Generalized Weyl's theorem and tensor product
We give necessary and/or sufficient conditions ensuring the passage of generalized a-Weyl theorem and property $(gw)$ from $A$ and $B$ to $A \otimes B$.
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| author | Rashid, M. H. M. Рашид, М. Х. М. |
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UDC 515.14
M. H. M. Rashid (Mu’tah Univ., Al-Karak, Jordan)
GENERALIZED WEYL’S THEOREM AND TENSOR PRODUCT
УЗАГАЛЬНЕНА ТЕОРЕМА ВЕЙЛЯ ТА ТЕНЗОРНИЙ ДОБУТОК
We give necessary and/or sufficient conditions ensuring the passage of generalized a-Weyl theorem and property (gw) from
A and B to A⊗B.
Наведено необхiднi та/або достатнi умови, що гарантують поширення узагальненої а-теореми Вейля та властивостi
(gw) iз A та B на A⊗B.
1. Introduction. Given Banach spaces X and Y, let X⊗Y denote the completion (in some reasonable
uniform cross norm) of the tensor product of X and Y. For Banach space operators A ∈ L(X) and
B ∈ L(Y), let A⊗B ∈ L(X⊗Y) denote the tensor product of A and B. Recall that for an operator
S, the Browder spectrum σb(S) and the Weyl spectrum σw(S) of S are the sets
σb(S) = {λ ∈ C : S − λ is not Fredholm or asc(S − λ) 6= dsc(S − λ)} ,
σw(S) = {λ ∈ C : S − λ is not Fredholm or ind(S − λ) 6= 0} .
In the case in which X and Y are Hilbert spaces, Kubrusly and Duggal [15] proved that
if σb(A) = σw(A) and σb(B) = σw(B), then σb(A⊗B) = σw(A⊗B)
if and only if σw(A⊗B) = σ(A)σw(B) ∪ σw(A)σ(B).
In other words, if A and B satisfy Browder’s theorem, then their tensor product satisfies Browder’s
theorem if and only if the Weyl spectrum identity holds true. The same proof still holds in a Banach
space setting.
For a bounded linear operator S ∈ L(X), let σ(S), σp(S) and σa(S) denote, respectively, the
spectrum, the point spectrum and the approximate point spectrum of S and if G ⊆ C, then Giso
denote the isolated points of G. Let α(S) and β(S) denote the nullity and the deficiency of S,
defined by α(S) = dim ker(S) and β(S) = codim<(S).
If the range <(S) of S is closed and α(S) < ∞ (respectively β(S) < ∞), then S is called an
upper semi-Fredholm (respectively a lower semi-Fredholm) operator. If S ∈ L(X) is either upper
or lower semi-Fredholm, then S is called a semi-Fredholm operator, and ind (S), the index of S,
is then defined by ind(S) = α(S) − β(S). If both α(S) and β(S) are finite, then S is a Fredholm
operator. The ascent, denoted asc(S), and the descent, denoted dsc(S), of S are given by asc(S) =
= inf
{
n ∈ N : ker(Sn) = ker(Sn+1
}
, dsc(S) = inf
{
n ∈ N : <(Sn) = <(Sn+1
}
(where the infi-
mum is taken over the set of non-negative integers); if no such integer n exists, then asc(S) = ∞,
respectively dsc(S) =∞.)
For S ∈ L(X) and a nonnegative integer n define S[n] to be the restriction of S to <(Sn)
viewed as a map from <(Sn) into <(Sn) (in particular, S[0] = S). If for some integer n the range
space <(Sn) is closed and S[n] is an upper (a lower) semi-Fredholm operator, then S is called
c© M. H. M. RASHID, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9 1289
1290 M. H. M. RASHID
an upper (a lower) semi-B-Fredholm operator. In this case the index of S is defined as the index
of the semi-B-Fredholm operator S[n], see [8]. Moreover, if S[n] is a Fredholm operator, then S is
called a B-Fredholm operator. A semi-B-Fredholm operator is an upper or a lower semi-B-Fredholm
operator. An operator S is said to be a B-Weyl operator [9] (Definition 1.1) if it is a B-Fredholm
operator of index zero. The B-Weyl spectrum σBW (S) of S is defined by σBW (S) = {λ ∈ C : S−λI
is not a B-Weyl operator}.
