Compact perturbations of operators with property (t)

Xinling Yu; Weijuan Shi; Guoxing Ji

doi:10.1515/math-2021-0114

Article Open Access

Compact perturbations of operators with property (t)

Xinling Yu , Weijuan Shi and Guoxing Ji

Published/Copyright: December 31, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

Let ℋ be an infinite dimensional complex Hilbert space and ℬ ( ℋ ) the algebra of all bounded linear operators on ℋ . For an operator T ∈ ℬ ( ℋ ) , we say property ( t ) holds for T if σ ( T ) ⧹ σ u w ( T ) = Π 00 ( T ) , where σ ( T ) and σ u w ( T ) denote the spectrum and the Weyl essential approximate point spectrum of T , respectively, and Π 00 ( T ) = { λ ∈ iso σ ( T ) : 0 < n ( T − λ ) < ∞ } . In this paper, we consider the stability of property ( t ) under (small) compact perturbations. Also, we explore the relations between the stability of property ( t ) and the stability of Weyl-type theorems. Moreover, we characterize those operators T satisfying that property ( t ) holds for f ( T ) for each function f analytic on some neighborhood of σ ( T ) .

Keywords: property (t); compact perturbation; Weyl-type theorem; function of operator

MSC 2010: 47B20; 47A10

1 Introduction

The study of Weyl’s theorem and its variants for bounded linear operators has a long history and has established many remarkable results in the past few decades (cf. [1,2,3, 4,5]). Recently, many scholars have paid considerable attention to Weyl-type theorems and their perturbations. For example, Cao et al., respectively, explored perturbations of Weyl’s theorem, property ( ω ), SVEP and so on (cf. [6,7,8, 9]). In 2014, Rashid first put forward property ( t ) and explored the relations between property ( t ) and Weyl-type theorems in [10]. Later, Shen and Ji in [11] established the conditions for property ( t ) to persist under commuting perturbations by nilpotent operators, finite rank operators, quasi-nilpotent operators and Riesz operators.

Inspired by [6,10, 11,12,13], in this paper, we will explore the stability of property ( t ) under (non-commuting) compact perturbations. Also, the relations between the stability of property ( t ) and the stability of Weyl-type theorems will be investigated. Moreover, it is an interesting topic to study Weyl-type theorems for functions of operators (cf. [9,14, 15,16]). Here, we will establish some necessary and sufficient conditions for an operator T satisfying that property ( t ) holds for f ( T ) for each function f analytic on some neighborhood of σ ( T ) .

Throughout this paper, we denote by ℋ an infinite dimensional complex Hilbert space. Let ℬ ( ℋ ) and K ( ℋ ) be the algebra of all bounded linear operators on ℋ and the ideal of compact operators in ℬ ( ℋ ) , respectively. For each λ 0 ∈ C and for some δ > 0 , let B δ ( λ 0 ) = { λ ∈ C : ∣ λ − λ 0 ∣ < δ } , B δ 0 ( λ 0 ) = B δ ( λ 0 ) ⧹ { λ 0 } , where C denotes the complex number field.

Let T ∈ ℬ ( ℋ ) . We denote by N ( T ) , ℛ ( T ) , σ ( T ) and ρ ( T ) the kernel, the range, the spectrum and the resolvent set of T , respectively. Let n ( T ) = dim N ( T ) and d ( T ) = codim ℛ ( T ) . We say n ( T ) and d ( T ) are the nullity and the deficiency of T , respectively. An operator T is called upper semi-Fredholm operator if n ( T ) < ∞ and ℛ ( T ) is closed, while T is called lower semi-Fredholm operator if d ( T ) < ∞ . In particular, if n ( T ) = 0 and ℛ ( T ) is closed, then we call T a bounded below operator. If both n ( T ) and d ( T ) are finite, then T is said to be a Fredholm operator. An operator T is said to be a semi-Fredholm operator if it is either upper semi-Fredholm or lower semi-Fredholm. In this case, the index of T is defined by ind ( T ) = n ( T ) − d ( T ) and the minimal index of T is defined by min ind ( T ) = min { n ( T ) , d ( T ) } . Let

ρ a ( T ) = { λ ∈ C : T − λ is a bounded below operator } ; ρ SF ( T ) = { λ ∈ C : T − λ is a semi-Fredholm operator } ; ρ u w ( T ) = { λ ∈ C : T − λ is a semi-Fredholm operator with ind ( T − λ ) ≤ 0 } ; ρ w ( T ) = { λ ∈ C : T − λ is a Fredholm operator of index zero } .

The approximate point spectrum, semi-Fredholm spectrum, Weyl essential approximate point spectrum and Weyl spectrum are defined by σ a ( T ) = C ⧹ ρ a ( T ) , σ SF ( T ) = C ⧹ ρ SF ( T ) , σ u w ( T ) = C ⧹ ρ u w ( T ) and σ w ( T ) = C ⧹ ρ w ( T ) , respectively. Denote

ρ SF + ( T ) = { λ ∈ ρ SF ( T ) : ind ( T − λ ) > 0 } , ρ SF − ( T ) = { λ ∈ ρ SF ( T ) : ind ( T − λ ) < 0 } .

Let σ 0 ( T ) denote the set of all normal eigenvalues (Riesz points) of T . From [19, Corollary 1.14], we can deduce

σ 0 ( T ) = { λ ∈ iso σ ( T ) : T − λ is Fredholm of index zero } .

Given a subset σ of C , we denote by iso σ , acc σ , int σ and ∂ σ the set of all isolated points, the set of all accumulation points, the set of all interior points and the set of all boundary points of σ , respectively. Let T ∈ ℬ ( ℋ ) . Then we denote

Π 00 ( T ) = { λ ∈ iso σ ( T ) : 0 < n ( T − λ ) < ∞ } , Π 00 a ( T ) = { λ ∈ iso σ a ( T ) : 0 < n ( T − λ ) < ∞ } .

