The Intersection of Upper and Lower Semi-Browder Spectrum of Upper-Triangular Operator Matrices

Abstract

When 𝐴∈𝐵(𝐻) and 𝐵∈𝐵(𝐾) are given, we denote by 𝑀_𝐶 the operator acting on the infinite-dimensional separable Hilbert space 𝐻⊕𝐾 of the form 𝑀_𝐶=^𝐴𝐶_0𝐵. In this paper, it is proved that there exists some operator 𝐶∈𝐵(𝐾,𝐻) such that 𝑀_𝐶 is upper semi-Browder if and only if there exists some left invertible operator 𝐶∈𝐵(𝐾,𝐻) such that 𝑀_𝐶 is upper semi-Browder. Moreover, a necessary and sufficient condition for 𝑀_𝐶 to be upper semi-Browder for some 𝐶∈𝐺(𝐾,𝐻) is given, where 𝐺(𝐾,𝐻) denotes the subset of all of the invertible operators of 𝐵(𝐾,𝐻).

1. Introduction

It is well known that if 𝐻 is a Hilbert space, 𝑇 is a bounded linear operator defined on 𝐻, and 𝐻₁ is an invariant closed subspace of 𝑇, then 𝑇 can be represented in the following form: 𝑇=∗∗0∗∶𝐻₁⊕𝐻^⟂₁⟶𝐻₁⊕𝐻^⟂₁, (1) which motivated the interest in 2×2 upper-triangular operator matrices. For recent investigations on this subject, see references [1–23].

Throughout this paper, let 𝐻 and 𝐾 be separable infinite-dimensional complex Hilbert spaces, and let 𝐵(𝐻,𝐾) be the set of all bounded linear operators from 𝐻 into 𝐾; when 𝐻=𝐾, we write 𝐵(𝐻,𝐻) as 𝐵(𝐻). For 𝐴∈𝐵(𝐻), 𝐵∈𝐵(𝐾), and 𝐶∈𝐵(𝐾,𝐻), we have 𝑀_𝐶=^𝐴𝐶_0𝐵∈𝐵(𝐻⊕𝐾). For 𝑇∈𝐵(𝐻,𝐾), let 𝑅(𝑇) and 𝑁(𝑇) denote the range and the kernel of 𝑇, respectively, and denote that 𝛼(𝑇)=dim𝑁(𝑇) and 𝛽(𝑇)=dim𝐾/𝑅(𝑇). If 𝑇∈𝐵(𝐻), the ascent asc(𝑇) of 𝑇 is defined to be the smallest nonnegative integer 𝑘 which satisfies and 𝑁(𝑇^𝑘)=𝑁(𝑇^𝑘+1). If such 𝑘 does not exist, then the ascent of 𝑇 is defined as infinity. Similarly, the descent des(𝑇) of 𝑇 is defined as the smallest nonnegative integer 𝑘 for which 𝑅(𝑇^𝑘)=𝑅(𝑇^𝑘+1) holds. If such 𝑘 does not exist, then des(𝑇) is defined as infinity, too. If the ascent and the descent of 𝑇 are finite, then they are equal (see [6]). For 𝑇∈𝐵(𝐻), if 𝑅(𝑇) is closed and 𝛼(𝑇)<∞, then 𝑇 is said to be an upper semi-Fredholm operator; if 𝛽(𝑇)<∞, which implies that 𝑅(𝑇) is closed, then 𝑇 is said to be a lower semi-Fredholm operator. If 𝑇∈𝐵(𝐻) is either upper or lower semi-Fredholm operator, then 𝑇 is said to be a semi-Fredholm operator. If both 𝛼(𝑇)<∞ and 𝛽(𝑇)<∞, then 𝑇 is said to be a Fredholm operator. For a semi-Fredholm operator 𝑇, its index ind(𝑇) is defined by ind(𝑇)=𝛼(𝑇)−𝛽(𝑇).

For a semi-Fredholm operator 𝑇∈𝐵(𝐻), its shift Samuel multiplicity 𝑠_mul(𝑇) and backward shift Samuel multiplicity 𝑏.𝑠._mul(𝑇) are defined, respectively, by the following (see [24]): 𝑠_mul(𝑇)=lim_𝑘→∞𝛽𝑇^𝑘𝑘,𝑏.𝑠._mul(𝑇)=lim_𝑘→∞𝛼𝑇^𝑘𝑘. (2) Moreover, it has been proved that 𝑠_mul(𝑇), 𝑏.𝑠._mul(𝑇)∈{0,1,2,…,∞} and that ind(𝑇)=𝑏.𝑠._mul(𝑇)−𝑠_mul(𝑇). These two invariants refine the Fredholm index and can be regarded as the stabilized dimensions of the kernel and the cokernel (see [24]).