An operator S ∈ L(X) is called Drazin invertible if it has a finite ascent and descent. The Drazin
spectrum σD(S) of an operator S is defined by σD(S) = {λ ∈ C : S − λI is not Drazin invertible}.
Define also the set LD(X) by LD(X) =
{
S ∈ L(X) : a(S) <∞ and <(T a(S)+1) is closed
}
and
σLD(S) = {λ ∈ C : S − λ /∈ LD(X)}. Following [10], an operator S ∈ L(X) is said to be left
Drazin invertible if S ∈ LD(X). We say that λ ∈ σa(T ) is a left pole of S if S − λI ∈ LD(X),
and that λ ∈ σa(S) is a left pole of S of finite rank if λ is a left pole of T and α(S − λI) < ∞.
Let πa(S) denotes the set of all left poles of S and let π0a(S) denotes the set of all left poles of S of
finite rank. From [10] (Theorem 2.8) it follows that if S ∈ L(X) is left Drazin invertible, then S is an
upper semi-B-Fredholm operator of index less than or equal to 0. Note that πa(S) = σa(S)\σLD(S)
and hence λ ∈ πa(S) if and only if λ /∈ σLD(S).
Following [9], we say that generalized Weyl’s theorem holds for S ∈ L(X) (in symbol S ∈ gW)
if ∆g(S) = σ(S) \ σBW (S) = E(S), where E(S) =
{
λ ∈ σiso(S) : 0 < α(S − λI)
}
is the set
of all isolated eigenvalues of S, and that generalized Browder’s theorem holds for S ∈ L(X)
(in symbol S ∈ gB) if ∆g(S) = π(S), where π(T ) is the set of poles of the resolvent of
T . It is proved in [5] (Theorem 2.1) that generalized Browder’s theorem is equivalent to Brow-
der’s theorem. In [10] (Theorem 3.9), it is shown that an operator satisfying generalized Weyl’s
theorem satisfies also Weyl’s theorem, but the converse does not hold in general. Nonetheless
and under the assumption E(S) = π(S), it is proved in [11] (Theorem 2.9) that generalized
Weyl’s theorem is equivalent to Weyl’s theorem. Let Ψ+(X) be the class of all upper semi-B-
Fredholm operators, Ψ−+(X) = {S ∈ Ψ+(X) : ind(S) ≤ 0} . The upper B-Weyl spectrum of S is
defined by σSBF−
+
(S) =
{
λ ∈ C : T − λI /∈ Ψ−+(X)
}
. We say that generalized a-Weyl’s theorem
holds for S ∈ L(X) (in symbol S ∈ gaW) if ∆g
a(S) = σa(S) \ σSBF−
+
(S) = Ea(S), where
Ea(S) =
{
λ ∈ σisoa (S) : α(S − λ) > 0
}
is the set of all eigenvalues of S which are isolated in
σa(S) and that S ∈ L(X) obeys generalized a-Browder’s theorem (S ∈ gaB) if ∆g
a(S) = πa(S).
It is proved in [5] (Theorem 2.2) that generalized a-Browder’s theorem is equivalent to a-Browder’s
theorem, and it is known from [10] (Theorem 3.11) that an operator satisfying generalized a-Weyl’s
theorem satisfies a-Weyl’s theorem, but the converse does not hold in general and under the as-
sumption Ea(S) = πa(S) it is proved in [11] (Theorem 2.10) that generalized a-Weyl’s theorem is
equivalent to a-Weyl’s theorem.