Let T ∈ ℬ ( ℋ ) . As in [10], we say that property ( t ) holds for T if

σ ( T ) ⧹ σ u w ( T ) = Π 00 ( T ) .

Denote

( t ) = { T ∈ ℬ ( ℋ ) : property ( t ) holds for T } .

An operator T is said to have the stability of property ( t ) under (small) compact perturbations if T + K ∈ ( t ) for all compact operator K ∈ K ( ℋ ) (with ‖ K ‖ < ε ).

2 Property ( t ) and its perturbations

In this section, we will study the stability of property ( t ) under (small) compact perturbations. Before giving our main results, we first give a useful lemma.

Lemma 2.1

Let T ∈ ℬ ( ℋ ) . If T ∈ ( t ) , then σ ( T ) = σ SF ( T ) ∪ ρ SF + ( T ) ∪ σ 0 ( T ) .

Proof

It suffices to show the inclusion “ ⊆ .” Take any λ which does not belong to the right side of the above formulation, then T − λ is a semi-Fredholm operator and ind ( T − λ ) ≤ 0 . We claim that λ ∉ σ ( T ) . Otherwise, we get that λ ∈ σ ( T ) ⧹ σ u w ( T ) . It follows from T ∈ ( t ) that λ ∈ σ 0 ( T ) , which contradicts with “ λ ∉ σ 0 ( T ) .”□

Next, we will characterize those operators for which property ( t ) is stable under small compact perturbations using Weyl essential approximate point spectrum.

Theorem 2.2

Let T ∈ ℬ ( ℋ ) . Then there exists ε > 0 such that T + K ∈ ( t ) for all K ∈ K ( ℋ ) with ‖ K ‖ < ε if and only if the following conditions hold:

σ ( T ) = σ SF ( T ) ∪ ρ SF + ( T ) ∪ σ 0 ( T ) ;
ρ u w ( T ) consists of finitely many connected components;
iso σ u w ( T ) = ∅ .

Proof

“ ⇒ .” From Lemma 2.1, we can easily get that ( i ) holds.

If ( i i ) does not hold, then let { Ω n } n = 1 ∞ be the enumeration of bounded connected components of ρ u w ( T ) . Obviously, ∑ n = 1 ∞ m ( Ω n ) ≤ m ( σ ( T ) ^ ) < ∞ , where σ ( T ) ^ is the polynomially convex hull of σ ( T ) . Then there exists k ∈ N (where N denotes the set of nonnegative integers) such that m ( Ω n ) < π ε 2 4 for any n > k . Fix n 0 > k . Since ∂ Ω n 0 ⊆ σ SF ( T ) , by [17, Lemma 3.2.6], there exists K 1 ∈ K ( ℋ ) with ‖ K 1 ‖ < ε 4 such that

T + K 1 = N ∗ 0 A H 1 H 1 ⊥ ,

where N is a normal operator and σ ( N ) = σ SF ( N ) = ∂ Ω n 0 . By [18, Theorem 3.1], there exists a compact operator K 2 ′ with ‖ K 2 ′ ‖ < m ( Ω n 0 ) / π + ε 4 = ε 2 such that σ ( N + K 2 ′ ) = Ω ¯ n 0 . Let

K 2 = K 2 ′ 0 0 0 and K = K 1 + K 2 .

Then

T + K = N + K 2 ′ ∗ 0 A H 1 H 1 ⊥ ,

‖ K ‖ ≤ ‖ K 1 ‖ + ‖ K 2 ‖ < ε and Ω n 0 ⊆ σ ( N + K 2 ′ ) ⧹ σ w ( N + K 2 ′ ) . Since T + K ∈ ( t ) , there exists a λ ∈ Ω n 0 such that T + K − λ is invertible. Therefore, N + K 2 ′ − λ is a bounded below operator. Then N + K 2 ′ − λ is invertible since N + K 2 ′ − λ is a Fredholm operator with index zero. It contradicts with “ σ ( N + K 2 ′ ) = Ω ¯ n 0 .”

Suppose iso σ u w ( T ) ≠ ∅ . Let λ 0 ∈ iso σ u w ( T ) . For given ε > 0 , there exists a δ such that 0 < δ < ε 4 and B δ 0 ( λ 0 ) ∩ σ u w ( T ) = ∅ . Put σ 1 = B δ ( λ 0 ) ∩ σ ( T ) . For any μ ∈ B δ 0 ( λ 0 ) ∩ σ ( T ) , we have μ ∈ σ ( T ) ⧹ σ u w ( T ) . It follows from T ∈ ( t ) that σ 1 consists of λ 0 and at most countable Riesz points of T . Hence, σ 1 is a clopen subset of σ ( T ) . Moreover, if σ 1 ∩ σ 0 ( T ) is an infinite set, then λ 0 is the unique limit point of σ 1 ∩ σ 0 ( T ) . By [16, Corollary 2.2], T can be represented as

T = A ∗ 0 B ℋ ( σ 1 , T ) ℋ ( σ 1 , T ) ⊥ ,

where σ ( A ) = σ 1 and σ ( B ) = σ ( T ) ⧹ σ 1 . Note that λ 0 ∈ σ SF ( T ) . We can check that σ SF ( A ) = { λ 0 } , σ ( A ) = σ SF ( A ) ∪ σ 0 ( A ) , σ 0 ( A ) ⊆ σ 0 ( T ) and max { dist [ λ , ∂ ρ SF ( A ) ] : λ ∈ σ 0 ( A ) } < ε 4 . Then by [19, Theorem 3.48] there exists a compact operator K ¯ 1 acting on ℋ ( σ 1 ; T ) with ‖ K ¯ 1 ‖ < ε 2 such that

min ind ( A + K ¯ 1 − λ ) = 0 , ∀ λ ∈ ρ SF ( A + K ¯ 1 ) .