In this paper, the sets of invertible operators and left invertible operators from 𝐻 into 𝐾 are denoted by 𝐺(𝐻,𝐾) and 𝐺_𝑙(𝐻,𝐾), respectively; the sets of all Fredholm operators, upper semi-Fredholm operators, and lower semi-Fredholm operators from 𝐻 into 𝐾 are denoted by Φ(𝐻,𝐾), Φ₊(𝐻,𝐾), and Φ₋(𝐻,𝐾), respectively; the sets of all Browder operators, upper semi-Browder operators, and lower semi-Browder operators, on 𝐻 are defined, respectively, by the following: Φ_𝑏(𝐻){(𝐻)(𝑇)(𝑇)},Φ∶=𝑇∈Φ∶asc=des<∞_𝑎𝑏(𝐻)∶=𝑇∈Φ₊(𝐻)(𝑇),Φ∶asc<∞_𝑠𝑏(𝐻)∶=𝑇∈Φ₋(𝐻)∶des(𝑇)<∞. (3) Moreover, for 𝑇∈𝐵(𝐻), we introduce its corresponding spectra as follows. The spectrum is given as 𝜎(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼∉𝐺(𝐻)}. The left spectrum is given as 𝜎_𝑙(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼∉𝐺_𝑙(𝐻)}. The essential spectrum is defined as 𝜎_𝑒(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼∉Φ(𝐻)}. The upper semi-Fredholm spectrum is defined as 𝜎_𝑆𝐹+(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼∉Φ₊(𝑋)}. The lower semi-Fredholm spectrum is presented as 𝜎_𝑆𝐹−(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼∉Φ₋(𝑋)}. The Browder spectrum is presented as 𝜎_𝑏(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼∉Φ_𝑏(𝐻)}. The upper semi-Browder spectrum is defined as 𝜎_𝑎𝑏(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼∉Φ_𝑎𝑏(𝑋)}. The lower semi-Browder spectrum is presented as 𝜎_𝑠𝑏(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼∉Φ_𝑠𝑏(𝑋)}.

Using the Samuel multiplicities, Zhang and Wu (see [20]) gave a necessary and sufficient condition for which 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾) for some 𝐶∈𝐵(𝐾,𝐻) and characterized the set of ∩_{𝐶∈𝐵(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶). In this paper, our main goal is to characterize the intersection of ∩_{𝐶∈𝐺𝑙(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶) and ∩_{𝐶∈𝐺(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶). This paper is organized as follows. In Section 2, we give a necessary and sufficient condition for which 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾) for some 𝐶∈𝐺_𝑙(𝐾,𝐻) and get _{𝐶∈𝐺𝑙()𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶. (4) In Section 3, we give a necessary and sufficient condition for which 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾) for some 𝐶∈𝐺(𝐾,𝐻) and get _{()𝐶∈𝐺𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=_{()𝐶∈Φ𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶∪𝜆∈ℂ∶𝐵−𝜆iscompact. (5)

For the sake of convenience, we now present some lemmas which will be used in the sequel.

Lemma 1 (see [20, 24]). An operator 𝑇∈𝐵(𝐻) is semi-Fredholm if and only if  𝑇 can be decomposed into the following form with respect to some orthogonal decomposition 𝐻=𝐻₁⊕𝐻₂⊕𝐻₃∶𝑇𝑇=¹𝑇₁₂𝑇₁₃0𝑇₂𝑇₂₃00𝑇₃∶𝐻₁⊕𝐻₂⊕𝐻₃⟶𝐻₁⊕𝐻₂⊕𝐻₃, (6) where dim(𝐻₃)<∞, 𝑇₁ is a right invertible operator, 𝑇₃ is a finite nilpotent operator, 𝑇₂ is a left invertible operator, and min{ind(𝑇₁),−ind(𝑇₂)}<∞. Moreover, ind(𝑇₁)=𝛼(𝑇₁)=𝑏.𝑠._mul(𝑇), ind(𝑇₂)=−𝛽(𝑇₂)=−𝑠_mul(T), and ind(𝑇)=𝛼(𝑇₁)−𝛽(𝑇₂).

Lemma 2 (see [18]). Let 𝐴∈𝐵(𝐻), 𝐵∈𝐵(𝐾), and 𝐶∈𝐵(𝐾,𝐻).(1)If 𝐴∈Φ_𝑏(𝐻), then 𝐵∈Φ_𝑎𝑏(𝐾) if 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾) for some 𝐶∈𝐵(𝐾,𝐻).(2)If 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾) for some 𝐶∈𝐵(𝐾,𝐻), then 𝐴∈Φ_𝑎𝑏(𝐻).(3)If 𝐴∈Φ_𝑎𝑏(𝐻) and 𝐵∈Φ_𝑎𝑏(𝐾), then 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾) for any 𝐶∈𝐵(𝐾,𝐻).(4)If 𝐵∈Φ_𝑏(𝐾), then 𝐴∈Φ_𝑎𝑏(𝐻) if 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾) for some 𝐶∈𝐵(𝐾,𝐻);𝐴∈Φ_𝑠𝑏(𝐻) if 𝑀_𝐶∈Φ_𝑠𝑏(𝐻⊕𝐾) for some 𝐶∈𝐵(𝐾,𝐻).(5)If 𝑀_𝐶∈Φ_𝑏(𝐻⊕𝐾) for some 𝐶∈𝐵(𝐾,𝐻), then 𝐴∈Φ_𝑎𝑏(𝐻) and 𝐵∈Φ_𝑠𝑏(𝐾).(6)If two of 𝐴, 𝐵, and 𝑀_𝐶 are Browder, then so is the third.

