Self-adjoint perturbations of spectra for upper triangular operator matrices

Linear Algebra and its Applications 531 (2017) 1–21

Contents lists available at ScienceDirect

Linear Algebra and its Applications www.elsevier.com/locate/laa

Self-adjoint perturbations of spectra for upper triangular operator matrices ✩ Xiufeng Wu, Junjie Huang ∗ , Alatancang Chen School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

a r t i c l e

i n f o

Article history: Received 28 August 2016 Accepted 9 May 2017 Available online 19 May 2017 Submitted by P. Semrl MSC: 47A53 47A55 47B99 Keywords: Upper triangular operator matrix Right (left) Fredholm operator Fredholm operator Weyl operator

a b s t r a c t This paper is concerned with the self-adjoint perturbations of the spectra for the upper triangular partial operator matrix with given diagonal entries. A necessary and sufficient condition is given under which such operator matrix admits a Weyl (Fredholm) operator completion by choosing some bounded self-adjoint operator. It is shown that the self-adjoint perturbation of the Weyl (essential) spectrum can be the proper set of the general perturbation. Combining the spectral properties, we further characterize the perturbation of the Weyl (essential) spectrum for Hamiltonian operators. © 2017 Elsevier Inc. All rights reserved.

1. Introduction The partial operator matrix is the operator matrix with some unknown entries, and fixing the corresponding unknown elements gives a definite operator matrix. The partial operator matrix was shown to be very useful in various research fields such as operator ✩

This work is supported by the NNSF of China (No. 11461049).

* Corresponding author. E-mail address: [email protected] (J. Huang). http://dx.doi.org/10.1016/j.laa.2017.05.017 0024-3795/© 2017 Elsevier Inc. All rights reserved.

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

2

theory, numerical analysis, optimal theory, systems theory and engineering problems (cf. [11] and references therein). We assume throughout that H and K are complex separable infinite dimensional Hilbert spaces. If T is a bounded linear operator from H to K, we write T ∈ B(H, K) and, if H = K, T ∈ B(H). By S(H) denote the subset of B(H) whose elements are self-adjoint. The identity operator on H is denoted by IH and simply by I if the underlying space is clear from the context. Let T ∈ B(H, K). Then N (T ), R(T ) and T ∗ are, respectively, used to denote the kernel, the range and the adjoint of T , and we write n(T ) := dim N (T ) and d(T ) := dim N (T ∗ ). For T ∈ B(H, K) with closed range R(T ), T is said to be left Fredholm, if n(T ) < ∞; while if d(T ) < ∞, we say T is right Fredholm. In particular, the operator T is called left invertible if, in addition, n(T ) = 0; while T is said to be right invertible for the case when d(T ) = 0. According to the Fredholm alternative theorem, T is right Fredholm (invertible) if and only if T ∗ is left Fredholm (invertible). If T is both left and right Fredholm (invertible), then it is Fredholm (invertible). Write ind(T ) := n(T ) − d(T ) for the index of T . We say T is Weyl, if T is Fredholm with ind(T ) = 0. Let T ∈ B(H) and λ ∈ C. The fact, T − λ is not left (right) Fredholm, means that λ belongs to the left (right) essential spectrum σle (T )(σre (T )) of T . Similarly, we have the left (right) spectrum σl (T )(σr (T )), the essential spectrum σe (T ), the Weyl spectrum σw (T ) and the spectrum σ(T ). Write ρ(T ) := C \σ(T ) and, for convenience, we define ρ (T ) := C \σ (T ) in which σ ∈ {σle , σre , σe , σl , σr } and ρ ∈ {ρle , ρre , ρe , ρl , ρr }. When the diagonal entries A ∈ B(H) and B ∈ B(K) are fixed, the authors have extensively studied the operator  MX :=

A 0

X B

 ∈ B(H ⊕ K)

with an unknown X ∈ B(H, K). See, e.g., [2–6,8–10,13,15–18,21,22]. In [9], [6] and [4], the perturbations of the spectra, Weyl spectra and essential spectra of MX were, respectively, given by 

σ(MX ) = σl (A) ∪ σr (B) ∪ {λ ∈ C : n(B − λ) = d(A − λ)},

X∈B(H)



σw (MX ) = σle (A) ∪ σre (B)

X∈B(H)

∪{λ ∈ C : n(A − λ) + n(B − λ) = d(A − λ) + d(B − λ)}, 

(1.1)

(1.2)

σe (MX ) = σle (A) ∪ σre (B) ∪ {λ ∈ C : n(B − λ) < ∞, d(A − λ) = ∞}

X∈B(H)

∪{λ ∈ C : n(B − λ) = ∞, d(A − λ) < ∞}. In [16], the self-adjoint perturbation of the spectra



(1.3)

σ(MX ) of MX was described

X∈S(H)

under some assumptions. Hai and Chen made further investigations later in [13], and they obtained a necessary and sufficient condition under which MX is invertible for

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

3

some X ∈ S(H). This paper mainly aims to characterize the Weylness and Fredholmness of MX for some X ∈ S(H). A second aim is to describe the following self-adjoint perturbations 



σw (MX ),

X∈S(H)



σe (MX ),

X∈S(H)

σ(MX ),

X∈S(H)

and explore the relationship between 

σ (MX )

and

X∈B(H)



σ (MX )

X∈S(H)

where σ ∈ {σw , σe }. Finally, a third aim is to develop the analogues for Hamiltonian operators, which is actually our original motivation for considering such self-adjoint perturbations. 2. Preliminaries In this section we collect some well-known fundamental results, and obtain our auxiliary Theorems 2.12, 2.13 and 2.14 which are all necessary for later proofs and of interest by themselves. Let M be a linear subspace of a Hilbert space, M and M⊥ stand for the closure and the orthogonal complement of M, respectively. Write T |M for the restriction of T to M and PM for the orthogonal projection onto M along M⊥ when M is closed. Lemma 2.1 (see [19, Lemma 6]). Let T ∈ B(K, H). Then, for any  > 0, there exist the orthogonal decompositions K = K ⊕ K and H = H ⊕ H such that T (K ) ⊂ H , T x ≤ x for all x ∈ K , T (K ) ⊂ H , T x ≥ x for all x ∈ K .

(2.1)

It is not hard to see the result below. Lemma 2.2. Let T ∈ B(K, H). Then, for any 0 <  < T , there exist the orthogonal decompositions K = K ⊕ K and H = H ⊕ H under which the relations in (2.1) are fulfilled, and T further admits the matrix representation  T =

T |K 0

0 T |K

     K H : → , K H

where T |K is left invertible. The following are well-known semi-Fredholm perturbation properties.

