Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1
If T or T∗ is an algebraically wF(p,r,q) operator with p,r>0 and q≥1 acting on an infinite-dimensional separable Hilbert space, then we prove that the Weyl theorem holds for f(T), for every f∈Hol(σ(T)), where Hol(σ(T)) denotes the set of all analytic functions in an open neighborhood of σ(T). Mor...
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nasplib_isofts_kiev_ua-123456789-1663572025-02-10T01:28:35Z Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1 Теорема Вейля для операторiв, що алгебраїчно належать класу wF(p,r,q) при p,q>0 і q≥1 Rashid, M.H.M. Статті If T or T∗ is an algebraically wF(p,r,q) operator with p,r>0 and q≥1 acting on an infinite-dimensional separable Hilbert space, then we prove that the Weyl theorem holds for f(T), for every f∈Hol(σ(T)), where Hol(σ(T)) denotes the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T∗ is a wF(p,r,q) operator with p,r>0 and q≥1, then the a-Weyl theorem holds for f(T). Also, if T or T∗ is an algebraically wF(p,r,q) operators with p,r>0 and q≥1, then we establish spectral mapping theorems for the Weyl spectrum and essential approximate point spectrum of T for every f∈Hol(σ(T)), respectively. Finally, we examine the stability of the Weyl theorem and a-Weyl theorem under commutative perturbation by finite-rank operators. У випадку, коли T або T∗ — оператори, що алгебраїчно належать класу wF(p,r,q), де p,r>0,q≥1i дiють на нескiнченновимiрному сепарабельному гiльбертовому просторi, доведено, що теорема Вейля виконується для f(T) при кожному f∈Hol(σ(T)), де Hol(σ(T)) — множина всiх аналiтичних функцiй у вiдкритому околi σ(T). Крiм того, якщо T∗ — оператор класу wF(p,r,q), де p,r>0 i q≥1, то a-теорема Вейля виконується для f(T). У випадку, коли T або T∗ — оператори, що алгебраїчно належать класу wF(p,r,q) при p,r>0 i q≥1, встановлено теореми про спектральне вiдображення, вiдповiдно, для спектра Вейля та для iстотного наближеного точкового спектра оператора T для кожного f∈Hol(σ(T)). Дослiджено стiйкiсть теореми Вейля та a-теореми Вейля при комутативному збуреннi операторами скiнченного рангу. 2011 Article Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1 / M.H.M. Rashid // Український математичний журнал. — 2011. — Т. 63, № 8. — С. 1092–1102. — Бібліогр.: 25 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166357 517.9 en Український математичний журнал application/pdf Інститут математики НАН України |
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Статті Статті Rashid, M.H.M. Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1 Український математичний журнал |
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If T or T∗ is an algebraically wF(p,r,q) operator with p,r>0 and q≥1 acting on an infinite-dimensional separable Hilbert space, then we prove that the Weyl theorem holds for f(T), for every f∈Hol(σ(T)), where Hol(σ(T)) denotes the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T∗ is a wF(p,r,q) operator with p,r>0 and q≥1, then the a-Weyl theorem holds for f(T). Also, if T or T∗ is an algebraically wF(p,r,q) operators with p,r>0 and q≥1, then we establish spectral mapping theorems for the Weyl spectrum and essential approximate point spectrum of T for every f∈Hol(σ(T)), respectively. Finally, we examine the stability of the Weyl theorem and a-Weyl theorem under commutative perturbation by finite-rank operators. |
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Rashid, M.H.M. |
| author_facet |
Rashid, M.H.M. |
| author_sort |
Rashid, M.H.M. |
| title |
Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1 |
| title_short |
Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1 |
| title_full |
Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1 |
| title_fullStr |
Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1 |
| title_full_unstemmed |
Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1 |
| title_sort |
weyl's theorem for algebrascally wf(p,r,q) operators with p,q>0 and q≥1 |
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Інститут математики НАН України |
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2011 |
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https://nasplib.isofts.kiev.ua/handle/123456789/166357 |
| citation_txt |
Weyl's theorem for algebrascally wF(p,r,q) operators with p,q>0 and q≥1 / M.H.M. Rashid // Український математичний журнал. — 2011. — Т. 63, № 8. — С. 1092–1102. — Бібліогр.: 25 назв. — англ. |
| series |
Український математичний журнал |
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2025-12-02T11:39:39Z |
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UDC 517.9
M. H. M. Rashid (Mu’tah Univ., Al-Karak, Jordan)
WEYL’S THEOREM FOR ALGEBRAICALLY wF (p, r, q)
OPERATORS WITH p, r > 0 AND q ≥ 1
ТЕОРЕМА ВЕЙЛЯ ДЛЯ ОПЕРАТОРIВ, ЩО АЛГЕБРАЇЧНО
НАЛЕЖАТЬ КЛАСУ wF (p, r, q) ПРИ p, r > 0 I q ≥ 1
If T or T ∗ is an algebraically wF (p, r, q) operator with p, r > 0 and q ≥ 1 acting on an infinite-dimensional
separable Hilbert space, then we prove that the Weyl theorem holds for f(T ), for every f ∈ Hol(σ(T )),
where Hol(σ(T )) denotes the set of all analytic functions in an open neighborhood of σ(T ). Moreover, if
T ∗ is a wF (p, r, q) operator with p, r > 0 and q ≥ 1, then the a-Weyl theorem holds for f(T ). Also, if T
or T ∗ is an algebraically wF (p, r, q) operators with p, r > 0 and q ≥ 1, then we establish spectral mapping
theorems for the Weyl spectrum and essential approximate point spectrum of T for every f ∈ Hol(σ(T )),
respectively. Finally, we examine the stability of the Weyl theorem and a-Weyl theorem under commutative
perturbation by finite-rank operators.