Lemma 3 (see [20]). Let 𝑇∈𝐵(𝐻). Then, 𝑇 is upper semi-Browder if 𝑇 can be decomposed into the following form with respect to some orthogonal decomposition 𝐻=𝐻₁⊕𝐻₂∶𝑇𝑇=¹𝑇₁₂0𝑇₂, (7) where dim(𝐻₁)<∞, 𝑇₁ is nilpotent, 𝑇₂ is left invertible, and 𝛽(𝑇₂)=𝑠_mul(𝑇)=−ind(T).

Lemma 4 (see [20]). Let 𝑇∈𝐵(𝐻). Then, 𝑇 is lower semi-Browder if 𝑇 can be decomposed into the following form with respect to some orthogonal decomposition 𝐻=𝐻₁⊕𝐻₂: 𝑇𝑇=¹𝑇₁₂0𝑇₂, (8) where dim(𝐻₂)<∞, 𝑇₁ is right invertible, 𝑇₂ is nilpotent, and 𝛼(𝑇₁)=𝑏.𝑠._mul(𝑇)=ind(𝑇).

Lemma 5 (see [20]). For any given 𝐴∈𝐵(𝐻) and 𝐵∈𝐵(𝐾), 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾) for some 𝐶∈𝐵(𝐾,𝐻) if 𝐴∈Φ_𝑎𝑏(𝐻) and 𝑠_mul(𝐴)=∞𝑖𝑓𝐵∉Φ₊(𝐾),𝑏.𝑠._mul(𝐵)≤𝑠_mul(𝐴)𝑖𝑓𝐵∈Φ₊(𝐾). (9)

Lemma 6 (see [9]). For any given 𝐴∈𝐵(𝐻) and 𝐵∈𝐵(𝐾), 𝑀_𝐶 is left invertible for some 𝐶∈𝐵(𝐾,𝐻) if 𝐴 is left invertible and 𝑎(𝐵)(𝐴)(𝐵)𝛽(𝐴)(𝐵)≤𝛽𝑖𝑓𝑅𝑖𝑠𝑐𝑙𝑜𝑠𝑒𝑑,=∞𝑖𝑓𝑅𝑖𝑠𝑛𝑜𝑡𝑐𝑙𝑜𝑠𝑒𝑑. (10)

Lemma 7 (see [25]). Let 𝑉 be a linear subspace of 𝐻. Then, the following statements are equivalent.(1)Any bounded operator 𝐴∈𝐵(𝐻) with 𝑅(𝐴)⊆𝑉 is compact.(2)𝑉 contains no closed infinite-dimensional subspace.

2. ⋂_{𝐶∈𝐵(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶) and ⋂_{𝐶∈𝐺𝑙(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶)

In [1, 20], the authors have proved that _{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑏𝑀_𝐶=𝜎_𝑎𝑏(𝐴)∪𝜎_𝑠𝑏(𝐵)∪{()()()(.𝜆∈ℂ∶𝛼𝐴−𝜆+𝛼𝐵−𝜆≠𝛽𝐴−𝜆+𝛽𝐵−𝜆) (11) They, moreover, proved that _{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=𝜎_𝑎𝑏(𝐴)∪𝜆∈ℂ∶𝜆∈𝜎_𝑆𝐹+(𝐵),()}∪(𝐴)𝑠_mul𝐴−𝜆<∞𝜆∈Φ∩Φ₊(𝐵)()(.∶𝑏.𝑠._mul𝐵−𝜆>𝑠_mul𝐴−𝜆)} (12)

Comparing the above two kinds of spectra with the upper semi-Weyl spectrum and Weyl spectrum, one may expect that the following equality holds: _{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=𝜎_𝑎𝑏(𝐴)∪𝜆∈ℂ∶𝜆∈𝜎_𝑆𝐹+(𝐵)()∪()()()(.,𝛽𝐴−𝜆<∞𝜆∈ℂ∶𝛼𝐴−𝜆+𝛼𝐵−𝜆>𝛽𝐴−𝜆+𝛽𝐵−𝜆) (13) However, it is not that case, as the following example shows.

Example 8. Let 𝐴 be the unilateral shift on ℓ², that is, 𝑉∶ℓ²⟶ℓ²,𝑧₁,𝑧₂,…↦0,𝑧₁,𝑧₂,…, (14) and let the operators 𝐴 and 𝐵 be defined by 𝑉𝐴=𝑉,𝐵=^∗200𝑉⁵∶ℓ²⊕ℓ²⟶ℓ²⊕ℓ². (15) Then, we have 𝑏.𝑠._mul(𝐵)=2>𝑠_mul(𝐴)=1, while 𝛼(𝐴)+𝛼(𝐵)=2<𝛽(𝐴)+𝛽(𝐵)=6. Moreover, 0∈⋂_{𝐶∈𝐵(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶), while 0∉𝜎_𝑎𝑏(𝐴)∪{𝜆∈ℂ∶𝜆∈𝜎_𝑆𝐹+(𝐵) and 𝛽(𝐴−𝜆)<∞}∪{𝜆∈ℂ∶𝛼(𝐴−𝜆) +𝛼(𝐵−𝜆)>𝛽(𝐴−𝜆)+𝛽(𝐵−𝜆)}. Thus, (13) does not hold.