(2.2)

4

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

Lemma 2.3 (see [1, Remark 1.54]). Let T ∈ B(H, K) be a right (left) Fredholm operator. Then there exists  := (T ) > 0 such that S ∈ B(H, K) and S <  implies that T + S is also a right (left) Fredholm operator. Moreover, n(T + S) ≤ n(T ), d(T + S) ≤ d(T ), ind(T + S) = ind(T ). Lemma 2.4 (see [1, Remark 1.54]). Let T ∈ B(H, K) be right (left) Fredholm, and let S ∈ B(H, K) be a compact operator. Then T + S is a right (left) Fredholm operator with ind(T + S) = ind(T ). Lemma 2.5 (see [7, Lemma 5.8]). Let T ∈ B(H, K), then T is compact if and only if R(T ) contains no closed infinite dimensional subspaces. The result above is a characterization for a compact operator in terms of its range. For operators A ∈ B(H) and C ∈ B(K, H), write N (A | C) := {G ∈ B(K, H) : R(AG) ⊂ R(C)}. The two lemmas below are of interest, which reflect the basic properties of the kernel of the row operator (A C) : H ⊕ K → H. Lemma 2.6. Let A ∈ B(H) and C ∈ B(K, H). Then N (A | C) contains a non-compact operator if and only if PH (N ((A C))) contains a closed infinite dimensional subspace. Proof. Assume that PH (N ((A C))) contains a closed infinite dimensional subspace M. Then, there exists a left invertible operator G ∈ B(K, H) satisfying R(G) = M, which is clearly non-compact. It is easy to see R(AG) ⊂ R(C), and hence G ∈ N (A | C). Conversely, let G ∈ N (A | C) be a non-compact operator. Then R(G) contains a closed infinite dimensional subspace M by Lemma 2.5. Obviously, R(A|M ) ⊂ R(C). Hence PH (N ((A C))) contains a closed infinite dimensional subspace M. 2 Lemma 2.7. Let A ∈ B(H) be right Fredholm, and let C ∈ B(K, H) be a non-compact operator. Then (A C) is a right Fredholm operator and PH (N ((A C))) contains a closed infinite dimensional subspace. Proof. Since A is right Fredholm, as an operator from N (A)⊥ ⊕ N (A) ⊕ K to R(A) ⊕ R(A)⊥ , the row operator (A C) has the new matrix representation 

A1 0

0 C1 0 C2

 .

(2.3)

Evidently, A1 is an invertible operator. As an operator from N (A)⊥ ⊕ N (A) ⊕ N (C2 )⊥ ⊕ N (C2 ) to R(A) ⊕ R(A)⊥ , (A C) can be further written as

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21



0 C11 0 C21

A1 0

C12 0

5

 .

(2.4)

It is clear that C11 and C21 are of finite rank by dim N (C2 )⊥ = dim R(C2 ) ≤ d(A) < ∞. Then (A C) is a right Fredholm operator. In view of the non-compactness of C, we know that C12 is a non-compact operator. Hence R(C12 ) contains a closed infinite dimensional −1 subspace M. Since A1 is invertible, A−1 1 (M) is closed and dim A1 (M) = ∞. The desired result follows from A−1 1 (M) ⊂ PH (N ((A C))) immediately. 2 Combining [20, Theorem 1] with Lemma 2.6, we have Lemma 2.8. Let H1 , H2 , H3 and H4 be complex separable infinite dimensional Hilbert spaces, and let A ∈ B(H1 , H3 ), B ∈ B(H2 , H4 ) and C ∈ B(H2 , H3 ) be given operators. A C Then there exists X ∈ B(H1 , H4 ) such that MX = X : H1 ⊕ H2 → H3 ⊕ H4 is B C invertible if and only if (A C) : H1 ⊕ H2 → H3 is right invertible, B : H2 → H3 ⊕ H4 is left invertible and one of the following statements is fulfilled: C ⊥ (i) PH (N ((A C))) and PK (R( B ) ) contain closed infinite dimensional subspaces; A C  (ii) M0 = 0 B is a Weyl operator. In [6] and [4], the authors considered the Fredholmness and the Weylness of upper triangular operator matrices, respectively. Their main conclusions are listed as follows. Lemma 2.9 (see [6, Theorem 3.6]). Let A ∈ B(H) and B ∈ B(K) be given operators. Then there exists X ∈ B(K, H) such that MX is a Weyl operator if and only if A is left Fredholm, B is right Fredholm and one of the following statements is fulfilled: (i) n(B) = d(A) = ∞; (ii) n(B) < ∞, d(A) < ∞ and n(A) + n(B) = d(A) + d(B). In particular, if (ii) is satisfied, then MX is a Weyl operator for any X ∈ B(K, H). Lemma 2.10 (see [4, Theorem 2.4]). Let A ∈ B(H) and B ∈ B(K) be given operators. Then there exists X ∈ B(K, H) such that MX is a Fredholm operator if and only if A is left Fredholm, B is right Fredholm and one of the following statements is fulfilled: (i) n(B) = d(A) = ∞; (ii) n(B) < ∞ and d(A) < ∞. In what follows, we investigate the conditions under which  MX,Y :=

A X

C Y

 ∈ B(H ⊕ K)

is a Weyl (Fredholm) operator by choosing X ∈ B(H, K) and Y ∈ S(K). In this topic, the right (left) invertibility and idempotency of MX,Y with X ∈ B(H, K) and Y ∈ B(K) were considered in [12] and [14], respectively.

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

6

Theorem 2.11. Let A ∈ B(H) and C ∈ B(K, H) be given operators. If C is compact, then there exist X ∈ B(H, K) and Y ∈ S(K) such that MX,Y is a Weyl (Fredholm) operator if and only if A is a Weyl (Fredholm) operator. Proof. Since C is compact, it follows from Lemma 2.4 that MX,Y is Weyl (Fredholm) if A 0  and only if X is Weyl (Fredholm). So, the sufficiency remains trivial, and we now Y consider the necessity. By Lemma 2.9 (Lemma 2.10), Y is left Fredholm, which together with Y ∈ S(K) implies the Weylness of Y . Clearly, A is a Weyl (Fredholm) operator. 2 Theorem 2.12. Let A ∈ B(H) and C ∈ B(K, H) be given operators. If C is non-compact, then there exist X ∈ B(H, K) and Y ∈ S(K) such that MX,Y is a Weyl operator if and only if (A C) is right Fredholm and one of the following statements is fulfilled: (i) A is a Weyl operator; (ii) PH (N ((A C))) contains a closed infinite dimensional subspace. Proof. Necessity. Assume that there exist X ∈ B(H, K) and Y ∈ S(K) such that MX,Y is a Weyl operator. By Lemma 2.2, for any 0 <  < C, there exist the orthogonal decompositions K = K ⊕ K and H = H ⊕ H such that C(K ) ⊂ H , Cx ≤ x for all x ∈ K , C(K ) ⊂ H , Cx ≥ x for all x ∈ K , and  C=