У випадку, коли T або T ∗ — оператори, що алгебраїчно належать класу wF (p, r, q), де p, r > 0, q ≥ 1,
i дiють на нескiнченновимiрному сепарабельному гiльбертовому просторi, доведено, що теорема Вейля
виконується для f(T ) при кожному f ∈ Hol(σ(T )), де Hol(σ(T )) — множина всiх аналiтичних функцiй
у вiдкритому околi σ(T ). Крiм того, якщо T ∗ — оператор класу wF (p, r, q), де p, r > 0 i q ≥ 1, то
a-теорема Вейля виконується для f(T ). У випадку, коли T або T ∗ — оператори, що алгебраїчно
належать класу wF (p, r, q) при p, r > 0 i q ≥ 1, встановлено теореми про спектральне вiдображення,
вiдповiдно, для спектра Вейля та для iстотного наближеного точкового спектра оператора T для кожного
f ∈ Hol(σ(T )). Дослiджено стiйкiсть теореми Вейля та a-теореми Вейля при комутативному збуреннi
операторами скiнченного рангу.
1. Introduction. Throughout this paper let B(H), F(H), K(H), denote, respectively,
the algebra of bounded linear operators, the ideal of finite rank operators and the ideal of
compact operators acting on an infinite dimensional separable Hilbert space H. If T ∈
∈ B(H) we shall write ker(T ) andR(T ) for the null space and range of T , respectively.
Also, let α(T ) := dim ker(T ), β(T ) := dimR(T ), and let σ(T ), σa(T ), σp(T ) denote
the spectrum, approximate point spectrum and point spectrum of T , respectively. An
operator T ∈ B(H) is called Fredholm if it has closed range, finite dimensional null
space, and its range has finite codimension. The index of a Fredholm operator is given
by
i(T ) := α(T )− β(T ).
T is called Weyl if it is Fredholm of index 0, and Browder if it is Fredholm “of finite
ascent and descent”.
Recall that the ascent, a(T ), of an operator T is the smallest non-negative integer
p such that ker(T p) = ker(T p+1). If such integer does not exist we put a(T ) = ∞.
Analogously, the descent, d(T ), of an operator T is the smallest non-negative integer q
such that R(T q) = R(T q+1), and if such integer does not exist we put d(T ) =∞. The
essential spectrum σF (T ), the Weyl spectrum σW (T ) and the Browder spectrum σb(T )
of T are defined by
σF (T ) = {λ ∈ C : T − λ is not Fredholm},
σW (T ) = {λ ∈ C : T − λ is not Weyl},
and
σb(T ) = {λ ∈ C : T − λ is not Browder}
c© M. H. M. RASHID, 2011
1092 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8
WEYL’S THEOREM FOR ALGEBRAICALLY wF (p, r, q) OPERATORS . . . 1093
respectively. Evidently
σF (T ) ⊆ σW (T ) ⊆ σb(T ) ⊆ σF (T ) ∪ accσ(T ),
where we write accK for the accumulation points of K ⊆ C.
Following [1], we say that Weyl’s theorem holds for T if σ(T ) \ σW (T ) = E0(T ),
where E0(T ) is the set of all eigenvalues λ of finite multiplicity isolated in σ(T ). And
Browder’s theorem holds for T if σ(T ) \ σW (T ) = π0(T ), where π0 is the set of all
poles of T of finite rank.
Let SF (H) be the class of all semi-Fredholm operators on H. Let SF+(H) be the
class of all upper semi-Fredholm operators, SF−+ (H) be the class of all T ∈ SF+(H)
with i(T ) ≤ 0, and for any T ∈ B(H), let
σSF (T ) = {λ ∈ C : T − λI /∈ SF (H)} ,
σSF−
+
(T ) =
{
λ ∈ C : T − λI /∈ SF−+ (H)
}
,
ρSF (T ) = C \ σSF (T ) and ρSF−
+
(T ) = C \ σSF−
+
(T ).
In [2] Berkani define the class of B-Fredholm operators as follows. For each integer
n, define Tn to be the restriction of T to R(Tn) viewed as a map from R(Tn) into
R(Tn) (in particular T0 = T ). If for some n the range R(Tn) is closed and Tn is
Fredholm (resp. semi-Fredholm ) operator, then T is called a B-Fredholm (resp. semi-
B-Fredholm ) operator. In this case and from [2] Tm is a Fredholm operator and i(Tm) =
= i(Tn) for each m ≥ n. The index of a B-Fredholm operator T is defined as the index
of the Fredholm operator Tn, where n is any integer such that the range R(Tn) is closed
and Tn is Fredholm operator (see [2]). Let SBF (H) be the class of all semi-B-Fredholm
operators on H. For T ∈ B(H), let
σSBF (T ) = {λ ∈ C : T − λI /∈ SBF (H)} ,
ρSBF (T ) = C \ σSBF (T ).
