A note on property (gaR) and perturbations
We introduce a new property $(gaR)$ extending the property $(R)$ considered by Aiena. We study the property $(gaR)$ in connection with Weyl type theorems and establish sufficient and necessary conditions under which the property $(gaR)$ holds. In addition, we also study the stability of the property...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507604168474624 |
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| author | Chen, A. Shen, J. L. Чен, А. Шен, Ю.Л. |
| author_facet | Chen, A. Shen, J. L. Чен, А. Шен, Ю.Л. |
| author_sort | Chen, A. |
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| datestamp_date | 2019-12-05T09:25:34Z |
| description | We introduce a new property $(gaR)$ extending the property $(R)$ considered by Aiena. We study the property $(gaR)$ in connection with Weyl type theorems and establish sufficient and necessary conditions under which the property $(gaR)$ holds. In addition, we also study the stability of the property $(gaR)$ under perturbations by finite-dimensional operators,
by nilpotent operators, by quasinilpotent operators, and by algebraic operators commuting with $T$. The classes of operators
are considered as illustrating examples.
|
| first_indexed | 2026-03-24T02:11:57Z |
| format | Article |
| fulltext |
UDC 517.9
J. L. Shen (College Comput. and Inform. Technology, Henan Normal Univ., China),
A. Chen (School Math. Sci., Inner Mongolia Univ., China)
A NOTE ON PROPERTY (\bfitg \bfita \bfitR ) AND PERTURBATIONS*
ПРО ВЛАСТИВIСТЬ (\bfitg \bfita \bfitR ) ТА ЗБУРЕННЯ
We introduce a new property (gaR) extending the property (R) considered by Aiena. We study the property (gaR) in
connection with Weyl type theorems and establish sufficient and necessary conditions under which the property (gaR)
holds. In addition, we also study the stability of the property (gaR) under perturbations by finite-dimensional operators,
by nilpotent operators, by quasinilpotent operators, and by algebraic operators commuting with T . The classes of operators
are considered as illustrating examples.
Введено нову властивiсть (gaR), що узагальнює властивiсть (R), яку розглядав Айєна. Властивiсть (gaR) вивча-
ється у зв’язку з теоремами типу Вейля. Встановлено необхiднi та достатнi умови для того, щоб властивiсть (gaR)
виконувалася. Крiм того, вивчається стабiльнiсть властивостi (gaR) при збуреннях скiнченновимiрними, нiльпо-
тентними, квазiнiльпотентними та алгебраїчними операторами, що комутують з T . Цi класи операторiв розглянуто
як iлюстративнi приклади.
1. Introduction. Let X be an infinite-dimensional complex Banach space and L(X) be the algebra of
all bounded linear operators on X. For T \in L(X), we denote the null space, the range, the spectrum,
the approximate point spectrum, the surjective spectrum, the isolated points of the spectrum and
the isolated points of the approximate point spectrum of T by N(T ), R(T ), \sigma (T ), \sigma a(T ), \sigma s(T ),
\mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) and iso\sigma a(T ), respectively.
If R(T ) is closed and \alpha (T ) = dimN(T ) < \infty (resp. \beta (T ) = dimX/R(T ) < \infty ), then T
is called an upper (resp. a lower) semi-Fredholm operator. In the sequel \Phi +(X) (resp. \Phi - (X))
is written for the set of all upper (resp. lower) semi-Fredholm operators. The class of all semi-
Fredholm operators is defined by \Phi \pm (X) = \Phi +(X) \cup \Phi - (X), in this case the index of T is given
by i(T ) = \alpha (T ) - \beta (T ). Denote \Phi (X) = \Phi +(X) \cap \Phi - (X) the set of all Fredholm operators.
Define W+(X) = \{ T \in \Phi +(X) : i(T ) \leq 0\} the set of all upper semi-Weyl operators, while
W - (X) = \{ T \in \Phi - (X) : i(T ) \geq 0\} the set of all lower semi-Weyl operators. The set of all Weyl
operators is defined by W (X) = W+(X) \cap W - (X) = \{ T \in \Phi (X) : i(T ) = 0\} . The classes of
operators defined above generate the following spectrum: the Weyl spectrum of T is defined by
\sigma w(T ) = \{ \lambda \in \BbbC : T - \lambda I /\in W (X)\} , while the upper semi-Weyl spectrum and the lower semi-Weyl
spectrum of T are defined by \sigma uw(T ) = \{ \lambda \in \BbbC : T - \lambda I /\in W+(X)\} and \sigma lw(T ) = \{ \lambda \in \BbbC :
T - \lambda I /\in W - (X)\} , respectively.