The operator T ∈ L(X) is said to have the single valued extension property at λ0 ∈ C (abbrevi-
ated SVEP at λ0) if for every open disc D centred at λ0, the only analytic function f : D → which
satisfies the equation (T − λ)f(λ) = 0 for all λ ∈ D is the function f ≡ 0. An operator T ∈ L(X)
is said to have SVEP if T has SVEP at every point λ ∈ C.
Obviously, every T ∈ L(X) has SVEP at the points of the resolvent ρ(T ) := C\σ(T ). Moreover,
from the identity theorem for analytic function, it easily follows that T ∈ L(X), as well as its dual
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
GENERALIZED WEYL’S THEOREM AND TENSOR PRODUCT 1291
T ∗, has SVEP at every point of the boundary ∂σ(T ) = ∂σ(T ∗) of the spectrum σ(T ). In particular,
both T and T ∗ have SVEP at every isolated point of the spectrum, see [1, 4, 2, 3].
Let
Ψ+(S) = {λ ∈ C : S − λ is upper semi-B-Fredholm} ,
Ψ(S) = {λ ∈ C : S − λ is B-Fredholm} ,
σSBF+(S) = {λ ∈ σa(S) : λ /∈ Ψ+(S)} ,
σSBF−
+
(S) =
{
λ ∈ σa(S) : λ ∈ σSBF+(S) or ind(S − λ) > 0
}
,
H0(S) =
{
x ∈ X : lim
n−→∞
‖Snx‖1/n = 0
}
.
2. Main results. Let σs(S) = {λ ∈ σ(S) : S − λ is not surjective} denote, the surjectivity spec-
trum. Let Ψ−(X) be the class of all lower semi-B-Fredholm operators, Ψ+
−(X) = {S ∈ Ψ−(X) :
ind(S−λ) ≥ 0}. The lower semi-B-Weyl spectrum of S is defined by σSBF+
−
(S) = {λ ∈ C : S−λ /∈
/∈ Ψ+
−(X)}. Define RD(X) =
{
S ∈ L(X) : dsc(S) = d < ∞ and <(Sd+1) is closed
}
. The right
Drazin invertible is defined by σRD(S) = {λ ∈ C : S − λ /∈ RD(X)}. It is not difficult to see
that σD(S) = σLD(S) ∪ σRD(S). Moreover, σLD(S) = σRD(S∗) [7]. Then S satisfies generalized
s-Browder’s theorem if σSBF+
−
(S) = σRD(S). Apparently, S satisfies generalized s-Browder’s theo-
rem if and only if S∗ satisfies generalized a-Browder’s theorem. A necessary and sufficient condition
for S to satisfy generalized a-Browder’s theorem is that S has SVEP at every λ ∈ ∆g
a(S) [12]
(Theorem 3.1); by duality, S satisfies generalized s-Browder’s theorem if and only if S∗ has SVEP at
every λ ∈ σs(S) \σSBF+
−
(S). More generally, if either of S and S∗ has SVEP, then S and S∗ satisfy
both generalized a-Browder’s theorem and generalized s-Browder’s theorem. Either of generalized
a-Browder’s theorem and generalized s-Browder’s theorem implies generalized Browder’s theorem,
but the converse is false. generalized a-Browder’s theorem fails to transfer from A and B to A⊗ B
[13] (Example 1).
Lemma 2.1. Let A ∈ L(X) and B ∈ L(Y). Then 0 /∈ σa(A⊗B) \ σSBF+(A⊗B).
Proof. Suppose 0 ∈ σa(A ⊗ B) \ σSBF+(A ⊗ B). Then 0 ∈ σa(A ⊗ B) ∩ Ψ+(A ⊗ B). So,
there exists an integer n0 such that for any n ≥ n0, A ⊗ B − 1
n
I has closed range and 0 <
< α
(
A⊗B − 1
n
I
)
<∞. Since A⊗B − 1
n
I is injective if and only if A and B are injective, we
have α(A) > 0 or α(B) > 0. But then α
(
A⊗B − 1
n
I
)
=∞, and we have a contradiction.