C|K 0

0 C|K

     K H : → , K H

(2.5)

where C|K is left invertible. Then (A C), as an operator from H ⊕ K ⊕ K to R(C|K ) ⊕ (R(C|K )⊥ ⊕ H ), can be further written as  (A C) =

A1 () C  A2 () 0

0 C

 ,

(2.6)

where  C|K =

C 0



 : K →

R(C|K ) R(C|K )⊥



 , C =

0 C|K



 : K →

R(C|K )⊥ H

 (2.7)

with C  : K → R(C|K ) invertible and C  ≤ . Then MX,Y has the following new representation ⎛

MX,Y

⎞ ⎞ ⎛ ⎞ ⎛ R(C|K ) A1 () H C 0 0 C ⎠ : ⎝ K ⎠ → ⎝ R(C|K )⊥ ⊕ H ⎠ , = ⎝ A2 () X Y1 () Y2 () K K

(2.8)

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

7

and hence

⎛ ⎞ ⎛ ⎞

A1 () A1 () C 0 C 0

= C  ≤ .

⎝ A2 () ⎠ ⎝ ⎠ A 0 C () 0 0 −  2

X Y1 () Y2 () X Y1 () Y2 ()

(2.9)

The Weylness of MX,Y implies that ⎞ ⎛ ⎞ ⎛ ⎞ R(C|K ) A1 () H C 0 0 0 ⎠ : ⎝ K ⎠ → ⎝ R(C|K )⊥ ⊕ H ⎠ := ⎝ A2 () X Y1 () Y2 () K K ⎛

LX,Y

(2.10)

is Weyl for sufficiently small  > 0 by Lemma 2.3. Since C  is invertible, there exist the invertible operator matrices U ∈ B(R(C|K ) ⊕ (R(C|K )⊥ ⊕ H ) ⊕ K) and V ∈ B(H ⊕ K ⊕ K ) with ⎛

I 0 U := ⎝ −Y1 ()(C  )−1

0 I 0

⎞ 0 0⎠ , I

⎛

I

0 V := ⎝ −(C  )−1 A1 () I 0 0

⎞ 0 0⎠ I

(2.11)

such that ⎛

0 A2 () U LX,Y V = ⎝ X − Y1 ()(C  )−1 A1 ()

C 0 0

⎞ 0 0 ⎠. Y2 ()

This shows that 

Y2 () Z() 0 A2 ()

     K K : → H R(C|K )⊥ ⊕ H

(2.12)

is a Weyl operator, where Z() = X − Y1 ()(C  )−1 A1 (). According to Lemma 2.9, Y2 () : K → K is left Fredholm, A2 () : H → R(C|K )⊥ ⊕ H is right Fredholm and one of the following statements holds: (a) n(A2 ()) = d(Y2 ()) = ∞; (b) n(A2 ()) < ∞, d(Y2 ()) < ∞ and n(A2 ()) + n(Y2 ()) = d(A2 ()) + d(Y2 ()). It is not hard to see that (A C) is right Fredholm from the facts that A2 () is right Fredholm and C  is invertible. The observation N (A2 ()) ⊂ PH (N ((A C))) suggests the implication (a)⇒(ii). From assertion (b), we easily obtain that ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ R(C|K ) A1 () H C 0 0 0 ⎠ : ⎝ K ⎠ → ⎝ R(C|K )⊥ ⊕ H ⎠ L0,Y = ⎝ A2 () 0 Y1 () Y2 () K K is a Weyl operator. If C as in (2.8) is a compact operator, then, by Lemma 2.4, the  A C Weylness of L0,Y implies that M0,Y = is Weyl. By Lemma 2.9, Y is right Fred0 Y holm, which indicates that Y is Weyl in combination with Y ∈ S(K), and hence A

8

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

is a Weyl operator. Note that A2 () is right Fredholm. If C is non-compact, then PH (N (A2 (() C ))) contains a closed infinite dimensional subspace by Lemma 2.7. This, together with PH (N (A2 () C ))) ⊂ PH (N ((A C))) demonstrates that PH (N (A C))) contains a closed infinite dimensional subspace. Sufficiency. Take Y = I and we still use (2.5), (2.6), (2.7), (2.10) and (2.11) replacing Y1 () by I1 () and Y2 () by I2 (), where I1 () and I2 () are the corresponding block entries of the identity operator I. Then ⎛

0 C A2 () 0 U MX,I V = ⎝ X − I1 ()(C  )−1 A1 () 0

⎞ 0 C ⎠ . I2 ()

(2.13)

If assertion (i) holds, then taking X = 0 yields the sufficiency immediately. Assume now that (A C) is right Fredholm and assertion (ii) holds. Since



A1 () C 

A2 () 0

0 C



 −

A1 () A2 ()

C 0

 0

= C  ≤ , 0

(2.14)

we see from Lemma 2.3 that 



A1 () C A2 () 0

0 0



⎛

⎞   H R(C|K ) : ⎝ K ⎠ → R(C|K )⊥ ⊕ H K

(2.15)

is right Fredholm for sufficiently small  > 0. For such  > 0, A2 () is then a right  Fred C holm operator. Since C is non-compact, using Lemma 2.7, we see that PK (R( )⊥ ) I contains a closed infinite dimensional subspace. thenn(A2 ()) = d(I2 ()) = ∞ by assertion (ii). From Lemma 2.9, If C is compact,  it follows that

A2 ()

0

Z() I2 ()



is a Weyl operator for some Z() ∈ B(H, K), and so is

A2 () C Z() I2 ()

     R(C|K )⊥ ⊕ H H → . : K K

(2.16)

Define X := Z() + I1 ()(C  )−1 A1 () and we have from (2.13) that MX,I is a Weyl operator. Let us suppose, in the following, that C is a non-compact operator. In view of R(C|K )⊥ ⊕ H = R(A2 ()) ⊕ R(A2 ())⊥ , the operator matrix of the form (2.16) can be written as ⎛ ⎞ ⎛ ⎞   R(A2 ()) A21 () C1 H ⎝ 0 → ⎝ R(A2 ())⊥ ⎠ , (2.17) C2 ⎠ : K K Z() I2 () where A21 () : H → R(A2 ()) is right invertible and I2 () is left invertible. Obviously, C1 is non-compact and C2 is of finite rank. There are two cases to be investigated.