Let p = p(T ) be the ascent of T, i.e., the smallest nonnegative integer p such that N(T p) =
= N(T p+1). If such integer does not exist we put p(T ) = \infty . Analogously, let q = q(T ) be the
descent of T, i.e., the smallest nonnegative integer q such that R(T q) = R(T q+1), and if such integer
does not exist we put q(T ) = \infty [15] (Proposition 38.3). Moreover, 0 < p(\lambda I - T ) = q(\lambda I - T ) < \infty
precisely when \lambda is a pole of the resolvent of T, see Proposition 50.2 of Heuser [15]. The class of all
upper semi-Browder operators is defined by B+(X) = \{ T \in \Phi +(X) : p(T ) < \infty \} and the class of all
Browder operators is defined by B(X) = \{ T \in \Phi (X) : p(T ) = q(T ) < \infty \} . The Browder spectrum
* This work is supported by the National Natural Science Foundation of China (№ 11601130) and the Natural Science
Foundation of the Department of Education of Henan Province (16A110033,17A110005).
c\bigcirc J. L. SHEN, A. CHEN, 2017
974 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
A NOTE ON PROPERTY (gaR) AND PERTURBATIONS 975
of T is defined by \sigma b(T ) = \{ \lambda \in \BbbC : \lambda I - T /\in B(X)\} and the upper semi-Browder spectrum is
defined by \sigma ub(T ) = \{ \lambda \in \BbbC : \lambda I - T /\in B+(X)\} , respectively. Let \pi a
00(T ) = \{ \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) :
0 < \alpha (T - \lambda ) < \infty \} and p00(T ) = \sigma (T )\setminus \sigma b(T ).
For T \in L(X), an operator T is called B-Fredholm if there exists n \in \BbbN such that R(Tn) is
closed and the induced operator
T[n] : R(Tn) \ni x \rightarrow Tx \in R(Tn)
is Fredholm, i.e., R(T[n]) = R(Tn+1) is closed, \alpha (T[n]) = \mathrm{d}\mathrm{i}\mathrm{m}N(T[n]) < \infty and \beta (T[n]) =
= \mathrm{d}\mathrm{i}\mathrm{m}R(Tn)/R(T[n]) < \infty . Similarly, a B-Fredholm operator T is called B-Weyl if i(T[n]) = 0.
The B-Weyl spectrum \sigma BW (T ) is defined by
\sigma BW (T ) =
\bigl\{
\lambda \in \BbbC : T - \lambda is not B-Weyl
\bigr\}
.
We say that generalized Weyl’s theorem holds for T if
\sigma (T )\setminus \sigma BW (T ) = E(T ),
where E(T ) = \{ \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) : 0 < \alpha (T - \lambda )\} .
For T \in L(X), an operator T is called an upper semi-B-Weyl operator if there exists n \in \BbbN
such that R(Tn) is closed and the induced operator T[n] : R(Tn) \ni x \rightarrow Tx \in R(Tn) is upper
semi-Fredholm (i.e., R(T[n]) = R(Tn+1) is closed, \mathrm{d}\mathrm{i}\mathrm{m}N(T[n]) = \mathrm{d}\mathrm{i}\mathrm{m}N(T ) \cap R(Tn) < \infty ) and
i(T[n]) \leq 0 [10]. We define
\sigma SBF -
+
(T ) =
\bigl\{
\lambda \in \BbbC : T - \lambda is not upper semi-B-Weyl
\bigr\}
.
We say that generalized a-Weyl’s theorem holds for T if
\sigma a(T )\setminus \sigma SBF -
+
(T ) = Ea(T ),
where Ea(T ) = \{ \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) : 0 < \alpha (T - \lambda )\} .
If p(T ) < \infty and R(T p(T )+1) is closed, then T is called left Drazin invertible. If p(T ) =
= q(T ) < \infty , then T is called Drazin invertible. The Drazin spectrum \sigma D(T ) and left Drazin
spectrum \sigma LD(T ) of an operator T are defined by
\sigma D(T ) =
\bigl\{
\lambda \in \BbbC : T - \lambda I is not Drazin invertible
\bigr\}
and
\sigma LD(T ) =
\bigl\{
\lambda \in \BbbC : T - \lambda I is not left Drazin invertible
\bigr\}
.
Let
\Pi (T ) =
\bigl\{
\lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) : T - \lambda is Drazin invertible
\bigr\}
denote the set of all poles of T and
p00(T ) =
\bigl\{
\lambda \in \Pi (T ) : \alpha (T - \lambda ) < \infty
\bigr\}
be the set of all poles of T of finite rank,
\Pi a(T ) =
\bigl\{
\lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) : T - \lambda is left Drazin invertible
\bigr\}
denotes the set of all left poles of T and
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
976 J. L. SHEN, A. CHEN
pa00(T ) =
\bigl\{
\lambda \in \Pi a(T ) : \alpha (T - \lambda ) < \infty
\bigr\}
denotes the set of all left poles of T of finite rank. According to [15], the space R((T - \lambda I)p(T - \lambda I)+1)
is closed for each \lambda \in \Pi (T ). Hence we have \Pi (T ) \subseteq \Pi a(T ) and p00(T ) \subseteq pa00(T ). We say that
generalized a-Browder’s theorem holds for T if
\sigma a(T )\setminus \sigma SBF -
+
(T ) = \Pi a(T ).
According to [9] we say that generalized Browder’s theorem holds for T if
\sigma (T )\setminus \sigma BW (T ) = \Pi (T ).
Recall [12] that property (gaw) is said to hold for T if \sigma (T )\setminus \sigma BW (T ) = Ea(T ). Recall [5] that
property (R) holds for T if pa00(T ) = \pi 00(T ), where \pi 00(T ) = \{ \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) : 0 < \alpha (T - \lambda ) < \infty \} .