In spite of the above counter example, we have the following.

Proposition 9. For any given 𝐴∈𝐵(𝐻) and 𝐵∈𝐵(𝐾), one has _{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶⊇𝜎_𝑎𝑏(𝐴)∪𝜆∈ℂ∶𝜆∈𝜎_𝑆𝐹+(𝐵)()∪{()()()(.,𝛽𝐴−𝜆<∞𝜆∈ℂ∶𝛼𝐴−𝜆+𝛼𝐵−𝜆>𝛽𝐴−𝜆+𝛽𝐵−𝜆) (16)

Proof. From the proof of Theorem  2.3 in [20], we know that when 𝐴∈Φ_𝑎𝑏(𝐻), 𝑠_mul(𝐴)<∞ if and only if 𝛽(𝐴)<∞. Combining this fact with Corollary 2.5 of [20], it is easy to see that _{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶⊇𝜎_𝑎𝑏(𝐴)∪𝜆∈ℂ∶𝜆∈𝜎_𝑆𝐹+(𝐵),𝛽(𝐴−𝜆)<∞. (17) Noting that 𝛽(𝐵−𝜆)<∞ implies that 𝑅(𝐵−𝜆) is closed, it follows from corollary 2.5 of [2] that _{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶⊇_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑤𝑀_𝐶⊇()()()(,𝜆∈ℂ∶𝛼𝐴−𝜆+𝛼𝐵−𝜆>𝛽𝐴−𝜆+𝛽𝐵−𝜆) (18) where 𝜎_𝑎𝑤(𝑀_𝐶)={𝜆∈ℂ∶𝑀_𝐶−𝜆 is not uppersemi-Fredholm operator with index less than or equal to 0}.

Now, we are ready to present the main result of this section.

Theorem 10. For any given 𝐴∈𝐵(𝐻) and 𝐵∈𝐵(𝐾), one has _{𝐶∈𝐺𝑙()𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶. (19)

Proof. Since ⋂_{𝐶∈𝐺𝑙(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶)⊇⋂_{𝐶∈𝐵(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶) is obvious, it is sufficient to prove that if 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾), then there exists some left invertible operator 𝑄∈𝐵(𝐾,𝐻) such that 𝑀_𝑄∈Φ_𝑎𝑏(𝐻⊕𝐾).
Suppose that 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾). It follows from Lemma 5 that 𝐴∈Φ_𝑎𝑏(𝐻) and 𝑠_mul(𝐴)=∞if𝐵∉Φ₊(𝐾),𝑏.𝑠._mul(𝐵)≤𝑠_mul(𝐴)if𝐵∈Φ₊(𝐾). (20)
There are two cases to consider.
Case  1. Assume that 𝐴∈Φ_𝑎𝑏(𝐻), 𝑠_mul(𝐴)=∞, and 𝐵∉Φ₊(𝐾). Then, it follows from Lemma 3 that 𝐴 can be decomposed into the following form: 𝐴𝐴=¹𝐴₁₂0𝐴₂∶𝐻₁⊕𝐻₂⟶𝐻₁⊕𝐻₂, (21) where dim(𝐻₁)<∞, 𝐴₁ is nilpotent, 𝐴₂ is a left invertible operator, and 𝛽(𝐴₂)=𝑠_mul(𝐴)=∞. So, we can let 00𝑉𝑄=∶𝐾⟶𝐻₁⊕𝑅𝐴₂⊕𝐻₂⊖𝑅𝐴₂, (22) where 𝑉∈𝐵(𝐾,(𝐻₂⊖𝑅(𝐴₂))) is unitary. Obviously, 𝑄 is left invertible. Now, 𝑀_𝑄 can be rewritten as 𝑀_𝑄=𝐴₁𝐴₁₂00𝐴₂000𝑉00𝐵∶𝐻₁⊕𝐻₂⊕𝐾⟶𝐻₁⊕𝑅𝐴₂⊕𝐻₂⊖𝑅𝐴₂⊕𝐾. (23) Since 𝐴₂ is left invertible and 𝑉 is invertible, then there exist unique 𝐴^′₂ and 𝑉^′ such that 𝐴^′₂𝐴₂=𝐼_𝐻2 and 𝑉^′𝑉=𝐼_𝐾, and 𝐴^′₂000𝑉^′0𝐴₂00𝑉0𝐵=𝐼_𝐻2⊕𝐼_𝐾. (24) This implies that 𝐴₂00𝑉0𝐵 is left invertible. And, hence, Lemma 2 leads to 𝑀_𝑄∈Φ_𝑎𝑏(𝐻⊕𝐾).
Case  2. Assume that 𝐴∈Φ_𝑎𝑏(𝐻), 𝑏.𝑠._mul(𝐵)≤𝑠_mul(𝐴), and 𝐵∈Φ₊(𝐾). Then, it follows from Lemma 3 that 𝐴 can be decomposed into the following form: 𝐴𝐴=¹𝐴₁₂0𝐴₂∶𝐻₁⊕𝐻₂⟶𝐻₁⊕𝐻₂, (25) where dim(𝐻₁)<∞, 𝐴₁ is nilpotent, 𝐴₂ is a left invertible operator, and 𝛽(𝐴₂)=𝑠_mul(𝐴). By the assumption that 𝐵∈Φ₊(𝐾) and Lemma 1, we know that 𝐵 can be decomposed into the following form with respect to some orthogonal decomposition 𝐾=𝐾₁⊕𝐾₂⊕𝐾₃∶𝐵𝐵=¹∗∗0𝐵₂∗00𝐵₃, (26) where dim(𝐾₃)<∞, 𝐵₁ is a right invertible operator, 𝐵₂ is a left invertible operator, 𝐵₃ is a finite nilpotent operator, and the parts marked by ∗ can be any operators. Moreover, ∞>𝛼(𝐵₁)=𝑏.𝑠._mul(𝐵). Thus, 𝛽(𝐴₂)≥𝛼(𝐵₁), and then there exists some left invertible 𝐶₁∈𝐵(𝑁(𝐵₁),𝐻₂⊖𝑅(𝐴₂)). Noting that dim((𝐾₁⊖𝑁(𝐵₁))⊕𝐾₂⊕𝐾₃)=dim(𝐻₁⊕𝑅(𝐴₂))=∞, we can let 𝐶₂∈𝐺((𝐾₁⊖𝑁(𝐵₁))⊕𝐾₂⊕𝐾₃,𝐻₁⊕𝑅(𝐴)). Consider 𝐶𝑄=¹00𝐶₂∶𝑁𝐵₁⊕𝐾₁⊖𝑁𝐵₁⊕𝐾₂⊕𝐾₃⟶𝐻₂⊖𝑅𝐴₂⊕𝐻₁⊕𝑅𝐴₂. (27)
Obviously, 𝑄 is left invertible, and 𝑀_𝑄 can be rewritten as 𝑀_𝑄=𝐴₁𝐴₁₂𝐶₁₁0𝐶₁₂𝐶₁₃0𝐴₂₁𝐶₂₁0𝐶₂₂𝐶₂₃000𝐶₁0000𝐵₁₁0∗∗0000𝐵₂∗00000𝐵₃∶𝐻₁⊕𝐻₂⊕𝐾₁⊖𝑁𝐵₁⊕𝑁𝐵₁⊕𝐾₂⊕𝐾₃⟶𝐻₁⊕𝑅𝐴₂⊕𝐻₂⊖𝑑𝑅𝐴₂⊕𝐾₁⊕𝐾₂⊕𝐾₃, (28) where 𝐴₂₁ and 𝐵₁₁ are invertible and 𝐶₁ and 𝐵₂ are left invertible. Similar to the proof of Case 1, through direct calculation we can show that 𝐴₂₁𝐶₂₁000𝐶₁0𝐵₁₁0 is left invertible. Also since dim(𝐻₁)<∞ and dim(𝐾₃)<∞, we have 𝐴₁∈Φ_𝑏(𝐻₁) and 𝐵₃∈Φ_𝑏(𝐾₃). Thus, it follows from Lemma 2 that 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾).