Note that σ SF ( A ) = { λ 0 } . It implies that ρ SF ( A ) is connected. It follows from ρ SF ( A ) = ρ SF ( A + K ¯ 1 ) that ρ SF ( A + K ¯ 1 ) is connected. Since ρ ( A + K ¯ 1 ) ⊆ ρ SF ( A + K ¯ 1 ) and the index of every connected component of ρ SF ( A + K ¯ 1 ) is constant, we obtain that

ind ( A + K ¯ 1 − λ ) = 0 , ∀ λ ∈ ρ SF ( A + K ¯ 1 ) .

Hence, we get σ ( A + K ¯ 1 ) = σ SF ( A + K ¯ 1 ) = σ SF ( A ) = { λ 0 } . For A + K ¯ 1 , similar to the proof of [16, Lemma 4.8], we obtain that there exists a compact operator K ¯ 2 acting on ℋ ( σ 1 ; T ) with ‖ K ¯ 2 ‖ < ε 2 such that

σ ( A + K ¯ 1 + K ¯ 2 ) = σ SF ( A + K ¯ 1 + K ¯ 2 ) = { λ 0 } and Π 00 ( A + K ¯ 1 + K ¯ 2 ) = { λ 0 } .

Set

K = K ¯ 1 + K ¯ 2 0 0 0 and A ¯ = A + K ¯ 1 + K ¯ 2 .

Then K ∈ K ( ℋ ) with ‖ K ‖ < ε and

T + K = A ¯ ∗ 0 B ℋ ( σ 1 , T ) ℋ ( σ 1 , T ) ⊥ .

Since λ 0 ∈ Π 00 ( A ¯ ) , there exists a δ ′ such that 0 < δ ′ < δ and A ¯ − μ is invertible for any μ ∈ B δ ′ 0 ( λ 0 ) . Since B δ ′ ( λ 0 ) ∩ σ ( B ) = ∅ , it implies that T + K − μ is invertible for any μ ∈ B δ ′ 0 ( λ 0 ) . Thus, λ 0 ∈ iso σ ( T + K ) . Note that n ( T + K − λ 0 ) = n ( A ¯ − λ 0 ) . We could obtain that 0 < n ( T + K − λ 0 ) < ∞ . Therefore, λ 0 ∈ Π 00 ( T + K ) . It follows from T + K ∈ ( t ) that λ 0 ∈ σ ( T + K ) ⧹ σ u w ( T + K ) . Hence, λ 0 ∈ ρ u w ( T ) . It is in contradiction with “ λ 0 ∈ iso σ u w ( T ) .”

“ ⇐ .” Let { Ω i } i = 1 n be the enumeration of connected components of ρ u w ( T ) . It follows from ( i ) that σ ( T ) ⧹ σ u w ( T ) = σ 0 ( T ) . Hence, we can choose λ i ∈ Ω i ( 1 ≤ i ≤ n ) such that T − λ i ( 1 ≤ i ≤ n ) is invertible. Then there exists ε i > 0 such that T + K − λ i is invertible for all K ∈ K ( ℋ ) with ‖ K ‖ < ε i . Let ε = min { ε i : 1 ≤ i ≤ n } . For any K ∈ K ( ℋ ) with ‖ K ‖ < ε , the index of every connected components of ρ u w ( T + K ) is zero. Moreover, by [19, Corollary 1.14], we know that the function

λ ↦ min ind ( T + K − λ ) = n ( T + K − λ ) = 0

on every component Ω i ( 1 ≤ i ≤ n ) of ρ u w ( T + K ) except for an at most denumerable set without limit points in ρ u w ( T + K ) . Then

Ω i ⊆ ρ ( T + K ) ∪ σ 0 ( T + K ) .

Hence,

ρ u w ( T + K ) = ∪ i = 1 n Ω i = ρ ( T + K ) ∪ σ 0 ( T + K ) .

Thus,

σ ( T + K ) ⧹ σ u w ( T + K ) = σ 0 ( T + K ) ⊆ Π 00 ( T + K ) .

To show the opposite inclusion, we suppose that λ ∈ Π 00 ( T + K ) . Then there exists a δ > 0 such that T + K − μ is invertible for any μ ∈ B δ 0 ( λ ) . Hence, T − μ is a Fredholm operator with index zero for any μ ∈ B δ 0 ( λ ) . It follows from iso σ u w ( T ) = ∅ that λ ∈ ρ u w ( T ) . Hence, λ ∈ σ ( T + K ) ⧹ σ u w ( T + K ) . Thus, Π 00 ( T + K ) ⊆ σ ( T + K ) ⧹ σ u w ( T + K ) .

Therefore, T + K ∈ ( t ) for all K ∈ K ( ℋ ) with ‖ K ‖ < ε .□

According to the proof of the sufficiency of Theorem 2.2, we know that ε is determined by the quantities of connected components of ρ u w ( T ) and the value of ε decreases as the number of connected components of ρ u w ( T ) increases. In the following, we will give an example of the stability of property ( t ) under small compact perturbations.

Example

Let T ∈ ℬ ( ℓ 2 ) be defined by

T ( x 1 , x 2 , x 3 , … ) = ( x 2 , x 3 , x 4 , … ) .