According to [18], an operator T is said to satisfy property (aR) if p00(T ) = \pi a
00(T ).
The single valued extension property plays an important role in local spectral theory, see the
recent monograph of Laursen and Neumamn [16] and Aiena [1]. In this article we shall consider the
following local version of this property.
Let T \in L(X). The operator T is said to have the single valued extension property at \lambda 0 \in
\in \BbbC (abbrev. SVEP at \lambda 0), the only analytic function f : D \rightarrow X which satisfies the equation
(\lambda I - T )f(\lambda ) = 0 for all \lambda \in D is the function f \equiv 0. An operator T is said to have SVEP if T
has SVEP at every point \lambda \in \BbbC .
It is known that both Browder’s (equivalently, generalized Browder’s) theorem and a-Browder’s
(equivalently, generalized a-Browder’s) theorem hold for T if T or T \ast has SVEP. Precisely, we
have that a-Browder’s (equivalently, generalized a-Browder’s) theorem holds for T if and only if T
has SVEP at every \lambda /\in \sigma uw(T ), and dually, a-Browder’s (equivalently, generalized a-Browder’s)
theorem holds for T \ast if and only if T \ast has SVEP at every \lambda /\in \sigma lw(T ), see [3, 8].
From the identity theorem for analytic function it easily follows that T \in L(X), as well as its
dual T \ast , has SVEP at every point of the boundary of the spectrum \sigma (T ) = \sigma (T \ast ), so both T and
T \ast have SVEP at every isolated point of the spectrum.
According to [3] (Theorem 1.2), if T \in L(X) and suppose that \lambda 0I - T \in \Phi \pm (X). Then the
following statements are equivalent:
(i) T has SVEP at \lambda 0;
(ii) p(T - \lambda 0I) < \infty ;
(iii) \sigma a(T ) doesn’t cluster at \lambda 0.
Dually, if \lambda 0I - T \in \Phi \pm (X), then the following statements are equivalent:
(iv) T \ast has SVEP at \lambda 0;
(v) q(T - \lambda 0I) < \infty ;
(vi) \sigma s(T ) doesn’t cluster at \lambda 0.
A bounded operator T is said to be polaroid if every isolated point of \sigma (T ) is a pole of the
resolvent of T. T is said to be hereditarily polaroid if every part of T is polaroid. T is said to
be a-polaroid if every isolated point of \sigma a(T ) is a pole of the resolvent of T. And T is said to be
a-isoloid if every isolated point of \sigma a(T ) is an eigenvalue of T.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
A NOTE ON PROPERTY (gaR) AND PERTURBATIONS 977
Let T \in L(X) and d \in \BbbN . Then T has a uniform descent for n \geq d if R(T ) + N(Tn) =
= R(T ) + N(T d) for n \geq d. If in addition, R(T ) + N(T d) is closed, then T is said to have a
topological uniform descent for n \geq d, see [14].
If \lambda \in \Pi a(T ) or T - \lambda I is a semi-B-Fredholm operator, then T - \lambda is an operator of topological
uniform descent.
In Section 2, we introduce and study the new property (gaR) in connection with Weyl type
theorems. We prove that an operator T possessing property (gaR) possesses property (aR), but the
converse is not true in general as shown by Example 2.2. We prove also that if T \ast has SVEP at every
\lambda /\in \sigma uw(T ), then property (gaR), property (gaw), generalized Weyl’s theorem and generalized a-
Weyl’s theorem are equivalent. In Section 3, in Theorem 3.3 we prove that if T \in L(X) and M is a
nilpotent operator commuting with T, then T possesses property (gaR) if and only if T+M possesses
property (gaR). And we provide a condition under which the new property (gaR) is preserved under
commuting finite-dimensional operator, we prove in Theorem 3.2 that if \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) = \phi and K is a
finite-dimensional operator commuting with T, then T +K satisfies property (gaR). In the last part,
as a conclusion, we give a diagram summarizing the different relations between Weyl type theorems,
extending a similar diagram given [9].
2. Property (\bfitg \bfita \bfitR ).
Definition 2.1. An operator T is said to satisfy property (gaR) if \Pi (T ) = Ea(T ).
Lemma 2.1 [4, 9]. Suppose that T \in L(X). Then we have:
(i) T satisfies generalized Weyl’s theorem if and only if generalized Browder’s theorem holds for
T and \Pi (T ) = E(T ).
(ii) T satisfies generalized a-Weyl’s theorem if and only if generalized a-Browder’s theorem holds
for T and \Pi a(T ) = Ea(T ).
Lemma 2.2 [14]. Suppose that T is a bounded linear operator and that \lambda belongs to the
boundary of the spectrum of T. If T - \lambda has topological uniform descent, then \lambda is a pole of T.
Theorem 2.1. Let T satisfy property (gaR). Then Ea(T ) = \Pi a(T ) = E(T ) = \Pi (T ).
Proof. Observe that \Pi (T ) \subseteq E(T ) \subseteq Ea(T ) holds for every operator T. As T satisfies prop-
erty (gaR), Ea(T ) = \Pi (T ), and hence \Pi (T ) = E(T ) = Ea(T ). As \Pi (T ) \subseteq \Pi a(T ) \subseteq Ea(T )
holds for every operator T and Ea(T ) = \Pi (T ), then \Pi (T ) = \Pi a(T ) = Ea(T ), i.e., Ea(T ) =
= \Pi a(T ) = E(T ) = \Pi (T ).