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

9

Case 1: d(A2 ()) = 0. Since A21 () is right and I2 () : K → K is left  1 invertible  C 1 ⊥ invertible, PH (N ((A21 () C ))) and PK (R( ) ) clearly contain closed infinite diI2 ()

mensional subspaces. By Lemma 2.8, we can choose a suitable Z() ∈ B(H, K) such that the operator matrix (2.16) is invertible. From (2.13), it follows that MX,Y is Weyl for X = Z() + I1 ()(C  )−1 A1 () and Y = I. Case 2: d(A2 ()) > 0. In this case, there exist a closed infinite dimensional  sub ⊥ space M such that K = M ⊕ M⊥  with dim M = d(A2 ()) < ∞. Then

C1

I2 ()

admits the new representation 

C12 C11 I21 () I22 ()

     M R(A2 ()) : → , K M⊥ 

(2.18)

where C12 and I22 () are finite rank operators. Evidently, I21 () is a left invertible 11 11 operator.  Note  that C is non-compact. It is clear that PH (N ((A21 () C ))) and PK (R(

C11

I21 ()

)⊥ ) contain closed infinite dimensional subspaces. By Lemma 2.8, there

exists Z() ∈ B(H, K) such that 

A11 21 () Z()

C11 I21 ()

     R(A2 ()) H → : M K

is invertible. This together with dim M⊥  = d(A2 ()) implies that ⎛

⎞ ⎛ ⎞ ⎞ ⎛ R(A2 ()) H A11 C11 0 21 () 0 0 ⎠ : ⎝ M1 ⎠ → ⎝ R(A2 ())⊥ ⎠ QZ := ⎝ 0 M2 Z() I21 () 0 K is a Weyl operator. It is not hard to see that MX,Y is a Weyl operator for X = Z() + I1 ()(C  )−1 A1 () and Y = I. 2 With Lemma 2.10 taking the role of Lemma 2.9 in the proof of necessity of Theorem 2.12 yields Theorem 2.13. Let A ∈ B(H) and C ∈ B(K, H) be given operators. If C is non-compact, then there exist X ∈ B(H, K) and Y ∈ S(K) such that MX,Y is a Fredholm operator if and only if (A C) is right Fredholm and one of the following statements is fulfilled: (i) A is a Fredholm operator; (ii) PH (N ((A C))) contains a closed infinite dimensional subspace. Theorem 2.14. Let A ∈ B(H) and C ∈ B(K, H) be given operators. If C is non-compact, then there exist X ∈ B(H, K) and Y ∈ S(K) such that MX,Y is an invertible operator if and only if (A C) is right invertible and one of the following statements is fulfilled: (i) A is a Weyl operator; (ii) PH (N ((A C))) contains a closed infinite dimensional subspace.

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

10

3. Main results The following is the most important result of this section. Theorem 3.1. For given operators A, B ∈ B(H), MX is a Weyl operator for some X ∈ S(H) if and only if A is left Fredholm, B is right Fredholm and one of the following statements is fulfilled: (i) B|R(A) is a Fredholm operator with ind(B|R(A) ) = −n(A); (ii) B|R(A)⊥ is a non-compact operator and PR(A) (N (B)) contains a closed infinite dimensional subspace. In addition, the collection of all X ∈ S(H), completing MX as a Weyl operator, is further given by     ⊥ P X SW (A, B) = {X ∈ S(H) : R(A)⊥ : H → R(A) is Fredholm with H B   PR(A)⊥ X ind( ) = −n(A)}.

(3.1)

B

Proof. Necessity. Let MX be a Weyl operator for some X ∈ S(H). Clearly, A is left Fredholm and B is right Fredholm. Then ⎛

MX

0 = ⎝0 0

A1 0 0

⎞ ⎛ ⎞ ⎛ ⎞ N (A) R(A) X1 X2 ⎠ : ⎝ N (A)⊥ ⎠ → ⎝ R(A)⊥ ⎠ , B H H

(3.2)

where A1 : N (A)⊥ → R(A) is invertible. Hence there exists the invertible operator ⎛

I ⎝ V := 0 0

0 I 0

⎞ ⎛

⎞ ⎞ ⎛ N (A) N (A) ⎠ : ⎝ N (A)⊥ ⎠ → ⎝ N (A)⊥ ⎠ −A−1 1 X1 H H I 0

such that ⎛

0 MX V = ⎝ 0 0

A1 0 0

⎞ 0 X2 ⎠ . B

(3.3)

This, together with the Weylness of MX implies that 

X2 B



 :H→

R(A)⊥ H

 (3.4)

is Fredholm and   X2 ind ( ) = −n(A). B

(3.5)

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

11

Picking a finite dimensional subspace M of H satisfying H = M ⊕ M⊥ and dim M = n(A), we have  B=

B21 B22

B11 B12

    ⊥ R(A) M : → . R(A)⊥ M

(3.6)

Note that B12 and B22 are of finite rank. So, by (3.5), the operator 

B21 X22

B11 X21

     R(A) M⊥ : → R(A)⊥ R(A)⊥

(3.7)

∗ with X22 = X22 is Weyl. Thus there exists an invertible operator U ∈ B(M⊥ , R(A)) such that      U 0 B11 U B21 B11 B21 = (3.8) 0 IR(A)⊥ C21 U C22 C21 C22

is Weyl. By applying Theorem 2.11 and Theorem 2.12, the Weylness of (3.8) gives that one of the following statements is fulfilled: (a) B11 U is Weyl; (b) B21 is a non-compact operator and PM⊥ (N ((B11 U B21 ))) contains a closed infinite dimensional subspaces. From assertion (a), we know that 

B11 0



 : R(A) →

M⊥ M

 (3.9)

    B B is Fredholm and ind( 11 ) = −n(A). From Lemma 2.4, it follows that B11 is Fred  0 12  B B holm and ind( B11 ) = −n(A). The assertion (i) follows from B|R(A) = B11 right 12 12 away. Since dim M < ∞, assertion (b) implies that 

B21 B22

 is non-compact and PR(A) (N (



⊥

: R(A) →

B11 B21 B12 B22



M⊥ M

 (3.10)



)) contains a closed infinite-dimensional sub  B spaces. By the representation (3.6) and B|R(A)⊥ = B21 , assertion (ii) is proven. 22 Sufficiency. Let A is left Fredholm and B is right Fredholm, and let M be a finite dimensional subspace of H such that H = M ⊕ M⊥ and dim M = n(A). Then B still has the representation (3.6). From assertion (i), we easily see that B11 : R(A) → M⊥ is a Weyl operator. According to Theorem 2.11 and Theorem 2.12, there exist X21 ∈ B(R(A), R(A)⊥ ) and X22 ∈ S(R(A)⊥ ) such that the operator matrix (3.7) is Weyl. Define X ∈ S(H) by  X :=

X21 0

X22 ∗ X21

     R(A) R(A)⊥ : → R(A)⊥ R(A)

(3.11)

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

12

and we verify that MX is clearly Weyl. Now assume that assertion (ii) holds. It is not hard to see that B21 : R(A)⊥ → M⊥ is non-compact and PR(A) (N ((B11 B21 ))) contains a closed infinite dimensional subspace. Again, by Theorem 2.12, we have X21 ∈ B(R(A), R(A)⊥ ) and X22 ∈ S(R(A)⊥ ) such that the operator matrix (3.7) is Weyl. Defining X as the form (3.11) yields the desired completion MX . From the fact X2 = PR(A)⊥ X and the previous proof, the relation (3.1) is clearly valid. 2 Remark 3.2. In the theorem above, the assertion (i) holds if and only if BA is a Fredholm operator with ind(BA) = 0, since R(B|R(A) ) = R(BA) and n(BA) = n(A) + n(B|R(A) ). Furthermore, if n(B) < ∞ or d(A) < ∞, then we claim ind(B|R(A) ) = n(B) − d(A) − d(B).