By duality, we have the following.

Theorem 11. For any given 𝐴∈𝐵(𝐻) and 𝐵∈𝐵(𝐾), one has _{𝐶∈𝐺𝑟()𝐾,𝐻}𝜎_𝑠𝑏𝑀_𝐶=_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑠𝑏𝑀_𝐶. (29)

3. ⋂_{𝐶∈𝐺(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶) and ⋂_{𝐶∈Φ(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶)

In this section, we give the characterization of invertible and Fredholm perturbations of upper semi-Browder spectra of 2×2 upper-triangular matrices. We begin with some lemmas.

Lemma 12 (see [19]). For a given pair (𝐴,𝐵)∈𝐵(𝐻)×𝐵(𝐾), if either 𝐴 or 𝐵 is a compact operator, then, for each 𝐶∈Φ(𝐾,𝐻), 𝑀_𝐶 is not a semi-Fredholm operator.
In particular, if 𝐵 is not compact, then 𝑀_𝐶 is not semi-Browder for any invertible operator 𝐶.

Lemma 13. The following statements are equivalent. (i)𝐵 is not compact. (ii)For each given 𝐴∈Φ_𝑎𝑏(𝐻), if 𝛽(𝐴)=∞, then there exists an operator 𝐶∈𝐺(𝐾,𝐻) such that 𝑀_𝐶 is an upper semi-Browder operator.(iii)For each given 𝐴∈Φ_𝑎𝑏(𝐻), if 𝛽(𝐴)=∞, then there exists an operator 𝐶∈Φ(𝐾,𝐻) such that 𝑀_𝐶 is an upper semi-Browder operator.