Let Ea0 be the set of all eigenvalues of T of finite multiplicity which are isolated
in σa(T ). According to [3], we say that T satisfies a-Weyl’s theorem if σSF−
+
(T ) =
= σa(T ) \ Ea0 (T ) and a-Browder’s theorem holds for T if σSF−
+
(T ) = σab(T ). It
follows from [3] (Corollary 2.5) a-Weyl’s theorem implies Weyl’s theorem.
It follows from [3, 4] that
a-Weyl’s theorem =⇒Weyl’s theorem =⇒ Browder’s theorem,
a-Weyl’s theorem =⇒ a-Browder’s theorem =⇒ Browder’s theorem.
The investigation of operators obeying Weyl’s theorem, a-Weyl’s theorem, Brow-
der’s theorem or a-Browder’s theorem was studied by many mathematicians [1, 3 – 9]
and the references cited therein.
Following [10], we say that T ∈ B(H) has the single-valued extension property
(SVEP) at point λ ∈ C if for every open neighborhood Uλ of λ, the only analytic
function f : Uλ −→ H which satisfies the equation (T − µ)f(µ) = 0 is the constant
function f ≡ 0. It is well-known that T ∈ B(H) has SVEP at every point of the
resolvent ρ(T ) := C \ σ(T ). Moreover, from the identity theorem for analytic function
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8
1094 M. H. M. RASHID
it easily follows that T ∈ B(H) has SVEP at every point of the boundary ∂σ(T )
of the spectrum. In particular, T has SVEP at every isolated point of σ(T ). In [11]
(Proposition 1.8), Laursen proved that if T is of finite ascent, then T has SVEP.
Proposition 1.1 [12]. Let T ∈ B(H).
(i) If T has the SVEP, then i(T − λI) ≤ 0 for every λ ∈ ρSBF (T ).
(ii) If T ∗ has the SVEP, then i(T − λI) ≥ 0 for every λ ∈ ρSBF (T ).
Definition 1.1 [13]. Let T ∈ B(H) and n, d ∈ N. Then T has a uniform descent
for n ≥ d if R(T ) + ker(Tn) = R(T ) + ker(T d) for all n ≥ d. If, in addition,
R(T ) + ker(T d) is closed, then T is said to have topological uniform descent for
n ≥ d.
2. Properties of algebraically wF (p, r, q) operators with p, r > 0 and q ≥ 1.
A bounded linear operator T ∈ B(H) belongs to the class wF (p, q, r) for each p, r > 0
and q ≥ 1 if
(|T ∗|r|T |2p|T ∗|r)1/q ≥ |T ∗|2(p+r)/q
and
|T |2(p+r)(1−1/q) ≥
(
|T |p|T ∗|2r|T |p
)(1−1/q)
.
This class has been introduced by Yang and Yuan, see [14]. An operator T ∈ B(H)
is called isoloid if every isolated point of σ(T ) is an eigenvalue of T. An operator
T ∈ B(H) is called normaloid if r(T ) = ‖T‖, where r(T ) is the spectral radius of T.
T ∈ B(H) is called convexoid if conv σ(T ) = W (T ), where W (T ) is the numerical
range of T. X ∈ B(H) is called a quasiaffinity if it has trivial kernel and dense range.
S ∈ B(H) is said to be a quasiaffine transform of T ∈ B(H) (notation: S ≺ T ) if there
is a quasiaffinity X ∈ B(H) such that XS = TX. If both S ≺ T and T ≺ S then we
say that S and T are quasisimilar.
In general, the following implications hold:
class wF (p, r, q) =⇒ algebraically class wF (p, r, q) for each p, r > 0 and q ≥ 1.
The following facts follow from the above definition and some well known facts
about class wF (p, r, q) for each p, r > 0 and q ≥ 1.
(i) If T ∈ B(H) is algebraically class wF (p, r, q) for each p, r > 0 and q ≥ 1 then
so is T − λI for each λ ∈ C.
(ii) If T ∈ B(H) is algebraically class wF (p, r, q) for each p, r > 0 and q ≥ 1 and
M is a closed T -invariant subspace of H then T |M is algebraically class wF (p, r, q) for
each p, r > 0 and q ≥ 1.
Remark 2.1. In what follows, we use the notation wF to denote the class wF (p, r, q)
operators with p, r > 0 and q ≥ 1.
Lemma 2.1. Let T ∈ B(H) belong to class wF (p, r, q) with p, r > 0 and q ≥ 1.
Let λ ∈ C. Assume that σ(T ) = {λ}. Then T = λI.
Proof. We consider two cases:
Case I (λ = 0): Since T belongs class wF for each p, r > 0 and q ≥ 1, T is
normaloid. Therefore T = 0.
Case II (λ 6= 0): Here T is invertible, and since T belongs class wF for each
p, r > 0 and q ≥ 1, we see that T−1 is also belongs to class wF for each p, r > 0
and q ≥ 1. Therefore T−1 is normaloid. On the other hand, σ(T−1) = {1/λ}, so
‖T‖‖T−1‖ = |λ| |1/λ| = 1. It follows that T is convexoid, so W (T ) = {λ}. Therefore
T = λ.
Lemma 2.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8
WEYL’S THEOREM FOR ALGEBRAICALLY wF (p, r, q) OPERATORS . . . 1095
Proposition 2.1. Let T be a quasinilpotent algebraically wF operator. Then T is
nilpotent.