Then σ ( T ) = σ u w ( T ) = { λ ∈ C : ∣ λ ∣ ≤ 1 } , σ SF ( T ) = { λ ∈ C : ∣ λ ∣ = 1 } and ρ SF + ( T ) = { λ ∈ C : ∣ λ ∣ < 1 } . Thus,

σ ( T ) = σ SF ( T ) ∪ ρ SF + ( T ) ∪ σ 0 ( T ) ;
ρ u w ( T ) consists of one connected component;
iso σ u w ( T ) = ∅ .

By Theorem 2.2, there exists ε > 0 such that T + K ∈ ( t ) for all K ∈ K ( ℋ ) with ‖ K ‖ < ε .

As we know, Weyl spectrum is closely related to Weyl essential approximate point spectrum. In the following, we will give another characterization on the operators for which property ( t ) is stable under small compact perturbations using Weyl spectrum.

Corollary 2.3

Let T ∈ ℬ ( ℋ ) . Then there exists ε > 0 such that T + K ∈ ( t ) for all K ∈ K ( ℋ ) with ‖ K ‖ < ε if and only if the following conditions hold:

σ ( T ) = σ w ( T ) ∪ σ 0 ( T ) and ρ SF − ( T ) = ∅ ;
ρ w ( T ) consists of finitely many connected components;
iso σ w ( T ) = ∅ .

Proof

By Theorem 2.2, it suffices to show that

σ ( T ) = σ SF ( T ) ∪ ρ SF + ( T ) ∪ σ 0 ( T ) ⇔ ρ SF − ( T ) = ∅ and σ ( T ) = σ w ( T ) ∪ σ 0 ( T ) .

“ ⇒ .” Suppose that ρ SF − ( T ) ≠ ∅ . Then there exists a λ such that T − λ is a semi-Fredholm operator and ind ( T − λ ) < 0 . Hence, λ ∈ σ ( T ) . It follows that λ ∈ σ SF ( T ) ∪ ρ SF + ( T ) ∪ σ 0 ( T ) , a contradiction. Next we show that σ ( T ) = σ w ( T ) ∪ σ 0 ( T ) . It suffices to prove the inclusion “ ⊆ .” Take any λ ∈ σ ( T ) . Then we have λ ∈ σ SF ( T ) ∪ ρ SF + ( T ) ∪ σ 0 ( T ) . Note that σ SF ( T ) ∪ ρ SF + ( T ) ⊆ σ w ( T ) . It follows that λ ∈ σ w ( T ) ∪ σ 0 ( T ) .

“ ⇐ .” It suffices to prove that the inclusion “ ⊆ .” Take any λ ∈ σ ( T ) , then we have λ ∈ σ w ( T ) ∪ σ 0 ( T ) . Since ρ SF − ( T ) = ∅ , it implies that σ w ( T ) = σ u w ( T ) . Hence, λ ∈ σ u w ( T ) ∪ σ 0 ( T ) . Thus, λ ∈ σ SF ( T ) ∪ ρ SF + ( T ) ∪ σ 0 ( T ) .□

Recall that an operator T ∈ ℬ ( ℋ ) is said to be finite-isoloid if iso σ ( T ) ⊆ { λ ∈ C : 0 < n ( T − λ ) < ∞ } . According to Theorem 2.2, we will have the following result.

Corollary 2.4

Let T ∈ ℬ ( ℋ ) be finite-isoloid and acc [ iso σ ( T ) ] = ∅ . Then the following statements are equivalent:

There exists ε > 0 such that T + K ∈ ( t ) for all K ∈ K ( ℋ ) with ‖ K ‖ < ε ,
T ∈ ( t ) and ρ u w ( T ) consists of finitely many connected components.

Proof

By Theorem 2.2, it suffices to prove that ( i i ) ⇒ ( i ) . Following the proof of Theorem 2.2, we know that there exists ε > 0 such that σ ( T + K ) ⧹ σ u w ( T + K ) ⊆ Π 00 ( T + K ) for all K ∈ K ( ℋ ) with ‖ K ‖ < ε . Now we will show the opposite inclusion. We first claim that

iso σ ( T + K ) ⊆ acc [ iso σ ( T ) ] ∪ iso σ ( T ) ∪ ρ ( T ) .

In fact, suppose λ 0 ∈ iso σ ( T + K ) . Then there exists a δ > 0 such that T + K − λ is invertible for any λ ∈ B δ 0 ( λ 0 ) . Hence, T − λ is a semi-Fredholm operator with ind ( T − λ ) = 0 for any λ ∈ B δ 0 ( λ 0 ) . We consider it from the following two cases.

Case 1

Suppose there exists a δ ′ such that 0 < δ ′ < δ and B δ ′ 0 ( λ 0 ) ⊆ ρ ( T ) . Then λ 0 ∈ ρ ( T ) ∪ iso σ ( T ) .

Case 2

Suppose for any δ i satisfying 0 < δ i < δ and B δ i 0 ( λ 0 ) ∩ σ ( T ) ≠ ∅ . Then there exists a sequence { λ i } i = 1 ∞ ⊆ σ ( T ) such that λ i → λ 0 ( i → ∞ ) . It follows from T ∈ ( t ) that λ i ∈ iso σ ( T ) ( i ≥ 1 ) . Hence, λ 0 ∈ acc [ iso σ ( T ) ] .

Thus, the claim is proved. Note that acc [ iso σ ( T ) ] = ∅ . Then iso σ ( T + K ) ⊆ iso σ ( T ) ∪ ρ ( T ) . Using the fact that T is finite-isoloid and T ∈ ( t ) , we get

iso σ ( T + K ) ⊆ σ ( T + K ) ⧹ σ u w ( T + K ) .

Hence, σ ( T + K ) ⧹ σ u w ( T + K ) = Π 00 ( T + K ) for all K ∈ K ( ℋ ) with ‖ K ‖ < ε .□

Furthermore, we will characterize those operators for which property ( t ) is stable under compact perturbations.