Proof. Obviously, we only need to prove the implications (i) ⇒ (ii) and (iii) ⇒ (i). (iii) ⇒ (i). If 𝐵 is compact, then it follows from Lemma 12 that 𝑀_𝐶 is not a semi-Fredholm operator for each 𝐶∈Φ(𝐾,𝐻), which contradicts with (iii). Thus, 𝐵 is not compact. (i) ⇒ (ii). Suppose that 𝐵 is not compact. Then, we consider the following two cases.
Case  1. Assume that 𝑅(𝐵) is closed. It follows from Lemma 3 that 𝐴 can be decomposed into the following form with respect to some orthogonal decomposition 𝐻=𝐻₁⊕𝐻₂∶𝐴𝐴=¹𝐴₁₂0𝐴₂∶𝐻₁⊕𝐻₂⟶𝐻₁⊕𝐻₂, (30) where dim(𝐻₁)<∞, 𝐴₁ is nilpotent, and 𝐴₂ is a left invertible operator. Noting that 𝛽(𝐴)=∞, we have 𝛽(𝐴₂)=∞. Since the assumption that 𝐵 is not compact, we have that dim𝑁(𝐵)^⟂=∞. Also since 𝛽(𝐴₂)=∞, let 𝑅(𝐴₂)^⟂=𝐻₃⊕𝐻₄ with dim(𝐻₃)=dim𝑁(𝐵) and dim(𝐻₄)=∞. Define an operator 𝐶∶𝐾→𝐻 by 𝐶𝐶=¹00𝐶₂∶𝑁(𝐵)⊕𝑁(𝐵)^⟂⟶𝐻₃⊕𝐻₁⊕𝑅𝐴₂⊕𝐻₄, (31) where 𝐶₁∈𝐵(𝑁(𝐵),𝐻₃) and 𝐶₂∈𝐵(𝑁(𝐵)^⟂,𝐻₁⊕𝑅(𝐴₂)⊕𝐻₄) are invertible operators. Obviously, 𝐶∈𝐵(𝐾,𝐻) is invertible. Next, we claim that 𝑀_𝐶 is an upper semi-Browder operator. To see this, 𝑀_𝐶 can be rewritten as 𝑀_𝐶=𝐴₁𝐴₁₂𝐶₁₁00𝐴₂₂𝐶₂₁0000𝐶₁00𝐶₄₁000𝐵₁0∶𝐻₁⊕𝐻₂⊕𝑁(𝐵)^⟂⊕𝑁(𝐵)⟶𝐻₁⊕𝑅𝐴₂⊕𝐻₃⊕𝐻₄⊕𝐾, (32) where 𝐴₂₂∈𝐵(𝐻₂,𝑅(𝐴₂)) is invertible and 𝐵₁∈𝐵(𝑁(𝐵)^⟂,𝑅(𝐵)) is left invertible. By Lemma 2 and the fact that 𝐴₁∈Φ_𝑏(𝐻₁), it is sufficient to prove that 𝑀₁𝐴=∶²²𝐶₂₁000𝐶₁0𝐶₄₁00𝐵₁0 (33) is semi-Browder. For this, we only need to show that 𝑀₁ is left invertible. In fact, since 𝐴₂₂ is invertible and 𝐵₁ and 𝐶₁ are left invertible, we can set 𝐴^′₂₂, 𝐵^′₁, and 𝐶^′₁ such that 𝐴^′₂₂𝐴₂₂=𝐼_𝐻2,𝐵^′₁𝐵₁=𝐼_𝑁(𝐵)^⟂,𝐶^′₁𝐶₁=𝐼_𝑁(𝐵). (34) Direct calculation shows that 𝐴^′₂₂00−𝐴^′₂₂𝐶₂₁𝐵^′₁000𝐵^′₁0𝐶^′₁𝐴00²²𝐶₂₁000𝐶₁0𝐶₄₁00𝐵₁0=𝐼_𝐻2000𝐼_𝑁(𝐵)^⟂000𝐼_𝑁(𝐵), (35) which implies that 𝑀₁ is left invertible. Noting that 𝐴₁∈Φ(𝐻₁), by Lemma 2 we have that 𝑀_𝐶 is upper semi-Browder.