(3.12)

Indeed, since B|R(A) is Fredholm, we have  B=

B11 0

0 B21 0 B22



⎛

⎞   N (B|R(A) )⊥ R(B|R(A) ) ⎝ ⎠ N (B|R(A) ) : , → R(B|R(A) )⊥ R(A)⊥

(3.13)

where B11 is invertible and B22 is a finite rank operator. Then there exists the unique invertible operator V ∈ B(N (B|R(A) )⊥ ⊕ N (B|R(A) ) ⊕ R(A)⊥ ) such that  BV =

0 0 0 B22

B11 0

 .

(3.14)

Evidently, 

B11 0

0 0

0 0

 (3.15)

is right Fredholm and ind(B) = ind(B|R(A) ) + d(A), which implies (3.12). As a direct consequence of Theorem 3.1 and the relation (3.12), one can obtain Corollary 3.3. For given operators A, B ∈ B(H), we have 

σw (MX )

X∈S(H)

= σle (A) ∪ σre (B) ∪ {λ ∈ C : n(A − λ) + n(B − λ) = d(A − λ) + d(B − λ)} ∪ {λ ∈ ρle (A) : (B − λ)|R(A−λ) is Fredholm, (B − λ)|R(A−λ)⊥ is compact, ind((B − λ)|R(A−λ) ) = −n(A − λ)}

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

13

∪ {λ ∈ ρle (A) : (B − λ)|R(A−λ) is Fredholm, ind((B − λ)|R(A−λ) ) = −n(A − λ), PR(A−λ) (N (B − λ)) contains no closed infinite dimensional subspaces} ∪ {λ ∈ ρle (A) : (B − λ)|R(A−λ) is not Fredholm, (B − λ)|R(A−λ)⊥ is compact} ∪ {λ ∈ ρle (A) : (B − λ)|R(A−λ) is not Fredholm, (B − λ)|R(A−λ)⊥ is non-compact, PR(A−λ) (N (B − λ)) contains no closed infinite dimensional subspaces}. 

Remark 3.4. In [5], a description of the set

σw (MX ) was given (see (1.2)). It is

X∈B(H)

clear that 



σw (MX ) ⊂

X∈B(H)

σw (MX ).

X∈S(H)

Based on Lemma 2.10 and Theorem 2.13, the following Fredholm results can be proved in the similar way as the previous results. Theorem 3.5. For given operators A, B ∈ B(H), MX is a Fredholm operator for some X ∈ S(H) if and only if A is left Fredholm, B is right Fredholm and one of the following statements is fulfilled: (i) B|R(A) is a Fredholm operator; (ii) B|R(A)⊥ is a non-compact operator and PR(A) (N (B)) contains a closed infinite dimensional subspace. In addition, the set of all X ∈ S(H), completing MX as a Fredholm operator, is further given by SF (A, B) = {X ∈ S(H) :



PR(A)⊥ X B



:H→



R(A)⊥ H

 is Fredholm}.

Corollary 3.6. For given operators A, B ∈ B(H), we have 

σe (MX )

X∈S(H)

= σle (A) ∪ σre (B) ∪ {λ ∈ C : n(B − λ) < ∞, d(A − λ) = ∞} ∪ {λ ∈ C : n(B − λ) = ∞, d(A − λ) < ∞} ∪ {λ ∈ ρle (A) : (B − λ)|R(A−λ) is not Fredholm, (B − λ)|R(A−λ)⊥ is compact} ∪ {λ ∈ ρle (A) : (B − λ)|R(A−λ) is not Fredholm, (B − λ)|R(A−λ)⊥ is non-compact, PR(A−λ) (N (B − λ)) contains no closed infinite dimensional subspaces}. Remark 3.7. A description of the set



σe (MX ) was given in [4] (see (1.3)). From

X∈B(H)

the corollary above, we clearly see  X∈B(H)

σe (MX ) ⊂

 X∈S(H)

σe (MX ).

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

14

The following corollary is consistent with [11, Theorem 2.1], and some other results associated with invertibility are also presented. Corollary 3.8. For given operators A, B ∈ B(H), MX is invertible for some X ∈ S(H) if and only if A is left invertible, B is right invertible and one of the following statements is fulfilled: (i) B|R(A) is a Fredholm operator and ind(B|R(A) ) = 0; (ii) B|R(A)⊥ is a non-compact operator and PR(A) (N (B)) contains a closed infinitedimensional subspaces. Furthermore, the set of all X ∈ S(H), completing MX as an invertible operator, is given by SI (A, B) = {X ∈ S(H) :



PR(A)⊥ X B



:H→



R(A)⊥ H

 is invertible}.

Corollary 3.9. For given operators A, B ∈ B(H), we have 

σ(MX )

X∈S(H)

= σl (A) ∪ σr (B) ∪ {λ ∈ C : n(B − λ) = d(A − λ)} ∪ {λ ∈ ρl (A) : (B − λ)|R(A−λ) is not Weyl, (B − λ)|R(A−λ)⊥ is compact} ∪ {λ ∈ ρl (A) : (B − λ)|R(A−λ) is not Weyl, (B − λ)|R(A−λ)⊥ is non-compact, PR(A−λ) (N (B − λ)) contains no closed infinite dimensional subspaces}. Remark 3.10. In [9], a description of the set



σ(MC ) was given (see (1.1)) and

C∈B(H)

clearly  C∈B(H)

σ(MC ) ⊂



σ(MC ).