Lemma 2.2. Let A ∈ L(X) and B ∈ L(Y). Then
σSBF−
+
(A⊗B) ⊆ σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B) ⊆
⊆ σa(A)σLD(B) ∪ σLD(A)σa(B) = σLD(A⊗B).
Proof. Since σSBF−
+
(S) ⊆ σLD(S) for every operator S, it follows that the inclusion
σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B) ⊆ σa(A)σLD(B) ∪ σLD(A)σa(B) is evident. To prove the
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1292 M. H. M. RASHID
inclusion σSBF−
+
(A ⊗ B) ⊆ σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B), take λ /∈ σa(A)σSBF−
+
(B) ∪
∪ σSBF−
+
(A)σa(B). Since
σSBF+(A⊗B) ⊆ σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B),
Lemma 2.1 implies that λ 6= 0. For every factorization λ = µν such that µ ∈ σa(A) and ν ∈ σa(B)
we have that µ ∈ σa\σSBF−
+
(A) and ν ∈ σa(B)\σSBF−
+
(B), i.e., µ ∈ Ψ+(A), ν ∈ Ψ+(B), ind(A−
− µ) ≤ 0 and ind(B − ν) ≤ 0. In particular, λ /∈ σSBF+(A⊗B).
We prove next that ind(A ⊗ B − λ) ≤ 0. Suppose ind(A ⊗ B − λ) > 0. Then there exists an
integer n0 such that for any n ≥ n0 we have α
(
A⊗B − λI − 1
n
I
)
< ∞. But this implies that
β
(
A⊗B − λI − 1
n
I
)
<∞, so that A⊗B − λ is B-Weyl. Let
F =
{
(µi, νi)
k
i=1 ∈ σ(A)σ(B) : µiνi = λ
}
.
Then F is a finite set. Furthermore
(i) if m > 1, then µi ∈ σiso(A) for 1 ≤ i ≤ m;
(ii) if k > m, then νi ∈ σiso(B) for m+ 1 ≤ i ≤ k;
(iii) ind(A⊗B−λ) =
∑k
j=m+1
ind(A−µi) dimH0(B−νi)+
∑m
j=1
ind(B−νi) dimH0(A−
− µi).
Since ind(A−µi) and ind(B−νi) are non-positive, we have a contradiction. Hence, ind(A⊗B−λ) ≤
≤ 0, and consequently, λ /∈ σSBF−
+
(A ⊗ B). This leaves us to prove the equality σLD(A ⊗ B) =
= σa(A)σLD(B) ∪ σLD(A)σa(B).
Suppose that λ /∈ σLD(A ⊗ B). Then λ 6= 0, λ ∈ LD(A ⊗ B), a = asc(A ⊗ B − λ) < ∞
and <(A ⊗ B − λ)a+1 is closed and hence λ ∈ πa(A ⊗ B). Observe that λ ∈ σisoa (A ⊗ B). Let
λ = µν be any factorization of λ such that µ ∈ σa(A) and ν ∈ σa(B); then µ ∈ LD(A) and
ν ∈ LD(B). Furthermore, since σisoa (A⊗ B) ⊆ σisoa (A) ∪ σisoa (B) ∪ {0} , A has SVEP at µ and B
has SVEP at ν. Consequently, µ ∈ πa(A), ν ∈ πa(B), that is, µ /∈ σLD(A) and ν /∈ σLD(B). But
then λ /∈ σa(A)σLD(B) ∪ σLD(A)σa(B). Hence σa(A)σLD(B) ∪ σLD(A)σa(B) ⊆ σLD(A⊗B).