Proof. Assume that p(T ) is wF operator for some nonconstant polynomial p. Since
σ(p(T )) = p(σ(T )), the operator p(T )−p(0) is quasinilpotent. Thus Lemma 2.1 would
imply that
cTm(T − λ1I) . . . (T − λnI) ≡ p(T )− p(0) = 0,
where m ≥ 1. Since T − λjI is invertible for every λj 6= 0, we must have Tm = 0.
Proposition 2.1 is proved.
An operator T ∈ B(H) is said to be polaroid if isoσ(T ) ⊆ π(T ), where π(T ) is the
set of all poles of T. In general, if T is polaroid then it is isoloid. However, the converse
is not true. Consider the following example. Let T ∈ `2(N) be defined by
T (x1, x2, . . .) =
(x2
2
,
x3
3
, . . .
)
.
Then T is a compact quasinilpotent operator with α(T ) = 1, and so T is isoloid.
However, since T does not have finite ascent, T is not polaroid.
In [15] they showed that every wF operator is isoloid. We can prove more:
Proposition 2.2. Let T be an algebraically wF operator. Then T is polaroid.
Proof. Suppose T is an algebraically wF operator. Then p(T ) is wF for some
nonconstant polynomial p. Let λ ∈ iso(σ(T )). Using the spectral projection P :=
:=
1
2iπ
∫
∂D
(µ−T )−1 dµ, where D is a closed disk of center λ which contains no other
points of σ(T ), we can represent T as the direct sum
T =
(
T1 0
0 T2
)
, σ(T1) = {λ} , and σ(T2) = σ(T ) \ {λ} .
Since T1 is algebraically class wF and σ(T1) = {λ} . But σ(T1 − λI) = {0} it follows
from Proposition 2.1 that T1 − λI is nilpotent. Therefore T1 − λ has finite ascent and
descent. On the other hand, since T2 − λI is invertible, clearly it has finite ascent and
descent. Therefore T − λI has finite ascent and descent. Therefore λ is a pole of the
resolvent of T. Thus if λ ∈ iso(σ(T )) implies λ ∈ π(T ), and so iso(σ(T )) ⊂ π(T ).
Hence T is polaroid.
Proposition 2.2 is proved.
Corollary 2.1. Let T be an algebraically wF operator. Then T is isoloid.
For T ∈ B(H), λ ∈ σ(T ) is said to be a regular point if there exists S ∈ B(H)
such that T − λI = (T − λI)S(T − λI). T is is called reguloid if every isolated
point of σ(T ) is a regular point. It is well known [16] (Theorems 4.6.4 and 8.4.4) that
T − λI = (T − λI)S(T − λI) for some S ∈ B(H)⇐⇒ T − λI has a closed range.
Theorem 2.1. Let T be an algebraically wF operator. Then T is reguloid.
Proof. Suppose T is an algebraically wF operator. Then p(T ) is wF for some
nonconstant polynomial p. Let λ ∈ iso(σ(T )). Using the spectral projection P :=
:=
1
2iπ
∫
∂D
(µ−T )−1 dµ, where D is a closed disk of center λ which contains no other
points of σ(T ), we can represent T as the direct sum
T =
(
T1 0
0 T2
)
, σ(T1) = {λ} , and σ(T2) = σ(T ) \ {λ} .
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 8
1096 M. H. M. RASHID
Since T1 is algebraically class wF and σ(T1) = {λ}, it follows from Lemma 2.1 that
T1 = λI. Therefore by [15] (Theorem 2.10),
H = E(H)⊕ E(H)⊥ = ker(T − λI)⊕ ker(T − λI)⊥. (2.1)
Relative to decomposition (2.1), T = λI ⊕ T2. Therefore T − λI = 0 ⊕ T − λI and
hence ran(T − λI) = (T − λI)(H) = 0⊕ (T2 − λI)(ker(T − λI)⊥). Since T2 − λI is
invertible, T − λI has closed range.
Theorem 2.1 is proved.
3. Weyl’s theorem for algebraically wF operators with p, r > 0 and q ≥ 1.
Theorem 3.1. Suppose T or T ∗ is an algebraically class wF (p, r, q) operator for
p, r > 0 and q ≥ 1. Then Weyl’s theorem holds for f(T ) for every f ∈ Hol(σ(T )).
Proof. Suppose that T is algebraically class wF. We first show that Weyl’s theorem
holds for T. Let λ ∈ σ(T ) \ σW (T ). Then T − λI is Weyl but not invertible. We
claim that λ ∈ ∂σ(T ). Assume to the contrary that λ is an interior point of σ(T ).
Then there exists a neighborhood U of λ such that dim ker(T − µ) > 0 for all µ ∈ U.
Then it follows from [10] (Theorem 10) that T does not have SVEP. On the other hand,
since p(T ) is of class wF (p, r, q) for some nonconstant polynomial p, it follows from
[15] and [10] (Proposition 1.8) that p(T ) has SVEP. Hence by [17], T has SVEP. This
is a contradiction. Therefore λ ∈ ∂σ(T ) \ σW (T ), and it follows from the punctured
neighborhood theorem that λ ∈ E0(T ). Conversely, suppose that λ ∈ E0(T ). Using the
spectral projection P :=
1
2iπ
∫
∂D
(µ − T )−1 dµ, where D is a closed disk of center λ
which contains no other points of σ(T ), we can represent T as the direct sum
T =
(
T1 0
0 T2
)
, σ(T1) = {λ} , and σ(T2) = σ(T ) \ {λ} .
