi \left(T\right)=AT{A}^{-1}+\chi \left(T\right)$ for every $T\in B\left(H\right)$, where A is an invertible operator in $B\left(H\right)$ and $\chi :B\left(H\right)\to \mathcal{K}\left(H\right)$ is a linear map.