Theorem 2.5

Let T ∈ ℬ ( ℋ ) . Then T + K ∈ ( t ) for all K ∈ K ( ℋ ) if and only if the following conditions hold:

ρ u w ( T ) is connected,
iso σ u w ( T ) = ∅ .

Proof

“ ⇒ .” If ( i ) does not hold, then we can choose a bounded connected component Ω n 0 . Note that ∂ Ω n 0 ⊆ σ SF ( T ) . Using a similar argument as in the proof of Theorem 2.2, we can get a contradiction. It is easy to see from Theorem 2.2 that ( i i ) holds.

“ ⇐ .” Since ρ u w ( T ) is connected and ρ u w ( T ) = ρ u w ( T + K ) for all K ∈ K ( ℋ ) , we have ρ u w ( T + K ) is connected. Similar to the proof of Theorem 2.2, we get ρ u w ( T + K ) = ρ ( T + K ) ∪ σ 0 ( T + K ) . Hence, σ ( T + K ) ⧹ σ u w ( T + K ) ⊆ Π 00 ( T + K ) . It follows from ( i i ) that Π 00 ( T + K ) ⊆ σ ( T + K ) ⧹ σ u w ( T + K ) . Therefore, T + K ∈ ( t ) for all K ∈ K ( ℋ ) .□

Next, we will give the relationships between isolated points of various spectrums when ρ u w ( T ) is connected in the following proposition.

Proposition 2.6

Let T ∈ ℬ ( ℋ ) . If ρ u w ( T ) is connected, then the following statements are equivalent:

iso σ u w ( T ) = ∅ ;
iso σ w ( T ) = ∅ ;
iso σ ( T ) = σ 0 ( T ) and acc σ w ( T ) = acc σ ( T ) .

Proof

( i ) ⇔ ( i i ) . Since ρ u w ( T ) is connected, similar to the proof of Theorem 2.2, we have ρ u w ( T ) = ρ ( T ) ∪ σ 0 ( T ) . It is clear that σ u w ( T ) = σ w ( T ) .

( i i ) ⇒ ( i i i ) . For iso σ ( T ) = σ 0 ( T ) , it suffices to prove that iso σ ( T ) ⊆ σ 0 ( T ) . Take any λ ∈ iso σ ( T ) . Then there exists a δ > 0 such that T − μ is invertible for any μ ∈ B δ 0 ( λ ) . It follows from iso σ w ( T ) = ∅ that λ ∈ ρ w ( T ) . Hence, λ ∈ σ 0 ( T ) .

For acc σ w ( T ) = acc σ ( T ) , it suffices to show that acc σ ( T ) ⊆ acc σ w ( T ) . Let λ 0 ∈ acc σ ( T ) ⧹ acc σ w ( T ) . Note that ρ u w ( T ) is connected. It implies that ρ w ( T ) = ρ u w ( T ) = ρ ( T ) ∪ σ 0 ( T ) . Hence, λ 0 ∈ iso σ w ( T ) , a contradiction. Therefore, acc σ ( T ) ⧹ acc σ w ( T ) = ∅ .

( i i i ) ⇒ ( i i ) . Suppose iso σ w ( T ) ≠ ∅ . Let λ 0 ∈ iso σ w ( T ) . Then there exists a δ > 0 such that B δ 0 ( λ 0 ) ⊆ ρ w ( T ) . It follows from ρ u w ( T ) is connected that B δ 0 ( λ 0 ) ⊆ ρ ( T ) ∪ σ 0 ( T ) . Next, we consider it from the following two cases.

Case 1

Suppose that there exists a δ ′ such that 0 < δ ′ < δ and B δ ′ 0 ( λ 0 ) ⊆ ρ ( T ) . Then λ 0 ∈ iso σ ( T ) = σ 0 ( T ) . It is in contradiction with “ λ 0 ∈ iso σ w ( T ) .”

Case 2

Suppose for any δ i satisfying 0 < δ i < δ and B δ i 0 ( λ 0 ) ∩ σ 0 ( T ) ≠ ∅ . Then there exists a sequence { λ i } i = 1 ∞ ⊆ σ 0 ( T ) such that λ i → λ 0 ( i → ∞ ) . Hence, λ 0 ∈ acc σ ( T ) = acc σ w ( T ) . It is in contradiction with “ λ 0 ∈ iso σ w ( T ) .”□

Based on Theorem 2.5 and Proposition 2.6, we will get the following corollary.

Corollary 2.7

Let T ∈ ℬ ( ℋ ) . Then the following statements are equivalent:

T + K ∈ ( t ) for all K ∈ K ( ℋ ) ;
ρ u w ( T ) is connected and iso σ w ( T ) = ∅ ;
ρ u w ( T ) is connected, iso σ ( T ) = σ 0 ( T ) and acc σ w ( T ) = acc σ ( T ) .

Note that property ( t ) is closely related to Weyl-type theorems. First, we recall [1,2, 3,4] that

Weyl’s theorem holds for T , denoted by T ∈ ( W ) , if σ ( T ) ⧹ σ w ( T ) = Π 00 ( T ) .
Browder’s theorem holds for T , denoted by T ∈ ( B ) , if σ ( T ) ⧹ σ w ( T ) ⊆ Π 00 ( T ) .
a-Weyl’s theorem holds for T , denoted by T ∈ ( a-W ) , if σ a ( T ) ⧹ σ u w ( T ) = Π 00 a ( T ) .
a-Browder’s theorem holds for T , denoted by T ∈ ( a-B ) , if σ a ( T ) ⧹ σ u w ( T ) ⊆ Π 00 a ( T ) .
Property ( ω ) holds for T , denoted by T ∈ ( ω ) , if σ a ( T ) ⧹ σ u w ( T ) = Π 00 ( T ) .