The following example shows neither of the two equalities Ea(T ) = \Pi a(T ), E(T ) = \Pi (T ) can
imply \Pi (T ) = Ea(T ).
Example 2.1. Let R : l2(\BbbN ) \rightarrow l2(\BbbN ) be the unilateral right shift operator defined by
R(x1, x2, . . .) = (0, x1, x2, . . .) for all x = (x1, x2, . . .) \in l2(\BbbN ) and Q(x1, x2, . . .) =
\biggl(
1
3
x1, x2,
x3, . . .
\biggr)
for all x = (x1, x2, . . .) \in l2(\BbbN ). Define T = R \oplus Q. Then \sigma (T ) = \sigma (T \ast ) = D,
\sigma a(T ) = \partial D \cup
\biggl\{
1
3
\biggr\}
and \sigma uw(T ) = \partial D, where D denotes the closed unit disc and \partial D denotes
the unit circle, and hence \Pi (T ) = E(T ) = \phi . We show that T does not satisfy property (gaR).
Since T has SVEP at the points of \partial D and T has SVEP at
1
3
. Hence T has SVEP and a-Browder’s
theorem holds for T, i.e., \sigma uw(T ) = \sigma ub(T ) = \partial D. It follows that pa00(T ) = \sigma a(T )\setminus \sigma ub(T ) =
\biggl\{
1
3
\biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
978 J. L. SHEN, A. CHEN
Observe that the operator T satisfies the equality \Pi a(T ) = Ea(T ). Indeed,
1
3
is an isolated point
of \sigma a(T ), and hence Ea(T ) =
\biggl\{
1
3
\biggr\}
= \Pi a(T ). While T does not satisfy property (gaR) since
Ea(T ) =
\biggl\{
1
3
\biggr\}
\not = \Pi (T ).
As noted in Example 2.1 the condition \Pi a(T ) = Ea(T ) is strictly weaker than property (gaR).
However, we have the following theorem.
Theorem 2.2. T satisfies property (gaR) if and only if the following two conditions hold:
(i) Ea(T ) \subseteq \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ),
(ii) \Pi a(T ) = Ea(T ).
Proof. If T satisfies property (gaR), then Ea(T ) = \Pi (T ) \subseteq \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) and by Theorem 2.1 we
have \Pi a(T ) = Ea(T ). Conversely, suppose that both (i) and (ii) hold. As \Pi (T ) \subseteq Ea(T ) holds for
every operator T. To show the opposite inclusion, let \lambda \in Ea(T ). Then \lambda \in Ea(T ) = \Pi a(T ), and
hence T - \lambda has topological uniform descent, since Ea(T ) \subseteq \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ), then \lambda is a pole of T, thus
\lambda \in \Pi (T ). Therefore \Pi (T ) = Ea(T ).
Theorem 2.3. Let T satisfy property (gaR). Then property (aR) holds for T.
Proof. Since p00(T ) \subseteq \pi a
00(T ) holds for every operator T. To show the opposite inclusion,
let \lambda \in \pi a
00(T ). Then \lambda \in Ea(T ) and \alpha (T - \lambda ) < \infty . Since T satisfies property (gaR), then
Ea(T ) = \Pi (T ). And hence \lambda \in \Pi (T ) and \alpha (T - \lambda ) < \infty , i.e., \lambda \in p00(T ).
The following example shows that property (aR) is weaker than property (gaR).
Example 2.2. Let R : l2(\BbbN ) \rightarrow l2(\BbbN ) be the unilateral right shift operator defined by R(x1,
x2, . . .) = (0, x1, x2, . . .) for all x = (x1, x2, . . .) \in l2(\BbbN ) and Q(x1, x2, . . .) =
\biggl(
1
2
x1,
1
2
x2, . . .
\biggr)
for all x = (x1, x2, . . .) \in l2(\BbbN ). Define T := R \oplus Q. Then \sigma (T ) = D, \sigma a(T ) = \partial D \cup
\biggl\{
1
2
\biggr\}
. It
follows that \Pi (T ) = \phi , Ea(T ) =
\biggl\{
1
2
\biggr\}
, then T does not satisfy property (gaR). But T satisfies
property (aR) since p00(T ) = \pi a
00(T ) = \phi .
In the following theorem we give a condition for the equivalence of property (gaR) and prop-
erty (gaw).
Theorem 2.4. T satisfies property (gaw) if and only if generalized Browder’s theorem holds for
T and T has property (gaR).
Proof. If generalized Browder’s theorem holds for T and T has property (gaR), then
\sigma (T )\setminus \sigma BW (T ) = \Pi (T ) and \Pi (T ) = Ea(T ), and hence \sigma (T )\setminus \sigma BW (T ) = Ea(T ).
Conversely, it is easy to prove that property (gaw) implies generalized Browder’s theorem by
[12] (Corollary 2.7, Theorem 3.5), then \sigma (T )\setminus \sigma BW (T ) = \Pi (T ). Since T satisfies property (gaw),
then \sigma (T )\setminus \sigma BW (T ) = Ea(T ), hence \Pi (T ) = Ea(T ), i.e., T has property (gaR).