C∈S(H)

We end this section by analyzing some special cases of our main results. Corollary 3.11. Let A ∈ B(H) be left Fredholm. If B|R(A) is right Fredholm and B|R(A)⊥ is non-compact, then MX is a Weyl (Fredholm) operator for some X ∈ S(H). Proof. For the proof, we need only use Lemma 2.7 and Theorem 3.1 (Theorem 3.5) directly. 2 Corollary 3.12. Let A ∈ B(H) be left Fredholm. If B|R(A)⊥ is right Fredholm, then MX is a Weyl (Fredholm) operator for some X ∈ S(H). Proof. Write B1 := B|R(A) and B2 := B|R(A)⊥ . Since B2 is right Fredholm, as an operator from R(A) ⊕ (R(A)⊥ N (B2 )) ⊕ N (B2 ) to R(B2 ) ⊕ R(B2 )⊥ , the operator B admits the new representation

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

 B=

B21 0

B11 B12

0 0

15

 ,

(3.16)

where B21 is invertible and B12 is of finite rank. It is clear that B is right Fredholm. Note that R(A) is an infinite dimensional closed subspace of H, since A is left Fredholm. Then we have n(B12 ) = ∞. Since PR(A) (N (B)) = N (B12 ), the result follows from Theorem 3.1. 2 Corollary 3.13. Let A ∈ B(H) be left Fredholm, and let B ∈ B(H) be right Fredholm. If B|R(A) is not left Fredholm and B|R(A)⊥ is non-compact, then MX is a Weyl (Fredholm) operator for some X ∈ S(H). Proof. Write B1 := B|R(A) and B2 := B|R(A)⊥ . By Lemma 2.2, for any 0 <  < B2 , ⊥ there exist the orthogonal decompositions R(A)⊥ = R(A)⊥ and H = H ⊕ H  ⊕ R(A) such that ⊥ B2 (R(A)⊥  ) ⊂ H , B2 x ≤ x for all x ∈ R(A) ,

B2 (R(A)⊥ ) ⊂ H , B2 x ≥ x for all x ∈ R(A)⊥ . Then we have  B=

B11 () B2 |R(A)⊥ B12 () 0

0



B2 |R(A)⊥ 

⎛

⎞   R(A) H : ⎝ R(A)⊥ ⎠ → , H ⊥ R(A)

where B2 |R(A)⊥ is left invertible. Note that



B11 () B2 |R(A)⊥

B12 () 0



0 B2 |R(A)⊥ 

 −

B11 () B12 ()

B2 |R(A)⊥ 0

 0

= B2 |R(A)⊥ ≤ .  0

By Lemma 2.3, the right Fredholmness of B implies that  (B1 B2 |R(A)⊥ ) =

B11 () B2 |R(A)⊥ B12 () 0

     R(A) H → : H R(A)⊥

is right Fredholm for sufficiently small  > 0. Suppose now n((B1 B2 |R(A)⊥ )) < ∞, then (B1 B2 |R(A)⊥ ) is Fredholm, which contradicts the assumption that B1 is not left Fredholm. This proves n(B1 B2 |R(A)⊥ ) = ∞. Since B2 |R(A)⊥ is left invertible, it follows that PR(A) (N ((B1 B2 |R(A)⊥ ))) is closed and dim PR(A) (N ((B1 B2 |R(A)⊥ ))) = n((B1 B2 |R(A)⊥ )). The observation PR(A) (N ((B1 B2 |R(A)⊥ ))) ⊂ PR(A) (N (B)) demonstrates the desired result by Theorem 3.1 (Theorem 3.5). 2

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

16

Corollary 3.14. Let A ∈ B(H) be left Fredholm, and let B ∈ B(H) be right Fredholm. Then MX is Weyl for some X ∈ S(H) if and only if PM⊥ B|R(A) + PM⊥ B|R(A)⊥ F is Weyl for some F ∈ B(R(A), R(A)⊥ ), where M is a finite dimensional subspace of H with dim M = n(A). Proof. Write B1 := B|R(A) and B2 := B|R(A)⊥ . Let M be a finite dimensional subspace of H with H = M ⊕ M⊥ and dim M = n(A). Assume that MX is Weyl for some X ∈ S(H). By Theorem 3.1, we have that PM⊥ B1 is a Weyl operator, or that PM⊥ B2 is a non-compact operator and PR(A) (N (B)) contains a closed infinite dimensional subspace. Note that R(A) is an infinite dimensional closed subspace of H. Then there exists an invertible operator W ∈ B(M⊥ , R(A)) such that PM⊥ B1 W is a Weyl operator or PM⊥ (N ((PM⊥ B1 W PM⊥ B2 ))) contains a closed infinite dimensional subspace. Since B is right Fredholm, (PM⊥ B1 W PM⊥ B2 ) : M⊥ ⊕ R(A)⊥ → M⊥ is clearly a right Fredholm operator. If PM⊥ B2 is a non-compact operator, then, from the proof of the sufficiency of Theorem 2.12, there exists F ∈ B(R(A), R(A)⊥ ) such that 

PM⊥ B1 W −F W

PM⊥ B2 IR(A)⊥

     M⊥ M⊥ : → R(A)⊥ R(A)⊥

(3.17)

is Weyl. Since 

PM⊥ B1 −F

PM⊥ B2 IR(A)⊥



W 0

0 IR(A)⊥



 =

PM⊥ B1 W −F W

PM⊥ B2 IR(A)⊥

 ,

(3.18)

it follows that 

PM⊥ B1 −F

PM⊥ B2 IR(A)⊥

     R(A) M⊥ : → R(A)⊥ R(A)⊥

(3.19)

is Weyl. This, together with 

−PM⊥ B2 IR(A)⊥

IM ⊥ 0



PM⊥ B1 −F   PM⊥ B1 + PM⊥ B2 F 0 = 0 I

PM⊥ B2 IR(A)⊥



IR(A) F

0 IR(A)⊥

 (3.20)

implies that PM⊥ B1 + PM⊥ B2 F is a Weyl operator. If PM⊥ B2 is a compact operator, then PM⊥ B1 is a Weyl operator. Then, for any F ∈ B(R(A), R(A)⊥ ), we see that 