Since T1 is of class wF and σ(T1) = {λ}, it follows from Lemma 2.1 that T1 − λI is
nilpotent. Since λ ∈ E0(T ), T −λI is a finite dimensional operator, so T −λI is Weyl.
Since T2 − λI is invertible, T2 − λI is Weyl. Thus Weyl’s theorem holds for T. Now
we claim that σW (f(T )) = f(σW (T )) for all f ∈ Hol(σ(T )). Let f ∈ Hol(σ(T )).
Since σW (f(T )) ⊆ f(σW (T )) with no other restriction on T , it suffices to show that
f(σW (T )) ⊆ σW (f(T )). Suppose that λ /∈ σW (f(T )). Then f(T )− λ is Weyl and
f(T )− λI = c(T − α1I)(T − α2I) . . . (T − αn)g(T ), (3.1)
where c, α1, α2, . . . , αn ∈ C and g(T ) is invertible. Since the operators in the right-hand
side of equation (3.1) commute, every T − αiI is Fredholm. Since T is algebraically
class wF (p, r, q) for each p, r > 0 and n ≥ 1, T has SVEP. It follows from Proposition
1.1 that i(T − αkI) ≤ 0 for each k = 1, . . . , n. Therefore λ /∈ f(σW (T )). Now recall
[17] that if T is isoloid then
f(σ(T ) \ E0(T )) = f(σ(T )) \ f(E0(T )) for every f ∈ Hol(σ(T )).
Since T is isoloid by Corollary 2.1 and Weyl’s theorem holds for T ,
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f(σ(T )) \ f(E0(T )) = f(σ(T ) \ E0(T )) = f(σW (T )) = σW (f(T ))
for every f ∈ Hol(σ(T )),
which implies that Weyl’s theorem holds for f(T ).
Now suppose that T ∗ is algebraically class wF (p, r, q) for each p, r > 0 and q ≥ 1.
We first show that Weyls theorem holds for T. Suppose that λ ∈ σ(T ) \ σW (T ).
Observe that σ(T ∗) = σ(T ) and σW (T ∗) = σW (T ). So λ̄ ∈ σ(T ) \ σW (T ), and hence
λ̄ ∈ E0(T ∗). Therefore λ is an isolated point of σ(T ), and so λ ∈ E0(T ). Conversely,
suppose that λ ∈ E0(T ) . Then λ is an isolated point of σ(T ) and 0 < α(T −λI) <∞.
Since λ̄ is an isolated point of σ(T ∗) and T ∗ is algebraically class wF (p, r, q) for each
p, r > 0 and q ≥ 1, it follows from that λ ∈ π(T ∗). Therefore there exists a natural
number n0 such that n0 = a(T ∗ − λ̄I) = d(T ∗ − λ̄I). Hence we have H = ker((T ∗ −
−λ̄I)n0)⊕ran((T ∗−λ̄I)n0) and ran((T ∗−λ̄I)n0) is closed. Therefore ran((T−λI)n0)
is closed and H = ker((T ∗ − λ̄I)n0)⊥ ⊕ ran((T ∗ − λ̄I)n0)⊥ = ker((T − λI)n0) ⊕
⊕ ran((T −λI)n0). So λ ∈ π(T ), and hence T −λI is Weyl. Consequently, λ ∈ σ(T )\
σW (T ). Thus Weyl’s theorem holds for T. Now we show that σW (f(T )) = f(σW (T ))
for each f ∈ Hol(σ(T )). Let f ∈ Hol(σ(T )). To show that σW (f(T )) = f(σW (T )) it
is sufficient to show that σW (f(T )) ⊇ f(σW (T )). Suppose that λ /∈ σW (f(T )). Then
f(T ) − λI is Weyl. Since T ∗ is algebraically class wF, it has SVEP. It follows from
Proposition 1.1 that i(T − αj) ≥ 0 for each j = 1, 2, . . . , n. Since
0 ≤
n∑
j=1
i(T − αj) = i(f(T )− λI) = 0,
T − αj is Weyl for each j = 1, . . . , n. Hence λ /∈ f(σW (T )), and so f(σW (T )) ⊆
⊆ σW (f(T )). Thus f(σW (T )) = σW (f(T )) for each f ∈ Hol(σ(T )). Since Weyl’s
theorem holds for T and T is isoloid, Weyl’s theorem holds for f(T ) for every f ∈
∈ Hol(σ(T )).
Theorem 3.1 is proved.
From the proof of the Theorem 3.1, we obtain the following useful consequence.
Corollary 3.1. Suppose T or T ∗ is an algebraically class wF (p, r, q) operator for
each p, r > 0 and q ≥ 1. Then σW (f(T )) = f(σW (T )) for every f ∈ Hol(σ(T )).
4. a-Weyl’s theorem for algebraically wF operators with p, r > 0 and q ≥ 1.
Let T ∈ B(H). It is well known that the inclusion σSF−
+
(f(T )) ⊆ f(σSF−
+
(T )) holds
for every f ∈ Hol(σ(T )) with no restriction on T [18]. The next theorem shows that
the spectral mapping theorem holds for the essential approximate point spectrum for
algebraically class wF.
Theorem 4.1. Suppose T ∗ or T is an algebraically class wF operator. Then
σSF−
+
(f(T )) = f(σSF−
+
(T )).