As seen in [10,11], we obtain

property ( t ) ⇒ a-Weyl’s theorem ⇒ Weyl’s theorem

and

property ( t ) ⇒ property ( ω ) .

Therefore, it is a natural question whether there are similar relationships between the stability of property ( t ) and the stability of Weyl-type theorems under compact perturbations. We have the following result.

Theorem 2.8

Let T ∈ ℬ ( ℋ ) . Then the following statements are equivalent:

T + K ∈ ( t ) for all K ∈ K ( ℋ ) ;
T + K ∈ ( ω ) for all K ∈ K ( ℋ ) ;
T + K ∈ ( a-W ) for all K ∈ K ( ℋ ) ;
T + K ∈ ( W ) for all K ∈ K ( ℋ ) and ρ SF − ( T ) = ∅ ;
T + K ∈ ( a-B ) for all K ∈ K ( ℋ ) and iso σ u w ( T ) = ∅ ;
T + K ∈ ( B ) for all K ∈ K ( ℋ ) , ρ SF − ( T ) = ∅ and iso σ w ( T ) = ∅ .

Proof

( i ) ⇒ ( i i ) . Assume that T + K ∈ ( t ) for all K ∈ K ( ℋ ) . It is obvious that σ a ( T + K ) ⧹ σ u w ( T + K ) ⊆ σ ( T + K ) ⧹ σ u w ( T + K ) = Π 00 ( T + K ) . Let λ ∈ Π 00 ( T + K ) . Then it follows from T + K ∈ ( t ) that λ ∈ σ ( T + K ) ⧹ σ u w ( T + K ) . Note that λ ∈ iso σ ( T + K ) . We have λ ∈ σ 0 ( T + K ) . It implies that λ ∈ σ a ( T + K ) . Therefore, Π 00 ( T + K ) ⊆ σ a ( T + K ) ⧹ σ u w ( T + K ) . Thus, T + K ∈ ( ω ) for all K ∈ K ( ℋ ) .

( i i ) ⇒ ( i ) . Assume that T + K ∈ ( ω ) for all K ∈ K ( ℋ ) . Then it is obvious that Π 00 ( T + K ) ⊆ σ ( T + K ) ⧹ σ u w ( T + K ) . To show the opposite inclusion, we let λ 0 ∈ σ ( T + K ) ⧹ σ u w ( T + K ) . If λ 0 ∉ σ a ( T + K ) , then ind ( T + K − λ 0 ) < 0 and we can choose a bounded component Ω of ρ u w ( T + K ) with negative index such that λ 0 ∈ Ω . Since ∂ Ω ⊆ σ SF ( T + K ) , by [17, Lemma 3.2.6], there exists K 1 ∈ K ( ℋ ) such that

T + K + K 1 = N ∗ 0 A H 1 H 1 ⊥ ,

where N is a normal operator and σ ( N ) = σ SF ( N ) = ∂ Ω . By [18, Theorem 3.1], there exists a compact operator K ¯ 2 such that σ ( N + K ¯ 2 ) = Ω ¯ . Let

K 2 = K ¯ 2 0 0 0 and K ′ = K 1 + K 2 .

Then

T + K + K ′ = N + K ¯ 2 ∗ 0 A and Ω ⊆ σ ( N + K ¯ 2 ) ⧹ σ w ( N + K ¯ 2 ) .

Since T + K + K ′ ∈ ( ω ) , there exists a λ ∈ Ω such that T + K + K ′ − λ is a bounded below operator. Therefore, N + K ¯ 2 − λ is also a bounded below operator. Then N + K ¯ 2 − λ is invertible. It is in contradiction with “ σ ( N + K ¯ 2 ) = Ω ¯ .” Hence, λ 0 ∈ σ a ( T + K ) ⧹ σ u w ( T + K ) . Then it follows from T + K ∈ ( ω ) that λ 0 ∈ Π 00 ( T + K ) . Therefore, σ ( T + K ) ⧹ σ u w ( T + K ) ⊆ Π 00 ( T + K ) . Thus, T + K ∈ ( t ) for all K ∈ K ( ℋ ) .

( i ) ⇒ ( i i i ) . Suppose that T + K ∈ ( t ) for all K ∈ K ( ℋ ) . Then

σ a ( T + K ) = σ ( T + K ) , ∀ K ∈ K ( ℋ ) .

Hence,

σ a ( T + K ) ⧹ σ u w ( T + K ) = σ ( T + K ) ⧹ σ u w ( T + K ) = Π 00 ( T + K ) = Π 00 a ( T + K ) .

Thus, T + K ∈ ( a-W ) for all K ∈ K ( ℋ ) .

( i i i ) ⇒ ( i ) . Suppose that T + K ∈ ( a-W ) for all K ∈ K ( ℋ ) . The inclusion Π 00 a ( T + K ) ⊆ σ ( T + K ) ⧹ σ u w ( T + K ) is obvious. To show the opposite inclusion, we let λ 0 ∈ σ ( T + K ) ⧹ σ u w ( T + K ) . The same argument in the above shows that λ 0 ∈ σ a ( T + K ) ⧹ σ u w ( T + K ) , then it follows from T + K ∈ ( a-W ) that λ 0 ∈ Π 00 a ( T + K ) . Hence, σ ( T + K ) ⧹ σ u w ( T + K ) ⊆ Π 00 a ( T + K ) . Thus, σ ( T + K ) ⧹ σ u w ( T + K ) = Π 00 a ( T + K ) . By [11, Theorem 1.2], we can get T + K ∈ ( t ) for all K ∈ K ( ℋ ) .