To prove the reverse inclusion we start by recalling the fact that if µ ∈ σisoa (A) and µ ∈ σisoa (B)
for every factorization λ = µν of λ 6= 0, then λ = µν ∈ σisoa (A ⊗ B). Let λ ∈ σa(A)σLD(B) ∪
∪ σLD(A)σa(B). Then λ 6= 0. Furthermore, if λ = µν is any factorization of λ such that µ ∈ σa(A)
and ν ∈ σa(B), then the following implications hold:
µ /∈ σLD(A) and ν /∈ σLD(B)⇒ µ ∈ πa(A) and ν ∈ πa(B)⇒
⇒ λ ∈ πa(A⊗B), µ ∈ σisoa (A) and ν ∈ σisoa (B)⇒
⇒ λ ∈ πa(A⊗B) and λ ∈ σisoa (A⊗B)⇒
⇒ λ /∈ σLD(A⊗B).
Hence σLD(A⊗B) ⊆ σa(A)σLD(B) ∪ σLD(A)σa(B).
Lemma 2.2 is proved.
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GENERALIZED WEYL’S THEOREM AND TENSOR PRODUCT 1293
Lemma 2.3. Let A ∈ L(X) and B ∈ L(Y). If A⊗B satisfies generalized a-Browder’s theorem,
then
σSBF−
+
(A⊗B) = σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B).
Proof. A⊗B satisfies generalized a-Browder’s theorem if and only if σSBF−
+
(A⊗B) = σLD(A⊗
⊗B). Thus the stated result is an immediate consequence of Lemma 2.2.
The next theorem, our main result, proves that A and B satisfy generalized a-Browder’s the-
orem implies A ⊗ B satisfies generalized a-Browder’s theorem if and only if σSBF−
+
(A ⊗ B) =
= σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B).
Theorem 2.1. Let A ∈ L(X) and B ∈ L(Y). If A and B satisfy generalized a-Browder’s
theorem, then the following are equivalent:
(i) A⊗B satisfies generalized a-Browder’s theorem;
(ii) σSBF−
+
(A⊗B) = σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B);
(iii) A has SVEP at every µ ∈ Ψ+(A) and B has SVEP at every ν ∈ Ψ+(B) such that (0 6=
6= λ) = µν ∈ σa(A⊗B) \ σSBF−
+
(A⊗B).
Proof. If A and B satisfy generalized a-Browder’s theorem, then σLD(A) = σSBF−
+
(A) and
σLD(B) = σSBF−
+
(B).
(i) ⇒ (ii). By Lemma 2.3 we have, without any extra conditions.
(ii) ⇒ (i). If (ii) is satisfied, then
σSBF−
+
(A⊗B) = σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B) =
= σa(A)σLD(B) ∪ σLD(A)σa(B) =
= σLD(A⊗B) (by Lemma 2.2).
Hence A⊗B satisfies generalized a-Browder’s theorem.
(ii)⇒ (iii). Suppose (ii) holds. Let λ ∈ σa(A⊗B) \ σSBF−
+
(A⊗B). Then λ 6= 0 and for every
factorization λ = µν such that µ ∈ σa(A) ∩ Ψ+(A) and ν ∈ σa(B) ∩ Ψ+(B). Hence µ ∈ πa(A)
and ν ∈ πa(B).So it follows from [10] (Remark 2.7) that µ ∈ σisoa (A) and ν ∈ σisoa (B). Therefore,
A and B have SVEP at (all such) µ and ν, respectively.
(iii) ⇒ (ii). In view of Lemma 2.2, we have to prove that σLD(A ⊗ B) ⊆ σSBF−
+
(A ⊗ B).
Suppose that (ii) is satisfied. Take a λ ∈ σa(A⊗B) \ σSBF−
+
(A⊗B). Then (0 6=)λ ∈ Ψ+(A⊗B)
and ind(A⊗B−λ) ≤ 0. The equality σSBF+(A⊗B) = σa(A)σSBF+(B)∪σSBF+(A)σa(B) implies
that for any factorization λ = µν (such that µ ∈ σa(A) and ν ∈ σa(B)) we have that µ ∈ Ψ+(A)
and ν ∈ Ψ+(B). The SVEP hypotheses on A and B implies that asc(A − µI) and asc(B − λ) are
finite. Hence, µ ∈ σisoa (A) and µ ∈ σisoa (B). So, it follows from Theorem 2.8 of [10] that µ ∈ πa(A)
and ν ∈ πa(B). Therefore, µ /∈ σLD(A) and ν /∈ σLD(B). But then λ /∈ σLD(A ⊗ B). Hence
σLD(A⊗B) ⊆ σSBF−
+
(A⊗B).