Proof. Assume first that T is algebraically wF and let f ∈ Hol(σ(T )). It suffices to
show that σSF−
+
(f(T )) ⊇ f(σSF−
+
(T )). Suppose that λ /∈ σSF−
+
(f(T )). Then f(T ) −
− λI ∈ SF−+ (H) and
f(T )− λI = c(T − µ1I)(T − µ2I) . . . (T − µnI)g(T ),
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1098 M. H. M. RASHID
where c, µ1, µ2, . . . , µn ∈ C, and g(T ) is invertible. Since T is algebraically class
wF , it has SVEP. It follows from [19] (Theorem 2.6) that i(T − µj) ≤ 0 for each
j = 1, 2, . . . , n. Therefore λ /∈ f(σSF−
+
(T )), and hence σSF−
+
(f(T )) = f(σSF−
+
(T )).
Suppose now that T ∗ is algebraically class wF. Then T ∗ has SVEP, and so by [19]
(Theorem 2.6) i(T − µjI) ≥ 0 for each j = 1, 2, . . . , n. Since
0 ≤
n∑
j=1
i(T − µjI) = i(f(T )− λI) ≤ 0,
T−µjI is Weyl for each j = 1, 2, . . . , n. Hence λ /∈ f(σSF−
+
(T )), and so σSF−
+
(f(T )) =
= f(σSF−
+
(T )).
Theorem 4.1 is proved.
An operator T ∈ B(H) is called a-isoloid if isoσa(T ) ⊆ σp(T ). Clearly, if T
is a-isoloid then it is isoloid. However, the converse is not true. Consider the fol-
lowing example: Let U ⊕ Q, where U is the unilateral forward shift on `2 and Q is
an injective quasinilpotent on `2, respectively. Then σ(T ) = {λ ∈ C : |λ| ≤ 1} and
σa(T ) = {λ ∈ C : |λ| = 1} ∪ {0} . Therefore T is isoloid but not a-isoloid.
It is easily seen that quasinilpotent operators do not satisfy a-Weyl’s theorem, in
general. for instance, if
T (x1, x2, . . .) =
(
0,
x2
2
,
x3
3
, . . .
)
, (xn) ∈ `2(N),
then T is quasinilpotent but a-Weyl’s theorem fails for T , since σ(T ) = σa(T ) =
= σSF−
+
(T ) = {0} = Ea0 (T ).
Theorem 4.2. Suppose T ∗ is an algebraically class wF (p, r, q). Then a-Weyl’s
theorem holds for f(T ) for every f ∈ Hol(σ(T )).
Proof. Suppose T ∗ is an algebraically class wF operator. We first show that a-
Weyl’s theorem holds for T. Suppose that λ ∈ σa(T ) \σSF−
+
(T ). Then T −λI is upper
semi-Fredholm and i(T − λI) ≤ 0. Since T ∗ is algebraically class wF , T ∗ has SVEP.
Therefore by [19] (Theorem 2.6) that i(T − λI) ≥ 0, and hence T − λI is Weyl. Since
T ∗ has SVEP, it follows from [10] (Corollary 7) that σa(T ) = σ(T ). Also, since Weyl’s
theorem holds for T by Theorem 3.1, λ ∈ πa0 (T ).
Conversely, suppose that λ ∈ πa0 (T ). Since T ∗ has SVEP, it follows from [10]
(Corollary 7) that σa(T ) = σ(T ). Therefore λ is an isolated point of σ(T ), and hence
λ̄ is an isolated point of σ(T ∗). But T ∗ is algebraically class wF operator, hence
by Proposition 2.2 that λ̄ ∈ π(T ∗). Therefore there exists a natural number n0 such
that n0 = a(T ∗ − λ̄I) = d(T ∗ − λ̄I). Hence we have H = ker((T ∗ − λ̄I)n0) ⊕
⊕ran((T ∗−λ̄I)n0) and ran((T ∗−λ̄I)n0) is closed. Therefore ran((T−λI)n0) is closed
andH = ker((T ∗−λ̄I)n0)⊥⊕ran((T ∗−λ̄I)n0)⊥ = ker((T−λI)n0)⊕ran((T−λI)n0).
So λ ∈ σp(T ), and hence T − λI is Weyl. Consequently, λ ∈ σa(T ) \ σSF−
+
(T ). Thus
a-Weyl’s theorem holds for T.
Now we show that T is a-isoloid. Let λ be an isolated point of σa(T ). Since T ∗
has SVEP, λ is an isolated point of σ(T ). But T ∗ is polaroid, hence T is also polaroid.
Therefore it is isoloid, and hence λ ∈ σp(T ). Thus T is a-isoloid.
Finally, we shall show that a-Weyl’s theorem holds for f(T ) for every f ∈ Hol(σ(T )).
Let f ∈ Hol(σ(T )). Since a-Weyl’s theorem holds for T , it satisfies a-Browder’s theo-
rem. Therefore σab(T ) = σSF−
+
(T ). It follows from Theorem 4.1 that
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σab(f(T )) = f(σab(T )) = f(σSF−
+
(T )) = σSF−
+
(f(T )),
and hence a-Browder’s theorem holds for f(T ). So σa()f(T ) \ σSF−
+
(f(T )) ⊂ πa0 (T ).