( i ) ⇒ ( i v ) . Assume that T + K ∈ ( t ) for all K ∈ K ( ℋ ) . Since σ u w ( T + K ) ⊆ σ w ( T + K ) , we can easily get that σ ( T + K ) ⧹ σ w ( T + K ) ⊆ Π 00 ( T + K ) . It suffices to prove the opposite inclusion. Let λ ∈ Π 00 ( T + K ) . It follows from T + K ∈ ( t ) that λ ∈ σ 0 ( T + K ) . Hence, we have λ ∈ σ ( T + K ) ⧹ σ w ( T + K ) . Therefore, Π 00 ( T + K ) ⊆ σ ( T + K ) ⧹ σ w ( T + K ) . Thus, T + K ∈ ( W ) for all K ∈ K ( ℋ ) .

Suppose ρ SF − ( T ) ≠ ∅ . Then there exists a λ such that T − λ is a semi-Fredholm operator and ind ( T − λ ) < 0 . Note that ρ SF − ( T ) = ρ SF − ( T + K ) for any K ∈ K ( ℋ ) . Then T + K − λ is also a semi-Fredholm operator and ind ( T + K − λ ) < 0 . So λ ∈ σ ( T + K ) ⧹ σ u w ( T + K ) . It follows from T + K ∈ ( t ) that λ ∈ Π 00 ( T + K ) . Hence, λ ∈ σ 0 ( T + K ) . Thus, ind ( T + K − λ ) = 0 , a contradiction.

( i v ) ⇒ ( i ) . Suppose that T + K ∈ ( W ) for all K ∈ K ( ℋ ) and ρ SF − ( T ) = ∅ . We have σ u w ( T ) = σ w ( T ) . Hence, σ ( T + K ) ⧹ σ u w ( T + K ) = σ ( T + K ) ⧹ σ w ( T + K ) = Π 00 ( T + K ) for all K ∈ K ( ℋ ) .

( i i i ) ⇔ ( v ) . By [12, Theorem 1.3], the necessity is obvious. It suffices to verify the sufficiency. In the following, we will prove the inclusion Π 00 a ( T + K ) ⊆ σ a ( T + K ) ⧹ σ u w ( T + K ) for all K ∈ K ( ℋ ) . Let λ ∈ Π 00 a ( T + K ) . Since λ ∈ iso σ a ( T + K ) , there exists a δ > 0 such that T + K − μ is a bounded below operator for any μ ∈ B δ 0 ( λ 0 ) . Hence, T + K − μ is a semi-Fredholm operator and ind ( T + K − μ ) ≤ 0 for any μ ∈ B δ 0 ( λ 0 ) . It follows from iso σ u w ( T ) = ∅ that λ ∈ ρ u w ( T + K ) . Thus, λ ∈ σ a ( T + K ) ⧹ σ u w ( T + K ) .

( i v ) ⇔ ( v i ) . It is similar to ( i i i ) ⇔ ( v i ) .□

Remark 2.9

Following the lines of the proof of Theorem 2.8, we get the similar results on the relations between the stability of property ( t ) and the stability of Weyl-type theorems under small compact perturbations.

3 Property ( t ) for functions of operators

Let Hol ( σ ( T ) ) be the set of all functions f which are analytic on some neighborhood of σ ( T ) . For given f ∈ Hol ( σ ( T ) ) , f ( T ) denotes the holomorphic functional calculus of T with respect to f . In this section, we will characterize those operators T satisfying that property ( t ) holds for f ( T ) for all f ∈ Hol ( σ ( T ) ) . Now we first give the following lemma.

Lemma 3.1

Let T ∈ ℬ ( ℋ ) . Then T ∈ ( t ) if and only if σ ( T ) ⧹ σ u w ( T ) ⊆ σ 0 ( T ) and Π 00 ( T ) ⊆ σ 0 ( T ) .

Proof

“ ⇐ .” By [20, Proposition 6.9], it is obvious that

σ 0 ( T ) ⊆ σ ( T ) ⧹ σ u w ( T ) and σ 0 ( T ) ⊆ Π 00 ( T ) .

Hence, we get σ ( T ) ⧹ σ u w ( T ) = σ 0 ( T ) = Π 00 ( T ) . Thus, T ∈ ( t ) .

“ ⇒ .” It follows from T ∈ ( t ) that

σ ( T ) ⧹ σ u w ( T ) ⊆ ρ u w ( T ) ∩ iso σ ( T ) ⊆ σ 0 ( T ) .

Hence, σ ( T ) ⧹ σ u w ( T ) ⊆ σ 0 ( T ) and Π 00 ( T ) ⊆ σ 0 ( T ) .□

Remark 3.2

Let T ∈ ℬ ( ℋ ) . Then T ∈ ( t ) if and only if σ ( T ) ⧹ σ u w ( T ) = σ 0 ( T ) and Π 00 ( T ) = σ 0 ( T ) .

Let

ρ SF + ( T ) = { λ ∈ C : T − λ is an upper semi-Fredholm operator } .

Suppose that T ∈ ( t ) . We claim that ind ( T − λ ) ≥ 0 for any λ ∈ ρ SF + ( T ) . Otherwise, λ ∈ σ ( T ) ⧹ σ u w ( T ) . It follows from Lemma 3.1 that λ ∈ σ 0 ( T ) . Then ind ( T − λ ) = 0 , a contradiction. In the following, we will establish some necessary and sufficient conditions for an operator T satisfying that property ( t ) holds for f ( T ) for all f ∈ Hol ( σ ( T ) ) .

Theorem 3.3

Let T ∈ ℬ ( ℋ ) . Then f ( T ) ∈ ( t ) for all f ∈ Hol ( σ ( T ) ) if and only if the following conditions hold:

T ∈ ( t ) ,
If σ 0 ( T ) ≠ ∅ , then iso σ ( T ) ⊆ σ p ( T ) .