Theorem 2.1 is proved.
The next theorem gives a sufficient condition for A ⊗ B to satisfy generalized a-Weyl theorem,
given that A and B satisfy generalized a-Weyl theorem. But before that a couple of technical lemmas.
Recall that an operator S is said to be a-isoloid if λ ∈ σisoa (S) implies λ ∈ σp(S).
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1294 M. H. M. RASHID
Lemma 2.4. Suppose that A,B and A ⊗ B satisfy generalized a-Browder’s theorem. If µ ∈
∈ πa(A) and ν ∈ πa(B), then λ = µν ∈ πa(A⊗B).
Proof. Since µ ∈ σa(A) \ σSBF−
+
(A), ν ∈ σa(B) \ σSBF−
+
(B) and σSBF−
+
(A ⊗ B) =
= σa(A)σSBF−
+
(B)∪σSBF−
+
(A)σa(B). Hence, λ = µν ∈ σa(A⊗B)\σSBF−
+
(A⊗B) = πa(A⊗B).
Theorem 2.2. Suppose that A ∈ L(X) and B ∈ L(Y) are a-isoloid which satisfy generalized
a-Weyl theorem. If σSBF−
+
(A ⊗ B) = σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B), then A ⊗ B satisfies
generalized a-Weyl theorem.
Proof. The hypotheses imply that A ⊗ B satisfies generalized a-Browder’s theorem, that is,
σa(A ⊗ B) \ σSBF−
+
(A ⊗ B) = πa(A ⊗ B). Since πa(A ⊗ B) ⊆ Ea(A ⊗ B), we have to prove
that Ea(A ⊗ B) ⊆ πa(A ⊗ B). Let λ ∈ Ea(A ⊗ B). Then 0 6= λ = µν for some µ ∈ σisoa (A) and
ν ∈ σisoa (B). The operators A and B being a-isoloid, it follows from λ = µν ∈ Ea(A ⊗ B) that
µ ∈ Ea(A) = πa(A) and ν ∈ Ea(B) = πa(B). By Lemma 2.4, λ ∈ πa(A⊗B).
Theorem 2.2 is proved.
Following [16], we say that S ∈ L(X) satisfies property (w) if σa(S) \ σaw(S) = E0(S). The
property (w) has been studied in [2, 3, 4, 16]. In [3] (Theorem 2.8), it is shown that property (w)
implies Weyl’s theorem, but the converse is not true in general. An operator S ∈ L(X) is said to be
satisfies property (gw) if σa(S)\σSBF−
+
(S) = E(S). Property (gw) has been introduced and studied
in [6]. Property (gw) extends property (w) to the context of B-Fredholm theory, and it is proved in
[6] that an operator satisfying property (gw) satisfies property (w) and generalized Weyl’s theorem
but the converse is not true in general.
The following theorem gives a necessary and sufficient condition for the transference of property
(gw) from isoloid A and B to A ⊗ B But before that a lemma and some observations, which will
often be used in the sequel. Let A ∈ L(X) and B ∈ L(Y). Then σiso(A⊗B) ⊆ σiso(A).σiso ∪ {0}.
If 0 is in the point spectrum of either of A and B, then α(A⊗B) = 0; in particular, 0 /∈ E(A⊗B)).
It is easily seen, see the argument of the proof of [15] (Proposition 2), that E(A⊗B) ⊆ E(A)E(B).