The following example shows that property (gaR) is weaker than property (gaw).
Example 2.3. Let R : l2(\BbbN ) \rightarrow l2(\BbbN ) be the unilateral right shift operator defined by R(x1,
x2, . . .) = (0, x1, x2, . . .) for all x = (x1, x2, . . .) \in l2(\BbbN ) and L : l2(\BbbN ) \rightarrow l2(\BbbN ) be the unilateral
left shift operator defined by L(x1, x2, . . .) = (x2, x3, . . .) for all x = (x1, x2, . . .) \in l2(\BbbN ). Define
T := R \oplus L. Then \sigma (T ) = \sigma a(T ) = D. It follows that \Pi (T ) = Ea(T ) = \phi , then T satisfies
property (gaR). While T doesn’t satisfy property (gaw), since 0 \in \sigma (T )\setminus \sigma BW (T ) \not = \phi = Ea(T ).
The following example shows property (gaR) for an operator is not transmitted to the dual T \ast .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
A NOTE ON PROPERTY (gaR) AND PERTURBATIONS 979
Example 2.4. Let L : l2(\BbbN ) \rightarrow l2(\BbbN ) be the unilateral left shift operator defined by
L(x1, x2, . . .) = (x2, x3, . . .) for all x = (x1, x2, . . .) \in l2(\BbbN )
and
Q(x1, x2, . . .) = (0, x2, x3, . . .) for all x = (x1, x2, . . .) \in l2(\BbbN ).
Define T := L \oplus Q. Then \sigma (T ) = \sigma (T \ast ) = \sigma a(T ) = D and \sigma a(T
\ast ) = \partial D \cup \{ 0\} . It follows that
\Pi (T ) = Ea(T ) = \phi , then T satisfies property (gaR). While T \ast doesn’t satisfy property (gaR),
since 0 \in Ea(T
\ast ) \not = \phi = \Pi (T \ast ).
The following example shows that generalized a-Weyl’s theorem does not entail property (gaR).
Example 2.5. Let T be defined as in Example 2.1. As already observed, T does not satisfy
property (gaR), while T has SVEP and hence generalized a-Browder’s theorem holds for T. Since
\Pi a(T ) = Ea(T ), by part (ii) of Lemma 2.1, then generalized a-Weyl’s theorem holds for T.
The following example shows that property (gaR) does not entail generalized a-Weyl’s theorem.
Example 2.6. Let T be defined as in Example 2.3. We have \alpha (T ) = \beta (T ) = 1 and p(T ) = \infty .
Therefore, 0 /\in \sigma w(T ), while 0 \in \sigma b(T ), so Browder’s theorem (and hence generalized a-Weyl’s
theorem) does not hold for T. On the other hand, since \sigma (T ) = \sigma a(T ) = D, we have \Pi (T ) =
= Ea(T ) = \phi , and hence property (gaR) holds for T.
Theorem 2.5. Let T satisfy both generalized a-Browder’s theorem and property (gaR). Then T
satisfies generalized a-Weyl’s theorem and \sigma a(T )\setminus \sigma SBF -
+
(T ) = \Pi (T ).
Proof. If T satisfies generalized a-Browder’s theorem and property (gaR), then Ea(T ) = \Pi a(T )
by Theorem 2.1. Therefore generalized a-Weyl’s theorem holds for T by (ii) of Lemma 2.1, i.e.,
\sigma a(T )\setminus \sigma SBF -
+
(T ) = Ea(T ). Property (gaR) then implies \sigma a(T )\setminus \sigma SBF -
+
(T ) = \Pi (T ).
In [11] an operator T is said to have property (gb) if \sigma a(T )\setminus \sigma SBF -
+
(T ) = \Pi (T ).
The following example shows that property (gaR) does not entail property (gb).
Example 2.7. Let T be defined as in Example 2.3. Then T satisfies property (gaR), while
property (gb) does not hold for T, since 0 \in \sigma a(T )\setminus \sigma SBF -
+
(T ) and \Pi (T ) = \phi . This example also
shows that without the assumption that T satisfies generalized a-Browder’s theorem, the result of
Theorem 2.5 does not hold.
The following example shows that property (gb) does not entail property (gaR).
Example 2.8. Let Q(x1, x2, . . .) =
\biggl(
x2
22
,
x3
23
, . . .
\biggr)
for all x = (x1, x2, . . .) \in l2(\BbbN ). Clearly,
Q is quasinilpotent and hence \sigma (Q) = \sigma a(Q) = \sigma SBF -
+
(Q) = \{ 0\} . We have \alpha (Q) = 1, then
0 \in Ea(Q), \Pi (Q) = \phi , it then follows that Q does not satisfy property (gaR). On the other hand,
since \sigma a(Q)\setminus \sigma SBF -
+
(Q) = \Pi (Q) = \phi , Q has property (gb).
The next result shows that the equivalence of property (gaR), property (gaw), generalized Weyl’s
theorem and generalized a-Weyl’s theorem is true whenever we assume that T \ast has SVEP at the
points \lambda /\in \sigma uw(T ).