Case  2. Assume that 𝑅(𝐵) is not closed. If 𝐵 is not compact, then by Lemma 7, 𝑅(𝐵) contains a closed infinite-dimensional subspace. Without loss of generality, suppose that 𝐾₁ is a closed subspace of 𝑅(𝐵) with dim𝐾₁=∞ and dim𝐾₁^⟂=∞. Let 𝐾₁={𝑥∈𝑁(𝐵)^⟂∶𝐵𝑥∈𝐾₁}. Thus, 𝐾₁ is a closed subspace of 𝑁(𝐵)^⟂, and dim(𝐾₁)=∞. Denote 𝐾₂=𝑁(𝐵)^⟂⊖𝐾₁. Without loss of generality, we may assume that dim(𝐾₂)=∞ (otherwise, suppose that {𝑒_𝑛}^∞_𝑛=1 is an orthonormal basis of 𝐾₁. Denote 𝐾^′₁=span{𝑒_𝑛∶𝑛=2,4,6,…} and 𝐾^′₁={𝐵𝑥∶𝑥∈𝐾^′₁}, then 𝐾₁ and 𝐾₁ can be replaced by 𝐾^′₁ and 𝐾^′₁, resp.). Since 𝛽(𝐴₂)=∞, let 𝑅(𝐴₂)^⟂=𝐻₃⊕𝐻₄ with dim(𝐻₃)=dim𝑁(𝐵) and dim𝐻₄=∞. Define an operator 𝐶∶𝐾→𝐻 by 𝐶𝐶=¹000𝐶₂000𝐶₃∶𝐾₁⊕𝐾₂(𝐵)⟶𝐻⊕𝑁₁⊕𝑅𝐴₂⊕𝐻₄⊕𝐻₃, (36) where 𝐶₁, 𝐶₂, and 𝐶₃ are unitary operators. Obviously, 𝐶 is invertible. 𝑀_𝐶 can be rewritten as 𝑀_𝐶=𝐴₁𝐴₁₂𝐶₁₁000𝐴₂₂𝐶₂₁000000𝐶₃000𝐶₂000𝐵₁₁𝐵₁₂0000𝐵₂₂0∶𝐻₁⊕𝐻₂⊕𝐾₁⊕𝐾₂⊕𝑁(𝐵)⟶𝐻₁⊕𝑅𝐴₂⊕𝐻₃⊕𝐻₄⊕𝐾₁⊕𝐾₁^⟂, (37) where 𝐴₂₂ and 𝐵₁₁ are invertible and 𝐶₁=^𝐶11_𝐶21.
Next, we prove that 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾). Noting that dim(𝐻₁)<∞, then, by Lemma 2, it is sufficient to prove that 𝑀₁𝐴=∶²²𝐶₂₁00000𝐶₃00𝐶₂00𝐵₁₁𝐵₁₂000𝐵₂₂0 (38) is left invertible. For this, let 𝐴^′₂₂, 𝐵^′₁₁, 𝐶^′₁, and 𝐶^′₂ be operators satisfying 𝐴^′₂₂𝐴₂₂=𝐼_𝐻2,𝐵^′₁₁𝐵₁₁=𝐼_𝐾1,𝐶^′₂𝐶₂=𝐼_𝐾2,𝐶^′₃𝐶₃=𝐼_𝑁(𝐵). (39) Direct calculation shows that 𝐴^′₂₂0𝐴^′₂₂𝐶₂₁𝐵^′₁₁𝐵₁₂𝐶^′₂−𝐴^′₂₂𝐶₂₁𝐵^′₁₁000𝐵^′₁₁𝐵₁₂𝐶^′₂𝐵^′₁₁000𝐶^′₂000𝐶^′₃×𝐴00₂₂𝐶₂₁00000𝐶₃00𝐶₂00𝐵₁₁𝐵₁₂000𝐵₂₂0=𝐼_𝐻20000𝐼_𝐾10000𝐼_𝐾20000𝐼_𝑁(𝐵), (40) which implies that 𝑀₁ is left invertible.
Combining Case 1 with Case 2, the lemma is proved.