Theorem 2.3. If A ∈ L(X) and B ∈ L(Y) are isoloid operators which satisfy property (gw),
then the following conditions are equivalent:
(i) A⊗B satisfies property (gw);
(ii) the generalized a-Weyl spectrum equality σSBF−
+
(A ⊗ B) = σa(A)σSBF−
+
(B) ∪
∪ σSBF−
+
(A)σa(B) is satisfied;
(iii) A⊗B satisfies generalized a-Browder’s theorem.
Proof. Since property (gw) implies generalized a-Browder’s theorem, the equivalence (ii)⇔ (iii)
and (i) ⇒ (iii) follows from Theorem 2.2. We prove (iii) ⇒ (i). The hypothesis A and B satisfy
property (gw) implies
σa(A) \ σSBF−
+
(A) = E(A), σa(B) \ σSBF−
+
(B) = E(B).
Observe that (iii) implies generalized a-Browder’s theorem transfers from A and B to A⊗B : hence
σSBF−
+
(A⊗B) = σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B). Let λ ∈ E(A⊗B); then λ 6= 0 and hence
there exist µ ∈ σiso(A) and ν ∈ σiso(B) such that λ = µν. By hypotheses A and B are isoloid;
hence µ is an eigenvalue of A and ν is an eigenvalue of B. Hence µ ∈ E(A) = σa(A) \ σSBF−
+
(A)
and ν ∈ E(B) = σa(B) \ σSBF−
+
(B). Consequently, λ ∈ σa(A ⊗ B) \ σSBF−
+
(A ⊗ B); hence
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
GENERALIZED WEYL’S THEOREM AND TENSOR PRODUCT 1295
E(A ⊗ B) ⊆ σa(A ⊗ B) \ σSBF−
+
(A ⊗ B). Conversely, if λ ∈ σa(A ⊗ B) \ σSBF−
+
(A ⊗ B), then
λ 6= 0. So, there exist µ ∈ σa(A) \ σSBF−
+
(A) = E(A) and ν ∈ σa(B) \ σSBF−
+
(B) such that
λ = µν. But then λ ∈ E(A⊗B). Hence σa(A⊗B) \ σSBF−
+
(A⊗B) ⊆ E(A⊗B). Therefore, the
proof is achieved.
An operator S ∈ L(X) is said to be polaroid (respectively, a-polaroid) if σiso(S) (respectively,
σisoa (S)) is empty or every isolated point of σ(S) (respectively, σa(S)) is a pole of the resolvent.
S ∈ L(X) is polaroid implies S∗ polaroid. It is well known that if S orS∗ has SVEP and S is polaroid,
then S and S∗ satisfy generalized Weyl’s theorem. Not as well known is the fact [6] (Theorem 2.10),
that if S is polaroid and S∗ (respectively, S ) has SVEP, then S (respectively, S∗) satisfies property
(gw). Here the SVEP hypotheses on S and S∗ can not be exchanged. The following theorem is the
tensor product analogue of this result.
Theorem 2.4. Suppose that the operators A ∈ L(X) and B ∈ L(Y) are polaroid.
(i) If A∗ and B∗ have SVEP, then A⊗B satisfies property (gw).
(ii) If A and B have SVEP, then A∗ ⊗B∗ satisfies property (gw).
Proof. (i) The hypothesis A∗ and B∗ have SVEP implies
σ(A) = σa(A), σ(B) = σa(B), σSBF−
+
(A) = σBW (A), σSBF−
+
(B) = σBW (B)
and
A∗, B∗ and A∗ ⊗B∗ satisfy generalized s-Browder’s theorem.
Thus generalized s-Browder’s theorem and generalized Browder’s theorem transform from A∗ and
B∗ to A∗ ⊗B∗. Hence
σSBF−
+
(A⊗B) = σSBF+
−
(A∗ ⊗B∗) = σs(A
∗)σSBF+
−
(B∗) ∪ σSBF+
−
(A∗)σs(B
∗) =
= σa(A)σSBF−
+
(B) ∪ σSBF−
+
(A)σa(B) = σ(A)σBW (B) ∪ σBW (A)σ(B),
and
σBW (A⊗B) = σBW (A∗ ⊗B∗) = σ(A∗)σBW (B∗) ∪ σBW (A∗)σ(B∗) =
= σ(A)σBW (B) ∪ σBW (A)σ(B).