Theorem 2.6. Let T \ast have SVEP at every \lambda /\in \sigma uw(T ). Then the following statements are
equivalent:
(i) E(T ) = \Pi (T );
(ii) Ea(T ) = \Pi a(T );
(iii) Ea(T ) = \Pi (T ).
Consequently, property (gaR), property (gaw), generalized Weyl’s theorem and generalized
a-Weyl’s theorem are equivalent for T.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
980 J. L. SHEN, A. CHEN
Proof. It is easy to see that \sigma (T ) = \sigma a(T ), then we have E(T ) = Ea(T ). The following
we would show \Pi a(T ) = \Pi (T ), observe first that \Pi (T ) \subseteq \Pi a(T ) holds for every operator T.
To show the opposite inclusion, let \lambda \in \Pi a(T ). Then T - \lambda has topological uniform descent and
\lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) = \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ), it follows from Lemma 2.2 that \lambda is a pole of T, i.e., \lambda \in \Pi (T ). From
which the equivalence of (i), (ii) and (iii) easily be obtained. To show the last statement observe that
the SVEP of T \ast at the points \lambda /\in \sigma uw(T ) entails that generalized a-Browder’s theorem (and hence
generalized Browder’s theorem) holds for T, see [3] (Theorem 2.3). By Lemma 2.1 and Theorem 2.4,
then property (gaR), property (gaw), generalized Weyl’s theorem and generalized a-Weyl’s theorem
are equivalent for T.
Dually, we have the following theorem.
Theorem 2.7. Let T have SVEP at every \lambda /\in \sigma lw(T ). Then the following statements are
equivalent:
(i) E(T \ast ) = \Pi (T \ast );
(ii) Ea(T
\ast ) = \Pi a(T
\ast );
(iii) Ea(T
\ast ) = \Pi (T \ast ).
Consequently, property (gaR), property (gaw), generalized Weyl’s theorem and generalized
a-Weyl’s theorem are equivalent for T \ast .
Proof. It is clear from Theorem 2.6.
Theorem 2.8. Let T be a-polaroid. Then T satisfies property (gaR).
Proof. Since \Pi (T ) \subseteq Ea(T ) holds for every operator T. To show the opposite inclusion, let
\lambda \in Ea(T ). Then \lambda is an isolated point of \sigma a(T ). Since T is a-polaroid, \lambda is a pole of the resolvent
of T, \lambda \in \Pi (T ), i.e., T satisfies property (gaR).
Corollary 2.1 [18]. Let T be a-polaroid. Then T satisfies property (aR).
The next example shows that under a weaker condition of being polaroid the result of Theorem 2.8
does not hold.
Example 2.9. Let R : l2(\BbbN ) \rightarrow l2(\BbbN ) be the unilateral right shift operator defined by R(x1,
x2, . . .) = (0, x1, x2, . . .) for all x = (x1, x2, . . .) \in l2(\BbbN ) and Q(x1, x2, . . .) =
\biggl(
x2
22
,
x3
23
, . . .
\biggr)
for
all x = (x1, x2, . . .) \in l2(\BbbN ). Define T := R \oplus Q. Then \sigma (T ) = D, hence \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) = \Pi (T ) = \phi .
Therefore, T is polaroid. Moreover, \sigma a(T ) = \partial D \cup \{ 0\} , \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) = \{ 0\} , 0 < \alpha (T ) = 1 < \infty
implies 0 \in Ea(T ), and hence Ea(T ) \not = \Pi (T ), thus T doesn’t satisfy property (gaR).
From the proof of Theorem 2.6 we know that if T \ast has SVEP, then \sigma (T ) = \sigma a(T ). Therefore if
T \ast has SVEP, then T is a-polaroid \leftrightarrow T is polaroid.
Corollary 2.2. Let T be polaroid and T \ast have SVEP. Then T satisfies property (gaR).
Note that the result of Corollary 2.2 does not hold if we replace the SVEP for T \ast by the SVEP
for T.
Example 2.10. Let T be defined as in Example 2.9. Then T has SVEP and is polaroid, while
T does not satisfy property (gaR).
3. Property (\bfitg \bfita \bfitR ) under perturbations.
Theorem 3.1 [13]. If T is a-isoloid and satisfies a-Weyl’s theorem, then T+K satisfies a-Weyl’s
theorem for every finite-dimensional operator K commuting with T.
The following example shows that an analogous result of Theorem 3.1 does not hold for prop-
erty (gaR), even with the class of a-isoloid operators.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
A NOTE ON PROPERTY (gaR) AND PERTURBATIONS 981
Example 3.1. Let T : l2(\BbbN ) \rightarrow l2(\BbbN ) be defined by
T (x1, x2, . . .) = (2x1, 2x2, 0, x3, x4, . . .) for all x = (x1, x2, . . .) \in l2(\BbbN )
and
K(x1, x2, . . .) = ( - 2x1, - 2x2, 0, 0, 0, . . .) for all x = (x1, x2, . . .) \in l2(\BbbN ).
Then \sigma (T ) = D \cup \{ 2\} and \sigma a(T ) = \partial D \cup \{ 2\} , it follows that Ea(T ) = \Pi (T ) = \{ 2\} . Therefore,
T is a-isoloid operator, KT = TK and satisfies property (gaR). While \sigma (T + K) = D and
\sigma a(T + K) = \partial D \cup \{ 0\} , it follows that \Pi (T + K) = \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T + K) = \phi \not = \{ 0\} = Ea(T + K).