PM⊥ B1 −F

     R(A) M⊥ : → IR(A)⊥ R(A)⊥ R(A)⊥ 0

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

17

is Weyl. Applying Lemma 2.4, we infer that (3.19) is Weyl. By the factorization (3.20), we conclude that PM⊥ B1 + PM⊥ B2 F is a Weyl operator. Conversely, if PM⊥ B1 + PM⊥ B2 F is Weyl for some F ∈ B(R(A), R(A)⊥ ), then the operator (3.17) is Weyl by (3.20), (3.19) and (3.18). Note that IR(A)⊥ is a self-adjoint operator. If PM⊥ B2 is a non-compact operator, then, by Theorem 2.12, we know that PM⊥ B1 W is a Weyl operator, or that PM⊥ (N ((PM⊥ B1 W PM⊥ B2 ))) contains a closed infinite dimensional subspace. Hence, either PM⊥ B1 is Weyl or PR(A) (N (B)) contains a closed infinite dimensional subspace. From Theorem 3.1, we see that MX is Weyl for some X ∈ S(H). If PM⊥ B2 is a compact operator, then we obtain that PM⊥ B1 W is Weyl by Theorem 2.11, and so the result follows from Theorem 3.1. 2 Corollary 3.15. Let A ∈ B(H) be left Fredholm, and let B ∈ B(H) be right Fredholm. Then MX is Fredholm for some X ∈ S(H) if and only if B|R(A) + B|R(A)⊥ F is Fredholm for some F ∈ B(R(A), R(A)⊥ ). Proof. This corollary is the Fredholm case of Corollary 3.14 and its proof is omitted here. 2 Corollary 3.16. Let A ∈ B(H) be left invertible, and let B ∈ B(H) be right invertible. If B|R(A) is right Fredholm and B|R(A)⊥ is non-compact, then MX is an invertible operator for some X ∈ S(H). Proof. By Lemma 2.7 and Corollary 3.8, the desired result follows immediately. 2 Corollary 3.17. Let A ∈ B(H) be left invertible, and let B ∈ B(H) be right invertible. If B|R(A) is not left Fredholm and B|R(A)⊥ is non-compact, then MX is an invertible operator for some X ∈ S(H). Proof. Since B|R(A) is not left Fredholm and B is right invertible, we have from the proof of Corollary 3.13 that PR(A) (N (B)) contains a closed infinite dimensional subspace. Note that B|R(A)⊥ is a non-compact operator. By Corollary 3.8, MX is an invertible operator for some X ∈ S(H). 2 Corollary 3.18. Let A ∈ B(H) be left invertible, and let B ∈ B(H) be right invertible. If B|R(A)⊥ is right Fredholm, then MX is an invertible operator for some X ∈ S(H). Proof. If B|R(A)⊥ is right Fredholm, then PR(A) (N (B)) contains a closed infinite dimensional subspace from the proof of Corollary 3.12. Also, the right Fredholmness of B|R(A)⊥ implies that B|R(A)⊥ is non-compact. Thus, the result follows from Corollary 3.8. 2 Corollary 3.19. Let A, B ∈ B(H) be given operators with n(B) < ∞ or d(A) < ∞. Then MX is Weyl for some X ∈ S(H) if and only if MX is Weyl for some X ∈ B(H).

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

18

Proof. Let MX be Weyl for some X ∈ B(H). Then, in combination with n(B) < ∞ or d(A) < ∞, we deduce that A is left Fredholm, B is right Fredholm and n(A) + n(B) = d(A) + d(B). Hence B|R(A) is Fredholm and ind(B|R(A) ) = −n(A). By Theorem 3.1, MX is Weyl for some X ∈ S(H). The opposite implication is trivial. 2 Corollary 3.20. Let A, B ∈ B(H) be given operators with n(B) < ∞ or d(A) < ∞. Then MX is Fredholm for some X ∈ S(H) if and only if MX is Fredholm for some X ∈ B(H). Proof. This corollary is the Fredholm version of Corollary 3.19, and its proof is omitted. 2 4. Examples and applications Let A ∈ B(H). We denote HX by the operator on H ⊕ H of the form  HX :=

 A X 0 −A∗

with X ∈ S(H) unknown, which is clearly the so-called Hamiltonian operator. As applications, we now present the analogues of Hamiltonian operators. Proposition 4.1. For given operator A ∈ B(H), HX is Weyl for some X ∈ S(H) if and only if A is left Fredholm. Proof. Let HX be Weyl for some X ∈ S(H). By Theorem 3.1, A is left Fredholm. Conversely, if A is left Fredholm, then R(−A∗ |R(A) ) = R(A∗ ) is closed and ind(−A∗ |R(A) ) = −n(A). By Theorem 3.1, HX is Weyl for some X ∈ S(H). 2 Similarly, we get the following three conclusions. Proposition 4.2. For given operator A ∈ B(H), HX is Fredholm for some X ∈ S(H) if and only if A is left Fredholm. Proposition 4.3. For given operator A ∈ B(H), we have  σw (HX ) X∈S(H)

= σle (A) ∪ (−σle (A)) ∪ {λ ∈ C : n(A − λ) + d(A + λ) = d(A − λ) + n(A + λ)} ∪ {λ ∈ ρle (A) : (−A∗ − λ)|R(A−λ) is Fredholm, (−A∗ − λ)|R(A−λ)⊥ is compact, ind((−A∗ − λ)|R(A−λ) ) = −n(A − λ)} ∪ {λ ∈ ρle (A) : (−A∗ − λ)|R(A−λ) is Fredholm, ind((−A∗ − λ)|R(A−λ)) = −n(A − λ), PR(A−λ) (N (−A∗ − λ)) contains no closed infinite dimensional subspaces}

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

19

∪ {λ ∈ ρle (A) : (−A∗ − λ)|R(A−λ) is not Fredholm, (−A∗ − λ)|R(A−λ)⊥ is compact} ∪ {λ ∈ ρle (A) : (−A∗ − λ)|R(A−λ) is not Fredholm, (−A∗ − λ)|R(A−λ)⊥ is noncompact, PR(A−λ) (N (−A∗ − λ)) contains no closed infinite dimensional subspaces}, where −σle (A) = {λ ∈ C : −λ ∈ σle (A)}. Proof. Note that σre (−A∗ ) = {λ ∈ C : −λ ∈ σle (A)} and n(−A∗ − λ) = d(A + λ). By Corollary 3.3, we directly obtain the result. 2 Proposition 4.4. For given operator A ∈ B(H), we have 

σe (HX )

X∈S(H)

= σle (A) ∪ (−σle (A)) ∪ {λ ∈ C : d(A + λ) < ∞, d(A − λ) = ∞} ∪ {λ ∈ C : d(A + λ) = ∞, d(A − λ) < ∞} ∪ {λ ∈ ρle (A) : (−A∗ − λ)|R(A−λ) is not Fredholm, (−A∗ − λ)|R(A−λ)⊥ is compact} ∪ {λ ∈ ρle (A) : −A∗ − λ |R(A−λ) is not Fredholm, PR(A−λ) (N (−A∗ − λ)) contains no closed infinite dimensional subspaces}. Remark 4.5. Unlike the general operator matrix case, Propositions 4.3 and 4.4 can not be derived from Propositions 4.1 and 4.2, respectively. We conclude this section with two illustrating examples of the previous results. Example 4.6. Let H = K = 2 , and let A, B ∈ B(2 ) be defined by Ax = (0, x2 , 0, x3 , 0, x4 , · · · ), Bx = (0, x1 , x2 , x5 , x6 , x9 , x10 , · · · ) for x = (x1 , x2 , x3 , · · · ) ∈ 2 . Then we claim that MX = X ∈ S(2 ).