Proof

“ ⇒ .” It suffices to prove ( i i ) . Let λ 1 ∈ σ 0 ( T ) . Assume that iso σ ( T ) ⊈ σ p ( T ) . Then there exists λ 2 ∈ iso σ ( T ) ⧹ σ p ( T ) . Hence, n ( T − λ 2 ) = 0 and λ 2 ∈ σ SF ( T ) .

Set f ( λ ) = ( λ − λ 1 ) ( λ − λ 2 ) . Then we can check that 0 ∈ σ SF ( f ( T ) ) and 0 < n ( f ( T ) ) < ∞ . Since λ 1 , λ 2 ∈ is o σ ( T ) , by [9, Corollary 2.10], we get that 0 ∈ iso σ ( f ( T ) ) . Thus, 0 ∈ Π 00 ( f ( T ) ) . Note that 0 ∈ σ SF ( f ( T ) ) . It implies that 0 ∉ σ 0 ( f ( T ) ) . By Lemma 3.1, we can conclude f ( T ) ∉ ( t ) , a contradiction.

“ ⇐ .” Take any f ∈ Hol ( σ ( T ) ) , then we consider it from the following two steps.

Step 1. σ ( f ( T ) ) ⧹ σ u w ( f ( T ) ) ⊆ Π 00 ( f ( T ) ) .

Let λ 0 ∈ σ ( f ( T ) ) ⧹ σ u w ( f ( T ) ) . Suppose that

f ( T ) − λ 0 = ( T − λ 1 ) k 1 ⋯ ( T − λ n ) k n g ( T ) ,

where λ i ≠ λ j ( i ≠ j ) and g ( T ) is invertible. Since λ 0 ∉ σ u w ( f ( T ) ) , we have

λ i ∈ ρ SF + ( T ) ( 1 ≤ i ≤ n ) and ∑ i = 1 n k i ⋅ ind ( T − λ i ) ≤ 0 .

Note that T ∈ ( t ) . It implies that ind ( T − λ i ) = 0 ( 1 ≤ i ≤ n ) . Then λ i ∈ ρ ( T ) ∪ σ 0 ( T ) . Without loss of generality, we could assume that λ i ∈ σ 0 ( T ) ( 1 ≤ i ≤ n ) . It follows that λ 0 ∈ σ 0 ( f ( T ) ) ⊆ Π 00 ( f ( T ) ) .

Step 2. Π 00 ( f ( T ) ) ⊆ σ ( f ( T ) ) ⧹ σ u w ( f ( T ) ) .

Suppose λ 0 ∈ Π 00 ( f ( T ) ) and f ( T ) − λ 0 has the same decomposition as above. Then λ 0 ∈ iso σ ( f ( T ) ) and 0 < n ( f ( T ) − λ 0 ) < ∞ . It implies that λ i ∈ iso σ ( T ) ∪ ρ ( T ) ( 1 ≤ i ≤ n ) and there exists 1 ≤ i 0 ≤ n such that 0 < n ( T − λ i 0 ) < ∞ . Therefore, λ i 0 ∈ Π 00 ( T ) . It follows from Lemma 3.1 that λ i 0 ∈ σ 0 ( T ) . Hence, σ 0 ( T ) ≠ ∅ . Then by ( i i ) we get that λ i ∈ σ p ( T ) ( 1 ≤ i ≤ n ) . Thus, 0 < n ( T − λ i ) < ∞ and λ i ∈ Π 00 ( T ) ( 1 ≤ i ≤ n ) . Note that T ∈ ( t ) . It implies that T − λ i is a semi-Fredholm operator and ind ( T − λ i ) ≤ 0 for any 1 ≤ i ≤ n . Hence, f ( T ) − λ 0 is a semi-Fredholm operator and ind ( f ( T ) − λ 0 ) = ∑ i = 1 n k i ⋅ ind ( T − λ i ) ≤ 0 . Therefore, λ 0 ∈ σ ( f ( T ) ) ⧹ σ u w ( f ( T ) ) .□

In the following, we give an example.

Example

Let T ∈ ℬ ( ℓ 2 ) be defined by

T ( x 1 , x 2 , x 3 , … ) = ( 0 , 0 , x 3 , x 4 , … ) .

Then σ ( T ) = { 0 , 1 } , σ u w ( T ) = { 1 } . Hence, σ ( T ) \ σ u w ( T ) = { 0 } = Π 00 ( T ) , σ 0 ( T ) = { 0 } and iso σ ( T ) ⊆ σ p ( T ) . By Theorem 3.3, we get that f ( T ) ∈ ( t ) for all f ∈ Hol ( σ ( T ) ) .

From the theorem above, we can easily obtain the following results.

Corollary 3.4

Let σ 0 ( T ) = ∅ . Then f ( T ) ∈ ( t ) for all f ∈ Hol ( σ ( T ) ) if and only if T ∈ ( t ) .

Corollary 3.5

Let T ∈ ℬ ( ℋ ) . Then f ( T ) ∈ ( t ) for all f ∈ Hol ( σ ( T ) ) if and only if p ( T ) ∈ ( t ) for each polynomial p .

Acknowledgements

The authors express their sincere gratitude to the referees for their valuable comments and suggestions.

Funding information: This research was supported by the National Natural Science Foundation of China (No. 11771261), the Fundamental Research Funds for the Central Universities (GK202103007, GK202107014) and the Young Talent fund of University Association for Science and Technology in Shaanxi, China (20210507).
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-09-18

Accepted: 2021-10-11

Published Online: 2021-12-31

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https://doi.org/10.1515/math-2021-0114

Keywords for this article

property (t); compact perturbation; Weyl-type theorem; function of operator

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