Therefore, T +K does not satisfy property (gaR).
Theorem 3.2. Let T \in L(X) and \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) = \phi . If K is a finite-dimensional operator com-
muting with T, then T +K satisfies property (gaR).
Proof. Since \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) = \phi and K is a finite-dimensional operator commuting with T, by the
proof of [2] (Theorem 2.8), \sigma a(T ) = \sigma a(T +K), then \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T +K) = \phi . Since \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T +K) \subseteq
\subseteq \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T +K), \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T +K) = \phi . It follows that \Pi (T +K) = Ea(T +K) = \phi , i.e., T +K
satisfies property (gaR).
Corollary 3.1 [18]. Let T \in L(X) and \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) = \phi . If K is a finite-dimensional operator
commuting with T, then T +K satisfies property (aR).
The next result shows that property (gaR) for T is transmitted to T +M, when M is a nilpotent
operator which commutes with T. Recall first that the equality \sigma a(T ) = \sigma a(T +Q) holds for every
quasinilpotent operator Q which commutes with T.
Theorem 3.3. Let T \in L(X) and M \in L(X) be a nilpotent operator which commutes with T.
Then we have:
(i) Ea(T +M) = Ea(T ).
(ii) T satisfies property (gaR) if and only if T +M satisfies property (gaR).
(iii) If T is a-polaroid, then T +M satisfies property (gaR).
Proof. (i) Let \lambda \in Ea(T + M). We can assume \lambda = 0. Clearly, 0 \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T + M) =
= \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ). Let p \in \BbbN be such that Mp = 0. If x \in N(T +M), then T px = ( - 1)pMpx = 0, thus
N(T +M) \subseteq N(T p), since by assumption \alpha (T +M) > 0, it then follows that \alpha (T p) > 0 and this
obviously implies that \alpha (T ) > 0. Therefore, 0 \in Ea(T ), and consequently Ea(T +M) \subseteq Ea(T ).
Ea(T ) \subseteq Ea(T +M) follows by symmetry.
(ii) Suppose that T has property (gaR). Then Ea(T +M) = Ea(T ) = \sigma (T )\setminus \sigma D(T ) = \sigma (T +
+M)\setminus \sigma D(T +M) = \Pi (T +M), therefore T +M has property (gaR). The converse follows by
symmetry.
(iii) Obviously, by part (ii), since T satisfies property (gaR) by Theorem 2.8.
This example shows that the commutativity hypothesis in (ii) of Theorem 3.3 is essential.
Example 3.2. Let Q : l2(\BbbN ) \rightarrow l2(\BbbN ) be defined by
Q(x1, x2, . . .) =
\Bigl(
0, 0,
x1
2
,
x2
22
,
x3
23
, . . .
\Bigr)
for all x = (x1, x2, . . .) \in l2(\BbbN )
and
M(x1, x2, . . .) =
\Bigl(
0, 0, - x1
2
, 0, 0, . . .
\Bigr)
for all x = (x1, x2, . . .) \in l2(\BbbN ).
Clearly M is a nilpotent operator and \Pi (Q) = Ea(Q) = \phi , i.e., Q satisfies property (gaR). While
\Pi (Q +M) = \phi and Ea(Q +M) = \{ 0\} , it follows that \Pi (Q +M) \not = Ea(Q +M), i.e., Q +M
does not satisfy property (gaR).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
982 J. L. SHEN, A. CHEN
The previous theorem does not extend to commuting quasinilpotent operators as shown by the
following example.
Example 3.3. Let Q : l2(\BbbN ) \rightarrow l2(\BbbN ) be defined by Q(x1, x2, . . .) =
\biggl(
x2
22
,
x3
23
,
x4
24
, . . .
\biggr)
for all
x = (x1, x2, . . .) \in l2(\BbbN ) and T = 0. Clearly T satisfies property (gaR). While Q is quasinilpotent
and TQ = QT, so \sigma (Q) = \sigma D(Q) = \{ 0\} and hence \{ 0\} = Ea(Q) \not = \sigma (Q)\setminus \sigma D(Q) = \Pi (Q) = \phi ,
i.e., T +Q = Q does not satisfy property (gaR).
Theorem 3.4. Let T be a-polaroid and finite-isoloid and Q be a quasinilpotent operator which
commutes with T. Then T +Q has property (gaR).
Proof. Clearly by the proof of [2] (Theorem 2.13).
Recall that a bounded operator T is said to be algebraic if there exists a nonconstant polynomial
h such that h(T ) = 0.
Theorem 3.5. Let T \in L(X) and K \in L(X) be an algebraic operator which commutes with T :
(i) If T is hereditarily polaroid and has SVEP, then T \ast +K\ast satisfies property (gaR).
(ii) If T \ast is hereditarily polaroid and has SVEP, then T+K satisfies property (gaR).
Proof. Since T \ast + K\ast is a-polaroid by the proof of [2] (Theorem 2.15), property (gaR) for
T \ast +K\ast follows from Theorem 2.8.
(ii) The proof is similar to (i).