Consequently,
σSBF−
+
(A⊗B) = σBW (A⊗B).
Already,
σa(A⊗B) = σa(A)σa(B) = σ(A)σ(B) = σ(A⊗B).
Evidently, A ⊗ B is polaroid by Lemma 2 of [14]; combining this with A ⊗ B satisfies generalized
Browder’s theorem, it follows that A ⊗ B satisfies generalized Weyl’s theorem, i.e., σ(A ⊗ B) \
σBW (A⊗B) = E(A⊗B). It follows then
σa(A⊗B) \ σSBF−
+
(A⊗B) = σ(A⊗B) \ σBW (A⊗B) = E(A⊗B),
that is, A⊗B satisfies property (gw).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
1296 M. H. M. RASHID
(ii) In this case σ(A) = σa(A∗), σ(B) = σa(B∗), σBW (A∗) = σSBF−
+
(A∗), σBW (B∗) =
= σSBF−
+
(B∗, σ(A∗ ⊗ B∗) = σa(A∗ ⊗ B∗), both generalized Browder’s theorem and generalized
s-Browder’s theorem transfer from A and B to A⊗B. Hence
σSBF−
+
(A∗ ⊗B∗) = σSBF+
−
(A⊗B) = σs(A)σSBF+
−
(B) ∪ σSBF+
−
(A)σs(B) =
= σa(A∗)σSBF−
+
(B∗) ∪ σSBF−
+
(A∗)σa(B∗) = σ(A)σBW (B) ∪ σBW (A)σ(B) =
= σBW (A⊗B) = σBW (A∗ ⊗B∗).
Thus, since A∗ ⊗ B∗ polaroid and A ⊗ B) satisfies generalized Browder’s theorem imply A∗ ⊗ B∗
satisfy generalized Weyl’s theorem,
σa(A∗ ⊗B∗) \ σSBF−
+
(A∗ ⊗B∗) = σ(A∗ ⊗B∗) \ σBW (A∗ ⊗B∗) = E(A∗ ⊗B∗),
that is, A∗ ⊗B∗ satisfies property (gw).
Theorem 2.4 is proved.
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Received 06.10.11,
after revision — 26.04.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
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| id | umjimathkievua-article-2659 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:27:46Z |
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| spelling | umjimathkievua-article-26592020-03-18T19:32:05Z Generalized Weyl's theorem and tensor product Узагальнена теорема Вейля та тензорний добуток Rashid, M. H. M. Рашид, М. Х. М. We give necessary and/or sufficient conditions ensuring the passage of generalized a-Weyl theorem and property $(gw)$ from $A$ and $B$ to $A \otimes B$. Наведено необхiднi та/або достатнi умови, що гарантують поширення узагальненої а-теореми Вейля та властивостi $(gw)$ iз $A$ та $B$ на $A \otimes B$. Institute of Mathematics, NAS of Ukraine 2012-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2659 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 9 (2012); 1289-1296 Український математичний журнал; Том 64 № 9 (2012); 1289-1296 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2659/2077 https://umj.imath.kiev.ua/index.php/umj/article/view/2659/2078 Copyright (c) 2012 Rashid M. H. M. |
| spellingShingle | Rashid, M. H. M. Рашид, М. Х. М. Generalized Weyl's theorem and tensor product |
| title | Generalized Weyl's theorem and tensor product |
| title_alt | Узагальнена теорема Вейля та тензорний добуток |
| title_full | Generalized Weyl's theorem and tensor product |
| title_fullStr | Generalized Weyl's theorem and tensor product |
| title_full_unstemmed | Generalized Weyl's theorem and tensor product |
| title_short | Generalized Weyl's theorem and tensor product |
| title_sort | generalized weyl's theorem and tensor product |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2659 |
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