Similarly, we have the following.

Lemma 14. The following statements are equivalent: (i)𝐴 is not compact. (ii)For each given 𝐵∈Φ_𝑠𝑏(𝐻), if 𝛼(𝐵)=∞, then there exists an operator 𝐶∈𝐺(𝐾,𝐻) such that 𝑀_𝐶 is a lower semi-Browder operator.(iii)For each given 𝐵∈Φ_𝑠𝑏(𝐻), if 𝛼(𝐵)=∞, then there exists an operator 𝐶∈Φ(𝐾,H) such that 𝑀_𝐶 is a lower semi-Browder operator.
One is now ready to prove the main result of this section.

Theorem 15. For a given pair (𝐴,𝐵)∈𝐵(𝐻)×𝐵(𝐾), one has _{()𝐶∈𝐺𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=_{()𝐶∈Φ𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶∪𝜆∈ℂ∶𝐵−𝜆𝑖𝑠𝑐𝑜𝑚𝑝𝑎𝑐𝑡. (41)

Proof. According to Lemma 12, it is clear that _{()𝐶∈𝐺𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶⊇_{()𝐶∈Φ𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶⊇_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶∪𝜆∈ℂ∶𝐵−𝜆iscompact. (42) For the conversion, without loss of generality, suppose that 0∉_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶∪𝜆∈ℂ∶𝐵−𝜆iscompact. (43)
Then, 𝐵 is not compact, and there exists some 𝐶∈𝐵(𝐾,𝐻) such that 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾), and, hence, 𝐴∈Φ_𝑎𝑏(𝐻).
Case  1. 𝛽(𝐴)=∞. It follows from Lemma 13 that there exists some 𝐶∈𝐺(𝐾,𝐻) such that 𝑀_𝐶 is an upper semi-Browder operator. This implies that 𝜆∉⋂_{𝐶∈𝐺(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶). In this case, we have proved. Consider that _{()𝐶∈𝐺𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶⊆_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶∪𝜆∈ℂ∶𝐵−𝜆iscompact. (44)
Case  2. Consider 𝛽(𝐴)<∞. This implies that 𝐴∈Φ(𝐻), and, thus, 𝐵∈Φ₊(𝐾) since 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾). It follows from Lemma 5 that 𝑏.𝑠._mul(𝐵)≤𝑠_mul(𝐴). Moreover, using Lemmas 1 and 3, we have 𝐴𝐴=¹𝐴₁₂0𝐴₂∶𝐻₁⊕𝐻₂↦𝐻₁⊕𝐻₂,𝐵𝐵=¹∗∗0𝐵₂∗00𝐵₃∶𝐾₁⊕𝐾₂⊕𝐾₃↦𝐾₁⊕𝐾₂⊕𝐾₃, (45) where dim(𝐻₁)<∞, 𝐴₁ is nilpotent, 𝐴₂ is a left invertible operator, dim(𝐾₃)<∞, 𝐵₁ is a right invertible operator, 𝐵₂ is a left invertible operator, 𝐵₃ is a finite nilpotent operator, and the parts marked by ∗ can be any operators. Moreover, 𝛽(𝐴₂)=𝑠_mul(𝐴), and 𝛼(𝐵₁)=𝑏.𝑠._mul(𝐵). Hence, 𝛼(𝐵₁)≤𝛽(𝐴₂). Now, put 𝐻₂=𝑅(𝐴₂)⊕𝐻₃⊕𝐻₄, where dim(𝐻₃)=𝛼(𝐵₁)<∞. Noting that dim𝐻₁⊕𝑅(𝐴₂)⊕𝐻₄=𝑁(𝐵₁)^⟂⊕𝐾₂⊕𝐾₃=∞, there exist unitaries 𝐶₃₃∈𝐵(𝑁(𝐵₁),𝐻₃) and 𝐶^′∈𝐵(𝐻₁⊕𝑅(𝐴₂)⊕𝐻₄,𝑁(𝐵₁)^⟂⊕𝐾₂⊕𝐾₃). Let 𝐶=^𝐶330_0𝐶′. Obviously, 𝐶=^𝐶330_0𝐶′∈𝐺(𝐻⊕𝐾).
Consider that operator 𝑀_𝐶==𝐴𝐴𝐶0𝐵∶𝐻⊕𝐾→𝐻⊕𝐾₁𝐴₁₂𝐶₁₁0𝐶₁₃𝐶₁₄0𝐴₂𝐶₂₁0𝐶₂₃𝐶₂₄000𝐶₃₃0000𝐶₄₁0𝐶₄₃𝐶₄₄00𝐵₁₁0∗∗0000𝐵₂∗00000𝐵₃∶𝐻₁⊕𝐻₂⊕𝑁𝐵₁^⟂⊕𝑁𝐵₁⊕𝐾₂⊕𝐾₃⟶𝐻₁⊕𝑅𝐴₂⊕𝐻₃⊕𝐻₄⊕𝐾₁⊕𝐾₂⊕𝐾₃, (46) where 𝐶^′=𝐶₁₁𝐶₁₃𝐶₁₄𝐶₂₁𝐶₂₃𝐶₂₄𝐶₄₁𝐶₄₃𝐶₄₄∶𝐻₁⊕𝐻₂⊕𝑁𝐵₁^⟂⟶𝐻₁⊕𝑅𝐴₂⊕𝐻₄,𝐵₁=𝐵₁₁0. (47) We claim that 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾). In fact, since 𝐴₁ and 𝐵₃ are Browder operators, then, by Lemma 2, it is sufficient to show that 𝐴𝑀=∶²𝐶₂₁0𝐶₂₃00𝐶₃₃00𝐶₄₁0𝐶₄₃0𝐵₁₁0∗000𝐵₂∶𝐻₂⊕𝑁𝐵₁^⟂⊕𝑁𝐵₁⊕𝐾₂⟶𝑅𝐴₂⊕𝐻₃⊕𝐻₄⊕𝐾₁⊕𝐾₂ (48) is upper semi-Browder. Observe that 𝐴₂ and 𝐵₂ are left invertible; 𝐶₃₃ and 𝐵₁₁ are invertible. Direct calculation shows that 𝑀 is injective. Since 𝐴∈Φ(𝐻) and 𝐵∈Φ₊(𝐾), we have 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾), and, hence, 𝑀 is an upper semi-Fredholm operator. Thus, 𝑀 is left invertible. Combining this with Lemma 2 yields 𝑀_𝐶∈Φ_𝑎𝑏(𝐻⊕𝐾), which means that 𝜆∉⋂_{𝐶∈𝐺(𝐾,𝐻)}𝜎_𝑎𝑏(𝑀_𝐶). Thus, _{()𝐶∈𝐺𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶⊆_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶∪𝜆∈ℂ∶𝐵−𝜆iscompact. (49)
Combining Case 1 with Case 2 leads to _{()𝐶∈𝐺𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=_{()𝐶∈𝜙𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶=_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑎𝑏𝑀_𝐶∪𝜆∈ℂ∶𝐵−𝜆iscompact. (50) This completes the proof.

By duality, we have

Theorem 16. For a given pair (𝐴,𝐵)∈𝐵(𝐻)×𝐵(𝐾), one has _{()𝐶∈𝐺𝐾,𝐻}𝜎_𝑠𝑏𝑀_𝐶=_{()𝐶∈Φ𝐾,𝐻}𝜎_𝑠𝑏𝑀_𝐶=_{()𝐶∈𝐵𝐾,𝐻}𝜎_𝑠𝑏𝑀_𝐶∪𝜆∈ℂ∶𝐴−𝜆𝑖𝑠𝑐𝑜𝑚𝑝𝑎𝑐𝑡. (51)

Acknowledgments

Long Long research is supported by the Freedom Explore Program of Central South University (Grant no. 2012QNZT040) and the Post Doctoral Program Foundation of Central South University. This work is also supported by the National Natural Science Foundation of China (11226113, 11301077, 11171066, and 10771191), the Foundation of the Education Department of Fujian Province (JA12074), and the Natural Science Foundation of Fujian Province (2012J05003).