Conversely, suppose that λ ∈ πa0 (f(T )). Then λ is an isolated point of σa(f(T )) and
0 < α(f(T ) − λI) < 1. Since λ is an isolated point of f(σa(T )), if µj ∈ σa(T ) then
µj is an isolated point of σa(T ). Since T is a-isoloid, 0 < α(T − µj) < 1 for each
j = 1, 2, . . . , n. Since a-Weyl’s theorem holds for T , T−µj is upper semi-Fredholm and
i(T − µj) ≤ 0 for each j = 1, 2, . . . , n. Therefore f(T ) − λI is upper semi-Fredholm
and f(T )− λI =
∑n
j=1
i(T − µjI) ≤ 0. Hence λ ∈ σa()f(T ) \ σSF−
+
(f(T )), and so
a-Weyl’s theorem holds for f(T ) for each f ∈ Hol(σ(T )).
Theorem 4.2 is proved.
From the proof of the Theorem 4.2, we obtain the following useful consequence.
Corollary 4.1. Suppose T ∗ is an algebraically class wF (p, r, q). Then T is a-
isoloid.
5. Finite rank perturbations for Hilbert space operators. For each nonnegative
integer n define Tn to be the restriction of T toR(Tn) viewed as a map fromR(Tn) into
R(Tn) (in particular T0 = T ). If for some n, R(Tn) is closed and Tn is an upper (resp.
lower) semi-Fredholm operator then T is called an upper (resp. lower) semi-B-Fredholm
operator. A semi-B-Fredholm operator is an upper or lower semi-B-Fredholm operators.
If moreover, Tn is a Fredholm operator then T is called a B-Fredholm operator. The
index of a semi-B-Fredholm is defined as the index of the semi-Fredholm operator Tn
(see [13]). In [13] it is proved that an operator T is a B-Fredholm operator if and only if
T = F ⊕N , where F is a Fredholm operator and N is a nilpotent operator. An operator
T ∈ B(H) is said to be a B-Weyl operator if it is a B-Fredholm operator of index zero.
The B-Weyl spectrum σBW (T ) of T is defined by
σBW (T ) = {λ ∈ C : T − λI is not a B-Weyl operator}.
Following [13] generalized Browder’s theorem holds for T if
σ(T ) \ σBW (T ) = π(T );
where π(T ) is the set of all poles of T. Recently, in [20] it is proved that
generalized Browder’s theorem⇔ Browder’s theorem.
Recall that an operator T ∈ B(H) is a Drazin invertible if and only if it has a finite
ascent and descent, which is also equivalent to the fact that T = T0 ⊕ T1, where T0 is
nilpotent operator and T1 is invertible operator (see [21] (Proposition A). The Drazin
spectrum is given by
σD(T ) := {λ ∈ C : T − λI is not Drazin invertible}.
We observe that σD(T ) = σ(T ) \ π(T ), where π(T ) is the set of all poles.
Theorem 5.1. Suppose T ∈ B(H) be an algebraically wF. If F is finite rank on
H such that TF = FT , then T + F satisfies generalized Browder’s theorem.
Proof. From the characterization of σBW (T ) [2] (Theorem 4.3), it follows that if F
is a finite rank operator, then σBW (T +F ) = σBW (T ). Moreover, if F commutes with
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1100 M. H. M. RASHID
T , it follows from [22] (Theorem 2.7) that σD(T + F ) = σD(T ). Since T has SVEP,
then it satisfies generalized Browder’s theorem by [12] (Theorem 1.5), then σBW (T ) =
= σD(T ). Hence σBW (T + F ) = σD(T + F ), and so T + F satisfies generalized
Browder’s theorem.
Theorem 5.1 is proved.
Corollary 5.1. Suppose T ∈ B(H) be an algebraically wF. If F is finite rank on
H such that TF = FT , then T + F satisfies Browder’s theorem.
Theorem 5.2. If T ∈ B(H) is an algebraically wF , if FT = TF, F ∈ F(H).
Then T + F satisfies Weyl’s theorem.
Proof. Since by Corollary 5.1 Browder’s theorem holds for T+F it suffices to prove
that E0(T + F ) = π0(T + F ). Let λ ∈ E0(T + F ) be given, then λ ∈ isoσ(T + F )
and λ ∈ σp(T +F ), hence λ /∈ acc(σ(T +F )) and λ /∈ acc(σ(T )). We distinguish two
cases:
Case I. If λ /∈ σ(T ), then T − λI is invertible and T − λI is Fredholm of index
zero, since F is a finite rank operator on H, it follows that T + F − λI is Fredholm
operator of index zero. Then λ /∈ σW (T + F ) and λ ∈ π0(T + F ).
Case II. If λ ∈ σ(T ), then λ ∈ iso(σ(T )) and since T is isoloid λ ∈ σp(T ).
Thus λ ∈ iso(σ(T )) ∩ σp(T ) = E0(T ). From the fact that T obeys Weyl’s theorem, it
follows that λ /∈ σW (T ) = σW (T + F ) and since λ ∈ iso(σ(T + F )), it follows that
λ ∈ π0(T + F ). Finally E0(T + F ) ⊂ π0(T + F ), and since the reverse inclusion is
always true, T + F obeys Weyl’s theorem.
Theorem 5.2 is proved.
Example 5.1. This example shows that the commutativity hypothesis in Theo-
rem 5.2 is essential. Let H = `2(N) and T and F be defined by
T (x1, x2, . . .) :=
(
0,
x1
2
,
x2
3
, . . .
)
, {xn} ∈ `2(N),
and
F (x1, x2, . . .) :=
(
0,
−x1
2
, 0, . . .
)
, {xn} ∈ `2(N).