A X  0 B

is Weyl for some

It is not hard to see that A is left Fredholm, B is right Fredholm, B|R(A)⊥ is non-compact and PR(A) (N (B)) contains a closed infinite dimensional subspace. By The X is Weyl for some X ∈ S(2 ). In fact, define the orem 3.1, we obtain that MX = A 0 B self-adjoint operator Xx = (x1 + x4 , 0, x3 , x1 , x5 + x8 , 0, x7 , x5 , x9 + x12 , 0, x11 , x9 , · · · ) for x = (x1 , x2 , x3 , · · · ) ∈ 2 . Then we can check that R(MX ) is closed and n(MX ) = d(MX ) = 1, and hence MX is a Weyl operator.

20

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

Example 4.7. Let H = K = 2 , and let A, B ∈ B(2 ) be defined by Ax = (0, x1 , 0, x2 , 0, x3 , · · · ), Bx = (0, x1 , x2 +

x3 2 , x5 , x4

for x = (x1 , x2 , x3 , · · · ) ∈ 2 . Then we claim that MX = X ∈ S(2 ).

+

x7 4 , x9 , x6

A X  0 B

+

x11 6 ,···)

is not Weyl for any

Indeed, write B1 := B|R(A) and B2 := B|R(A)⊥ . Direct calculations verify that A is left invertible, B is right Fredholm and B2 is a non-compact operator, but B1 is not Fredholm and PR(A) (N (B)) contains no closed infinite-dimensional subspaces. By Theorem 3.1, there does not exist any X ∈ S(2 ) such that MX is Weyl. In fact, suppose that MX is Weyl for some X ∈ S(2 ). Then the operator matrix 

PR(B1 )⊥ B2 X2

B1 X1

     R(A) 2 : → R(A)⊥ R(A)⊥

(4.1)

is Weyl by the compactness of PR(B1 ) B2 . Note that B1 : R(A) → 2 is left invertible, and so 

PR(B1 )⊥ B2 X2



⊥

: R(A) →



R(B1 )⊥ R(A)⊥

 (4.2)

is Weyl, where X2 ∈ S(R(A)⊥ ). The right Fredholmness of the operator (4.2) implies that X2 : R(A)⊥ → R(A)⊥ is right Fredholm, and hence X2 : R(A)⊥ → R(A)⊥ is Fredholm. Under the decompositions R(A)⊥ = N (X2 )⊥ ⊕ N (X2 ) and R(A)⊥ = R(X2 ) ⊕ R(X2 )⊥ , we may write the operator (4.2) as the following new matrix form ⎛

B21 ⎝ X21 0

⎛ ⎞ ⎞   R(B1 )⊥ B22 ⊥ ) N (X2 0 ⎠: → ⎝ R(X2 ) ⎠ , N (X2 ) 0 R(X2 )⊥

(4.3)

where X21 is invertible and B22 is a finite rank operator. It is not hard to see that ⎛

0 ⎝ X21 0

⎛ ⎞ ⎞   R(B1 )⊥ 0 ⊥ ) N (X 2 0⎠ : → ⎝ R(X2 ) ⎠ N (X2 ) 0 R(X2 )⊥

is Weyl. This is in contradiction to d(B1 ) = ∞. Therefore, MX is not Weyl for any X ∈ S(2 ). Acknowledgements We would like to thank the anonymous referees for their valuable comments and suggestions which have greatly improved this paper.

X. Wu et al. / Linear Algebra and its Applications 531 (2017) 1–21

21

References [1] P. Aienia, Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer Academic Publishers, Dordrecht, 2004. [2] M. Barraa, M. Boumazgour, On the perturbations of spectra of upper triangular operator matrices, J. Math. Anal. Appl. 347 (2008) 315–322. [3] C. Benhida, E.H. Zerouali, H. Zguitti, Spectra of upper triangular operator matrices, Proc. Amer. Math. Soc. 133 (2005) 3013–3020. [4] X.H. Cao, M.Z. Guo, B. Meng, Semi-Fredholm spectrum and Weyl’s theorem for operator matrices, Acta Math. Sin. (Engl. Ser.) 22 (2006) 169–178. [5] X.H. Cao, B. Meng, Essential approximate point spectra and Weyl’s theorem for operator matrices, J. Math. Anal. Appl. 304 (2005) 759–771. [6] D.S. Djordjević, Perturbations of spectra of operator matrices, J. Operator Theory 48 (2002) 467–486. [7] R.G. Douglas, Banach Algebra Technique in Operator Theory, 2nd ed., Springer-Verlag, New York, 2003. [8] S.V. Djordjević, Y.M. Han, A note on Weyl’s theorem for operator matrices, Proc. Amer. Math. Soc. 131 (2002) 2543–2547. [9] H.K. Du, J. Pan, Perturbation of spectrums of 2 × 2 operator matrices, Proc. Amer. Math. Soc. 121 (1994) 761–766. [10] S.V. Djordjević, H. Zguitti, Essential point spectra of operator matrices trough local spectral theory, J. Math. Anal. Appl. 338 (2008) 285–291. [11] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, Birkhäuser Verlag, Basel, 1990. [12] A. Chen, G.J. Hai, Perturbations of the right and left spectra for operator matrices, J. Operator Theory 67 (2012) 207–214. [13] G.J. Hai, A. Chen, On the invertibility of upper triangular operator matrices, Linear Multilinear Algebra 62 (2014) 538–547. [14] J.C. Hou, H. Radjavi, P. Rosenthal, Idempotent completions of operator partial matrices, Acta Math. Sinica (Chin. Ser.) 42 (1999) 227–232. [15] J.J. Huang, Y.G. Huang, H. Wang, Closed range and Fredholm properties of upper-triangular operator matrices, Ann. Funct. Anal. 6 (2015) 42–52. [16] J.J. Huang, A. Chen, H. Wang, Self-adjoint perturbation of spectra of upper triangular operator matrices, Acta Math. Sinica (Chin. Ser.) 53 (6) (2010) 1193–1200. [17] I.S. Hwang, W.Y. Lee, The boundedness below of 2 × 2 upper triangular operator matrices, Integral Equations Operator Theory 39 (2001) 267–276. [18] W.Y. Lee, Weyl spectra of operator matrices, Proc. Amer. Math. Soc. 129 (2001) 131–138. [19] A.S. Markus, V.R. Olshevsky, Complete controllability and spectrum assignment in infinite dimensional spaces, Integral Equations Operator Theory 17 (1993) 109–121. [20] K. Takahashi, Invertible completions of operator matrices, Integral Equations Operator Theory 21 (1995) 355–361. [21] S.F. Zhang, Z.Y. Wu, H.J. Zhong, Continuous spectrum, point spectrum and residual spectrum of operator matrices, Linear Algebra Appl. 433 (2010) 653–661. [22] E.H. Zerouali, H. Zguitti, Perturbation of spectra of operator matrices and local spectral theory, J. Math. Anal. Appl. 324 (2006) 992–1005.