4. Conclusion. In the last part, we give a summary of the known Weyl type theorems as in [9],
including the properties introduced in [5 – 7, 11, 12, 17, 18], and in this paper. We use the abbreviations
gW ; W ; (gw); (w); (gaw); (aw); (gR); (R); (gaR); (aR); (gS) and (S) to signify that an
operator T \in L(X) obeys generalized Weyl’s theorem, Weyl’s theorem, property (gw), property (w),
property (gaw), property (aw), property (gR), property (R), property (gaR), property (aR),
property (gS) and property (S). Similarly, the abbreviations gB; B; (gb); (b); (gab) and (ab) have
analogous meaning with respect to Browder’s theorem.
The following table summarizes the meaning of various theorems and properties.
gW \sigma (T )\setminus \sigma BW (T ) = E(T ) (aR) p00(T ) = \pi a
00(T )
W \sigma (T )\setminus \sigma w(T ) = \pi 00(T ) gB \sigma (T )\setminus \sigma BW (T ) = \Pi (T )
(gw) \sigma a(T )\setminus \sigma SBF -
+
(T ) = E(T ) B \sigma (T )\setminus \sigma w(T ) = p00(T )
(w) \sigma a(T )\setminus \sigma uw(T ) = \pi 00(T ) (gb) \sigma a(T )\setminus \sigma SBF -
+
(T ) = \Pi (T )
(gaw) \sigma (T )\setminus \sigma BW (T ) = Ea(T ) (b) \sigma a(T )\setminus \sigma uw(T ) = p00(T )
(aw) \sigma (T )\setminus \sigma w(T ) = \pi a
00(T ) (gab) \sigma (T )\setminus \sigma BW (T ) = \Pi a(T )
(gR) \Pi a(T ) = E(T ) (ab) \sigma (T )\setminus \sigma w(T ) = pa00(T )
(R) pa00(T ) = \pi 00(T ) (gS) \Pi (T ) = E(T )
(gaR) \Pi (T ) = Ea(T ) (S) \pi 00(T ) = p00(T )
In the following diagram, which extends the similar diagram presented in [9], arrows signify impli-
cations between various Weyl type theorems, Browder type theorems, property (gw), property (gaw),
property (gR), property (gaR), property (gS), property (w), property (aw), property (R), prop-
erty (aR) and property (S). The numbers near the arrows are references to the results in the present
paper (numbers without brackets) or to the bibliography therein (the numbers in square brackets).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
A NOTE ON PROPERTY (gaR) AND PERTURBATIONS 983
gB
B (ab)
(gab) (gb)
(b)
(gw)
(w)
(gR)
(gaR)
(gaw)
(R)
(gS)
(aR) (R)
(aw)
(S)
W
gW
[12]
[12]
[12]
[12] [11] [7]
[6]
[7]
[17]
2.1
2.4
[17]
[18]
[9]
[12]
[11]
[11]
[7]
[6]
[17]
2.3 [18]
[12]
[17]
References
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P. 491 – 508.
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P. 1899 – 1906.
Received 23.09.13,
after revision — 25.02.17
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
|
| id | umjimathkievua-article-1750 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:11:57Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f6/50391cad14339543298163adf02fc0f6.pdf |
| spelling | umjimathkievua-article-17502019-12-05T09:25:34Z A note on property (gaR) and perturbations Про властивiсть (\bfit g \bfit a \bfit R ) та збурення Chen, A. Shen, J. L. Чен, А. Шен, Ю.Л. We introduce a new property $(gaR)$ extending the property $(R)$ considered by Aiena. We study the property $(gaR)$ in connection with Weyl type theorems and establish sufficient and necessary conditions under which the property $(gaR)$ holds. In addition, we also study the stability of the property $(gaR)$ under perturbations by finite-dimensional operators, by nilpotent operators, by quasinilpotent operators, and by algebraic operators commuting with $T$. The classes of operators are considered as illustrating examples. Введено нову властивiсть $(gaR)$, що узагальнює властивiсть $(R)$, яку розглядав Айєна. Властивiсть $(gaR)$ вивчається у зв’язку з теоремами типу Вейля. Встановлено необхiднi та достатнi умови для того, щоб властивiсть $(gaR)$ виконувалася. Крiм того, вивчається стабiльнiсть властивостi (gaR) при збуреннях скiнченновимiрними, нiльпотентними, квазiнiльпотентними та алгебраїчними операторами, що комутують з $T$. Цi класи операторiв розглянуто як iлюстративнi приклади. Institute of Mathematics, NAS of Ukraine 2017-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1750 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 7 (2017); 974-983 Український математичний журнал; Том 69 № 7 (2017); 974-983 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1750/732 Copyright (c) 2017 Chen A.; Shen J. L. |
| spellingShingle | Chen, A. Shen, J. L. Чен, А. Шен, Ю.Л. A note on property (gaR) and perturbations |
| title | A note on property (gaR) and perturbations |
| title_alt | Про властивiсть (\bfit g \bfit a \bfit R ) та збурення |
| title_full | A note on property (gaR) and perturbations |
| title_fullStr | A note on property (gaR) and perturbations |
| title_full_unstemmed | A note on property (gaR) and perturbations |
| title_short | A note on property (gaR) and perturbations |
| title_sort | note on property (gar) and perturbations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1750 |
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