Clearly, F is a nilpotent operator and hence of finite rank operator, and T is a quasi-
nilpotent satisfying Weyl’s theorem since σ(T ) = σW (T ) = {0} and E0(T ) = ∅. Now
T and F do not commute, σ(T + F ) = σW (T + F ) = E0(T + F ) = {0} , and T + F
does not satisfy Weyl’s theorem.
Theorem 5.3. Let T be an algebraically wF. If F is an operator that commutes
with T and for which there exits a positive integer n such that Fn is finite rank, then
T + F satisfies Weyl’s theorem.
Proof. Form Corollary 2.1 and Theorem 3.1, T is isoloid and satisfies Weyl’s theo-
rem. Now the result follows at once from [9] (Theorem 2.4).
Theorem 5.3 is proved.
Theorem 5.4. If T ∗ is an algebraically wF, then for any finite rank operator
F ∈ B(H) commuting with T , the a-Weyl’s theorem holds for T + F.
Proof. (a) Firstly, We will prove that σa(T + F ) \ σSF−
+
(T + F ) ⊆ Ea0 (T + F ).
Let λ0 ∈ σa(T + F ) \ σSF−
+
(T + F ). Using the perturbation theorem of semi-
Fredholm operator, T − λ0I is upper semi-Fredholm with i(T − λ0I) ≤ 0 because F is
compact. Since a-Weyl’s theorem holds for T by Theorem 4.2 if T is an algebraically
class wF operator, it follows that λ0 ∈ σa(T ) \ σSF−
+
(T ), or λ0 ∈ ρ(T ). Then T − λ0I
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WEYL’S THEOREM FOR ALGEBRAICALLY wF (p, r, q) OPERATORS . . . 1101
has finite ascent and hence T + F − λ0I has finite ascent [23] (Theorem 1), [24]
(Lemma 2.4), λ0 ∈ Ea0 (T + F ).
(b) Secondly, We will prove that σa(T + F ) \ σSF−
+
(T + F ) ⊇ Ea0 (T + F ).
Let λ0 ∈ Ea0 (T +F ), that is λ0 ∈ iso(σa(T +F )) and 0 < dim ker(T +F −λ0I) <
<∞. Then dim ker(T − λ0I) <∞ [25] (Lemma 2.1) and there exists ε > 0 such that
T+F−λ0I is bounded below if 0 < |λ−λ0| < ε. Then T−λI is upper semi-Fredholm
if 0 < |λ− λ0| < ε.
If there exists {λn}∞n=1 ⊆ σa(T ) such that λi 6= λj and λn → λ0 as n → ∞,
without loss of generality, we suppose 0 < |λ− λ0| < ε. Let Mn = ker(T − λnI) and
let Fn = F |Mn
. Then Fn is linear and injective. In fact, if there exists x ∈ Mn such
that Fnx = 0, then (T +F −λnI)x = Fnx = 0. Since T +F −λnI is bounded below,
we have x = 0. We know that in finite dimensional linear space Mn, Fn is injective
if and only if Fn is surjective. Then ker(T − λnI) = Fn ker(T − λnI) ⊆ ran(F ),
thus
∑∞
n=1
⊕
ker(T − λnI) ⊆ ran(F ). We have that
∑∞
n=1
dim ker(T − λnI) ≤
≤ dim ran(F ). Since λn ∈ σa(T ) \ σaw(T ), we have dim ker(T − λnI) > 0 for any
n ∈ N. Then dim ran(F ) =∞; it is impossible because dim ran(F ) <∞.
From the proof above, we get there exists ε′ > 0 (ε′ should be less than ε) such that
T − λI is bounded below if 0 < |λ− λ0| < ε′. Then λ0 ∈ iso(σa(T )).
Since T is a-isoloid by Corollary 4.1, it follows that 0 < ker(T − λ0I)∞, which
means that λ0 ∈ Ea0 (T ). The a-Weyl’s theorem holds for T , then λ0 ∈ σa(T ) \
σSF−
+
(T ), and hence T+F−λ0I is upper semi-Fredholm operator with i(T+F−λ0I) ≤
≤ 0. Now we have that λ0 ∈ σa(T + F ) \ σSF−
+
(T + F ).
From (a) and (b), we get σa(T + F ) \ σSF−
+
(T + F ) = Ea0 (T + F ), which means
that the a-Weyl’s theorem holds for T + F.
Theorem 5.4 is proved.
In general, a-Weyl’s theorem is not transmitted under commuting finite rank pertur-
bation.
Example 5.2. Let S = `2 −→ `2 be an injective quasinilpotent operator which
is not nilpotent and let U : `2 −→ `2 be defined by U 〈x1, x2, . . .〉 := 〈−x1, 0, . . .〉,
xn ∈ `2(N). Define on H := `2 ⊕ `2 the operators T and K by T := I ⊕ S where I is
the identity on `2 and K := U ⊕ 0.
It is easily that σa(T ) = {0, 1}, Ea0 (T ) = {1} and σSF−
+
(T ) = {0}. Hence T
satisfies a-Weyl’s theorem. Now K is finite rank operator and TK = KT. Moreover,
σa(T +K) = {0, 1} and Ea0 (T +K) = {0, 1}. As σSF−
+
(T +K) = σSF−
+
(T ) = {0} ,
Then T +K does not satisfy a-Weyl’s theorem.
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Received 02.06.09,
after revision — 11.04.11
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