1. Preliminaries and Introduction
Let
be the Banach algebra of all bounded linear operators on a complex Banach space
We put
the identity operator, and the resolvent of an operator
is defined as
where
denotes spectrum of an operator
which is defined as
The other classical spectra of
such as the approximate point spectrum
, surjective spectrum
essential spectrum or Fredholm spectrum
, Weyl spectrum
, Browder spectrum
, Drazin spectrum
, upper semi-Fredholm spectrum
, lower semi-Fredholm spectrum
, upper semi-Browder spectrum
essential approximate point or upper semi-Weyl spectrum
and left Drazin spectrum
are defined analogously to
. In addition, we denote by
and
the ascent and descent of
, respectively. See [
1] for more details.
Some spectra mentioned above have been generalized by Berkani et al. ([
2,
3,
4]). So, we have B-Weyl spectrum
, upper semi B-Weyl spectrum
, B-Fredholm spectrum
and upper semi B-Fredholm spectrum
. Recall that the upper semi B-Browder spectrum coincides with the left Drazin invertible spectrum, similarly the B-Browder spectrum coincides with Drazin spectrum, see [
5] or [
6].
There are several studies which give the relationship between some of the above mentioned spectra, but they do not give conditions for the spectra to coincide in a non-trivial way. Furthermore, an important breakthrough in operator theory is due to the fact that the various spectral properties are defined in a special way as a function of spectra, such as the classical Weyl’s theorem. We mention some of the notations and those properties to be used in this article.
For if then denotes the complement of in i.e., , and we denote by the set of isolated points in the set
.
.
Next, we recall the definition of some spectral properties.
Definition 1. An operator is said to satisfy:
Browder’s theorem [
7]
, if a-Browder’s theorem [
8]
, say , if Weyl’s theorem [
9]
, if . Property [
10]
, if . Property [
11]
, say if
Recall that for every bounded operator
is true, see ([
12], Proposition 50.2), and an operator
is said to be a-polaroid if every point
is a pole of the resolvent of
, i.e.,
. In addition, it is denoted by
,
, the index of
is defined by
. So,
is an upper semi-Weyl operator if
and
is closed. See [
1] for more details.
Remark 1. By ([8], Theorem 3.3) for , we have that Note that by Theorems 6, 7 and 2 of [
13], we have the next theorem.
Theorem 1. Let . If , then
- 1.
.
- 2.
.
The analysis of the spectrum of an operator has seen some development thanks to a powerful property called the single-valued extension property, introduced by Dunford and Schwarz in [
14]. In fact, this property plays an important role in the Laursen and Neumann [
15] and Aiena [
1] books. Finch defines it in a local version in [
16].
Definition 2. ([
16])
. An operator is said to have the single valued extension property at (abbreviated at ) if for every open disc centered at , the only analytic function which satisfies the equationis f The operator
is said to have
if it has
at every point
Evidently, every
has
at each point of the resolvent set
From ([
1], Theorem 3.8), we have for
that:
On the other hand, given Banach spaces
and
let
be the tensor product of
and
as defined in [
17]. The tensor product between two operators is defined as follows.
Definition 3. ([
17])
. The tensor product of and on is the operator given by The study of the tensor product between two operators has influence on spectral theory. The tensor product of two operators and its stability with respect to Weyl’s and Browder’s theorem were initiated by Kubrusly and Duggal in [
18]. Then, Duggal [
19], Rashid [
20] and Rashid and Prasad [
21] continued these studies with Weyl and Browder type theorems. In addition, recently in [
22,
23] we can see a strong study linked to the tensor product of two operators.
The different spectra of the tensor product of two operators
and
are established as follows in [
17,
24] as follows:
.
.
.
.
.
.
Recall that the essential approximate point spectrum of
for
and
verifies the spectral identity if
In this paper, we consider the property
or equivalently
a-Browder’s theorem. This has been studied through the methods of the local spectral theory, through localized
, under a proper closed subspace of
and also under some topological conditions and others. So, it has a lot of influence on the development of the spectral theory because the class of operators satisfying the property
is stronger than the class of operators satisfying other properties, such as those seen in [
10,
22,
25].
In view of the many existing studies on a-Browder’s theorem, it is important to continue studying the property . Thus, for further development of operator theory, in this paper, combining local spectral theory with the properties of the tensor product of two operators, we give new results on the property . Namely:
In
Section 2, taking into account that new partitions of the spectrum of an operator are always investigated, we show that the spectrum of an operator satisfying property
can be partitioned into an essential approximate spectrum and a surjective spectrum. We also show that the spectrum of an a-polaroid operator satisfying property
coincides with its approximate point spectrum, and the essential spectrum coincides with its upper semi-Fredholm spectrum.
In
Section 3, since the question “are there infinite operators satisfying
a-Browder’s theorem?” has not been answered, we prove that
verifies property
if and only if every nonzero scalar multiple of
verifies property
In particular, the same is true for the two operators that are norm equivalent. Thus, it is observed that there exist infinite
in
verifying
a-Browder’s theorem.
In
Section 4, in view of the fact that the relationships between the different spectra are always under research, assuming that the essential approximate point spectrum verifies the spectral identity for
with
and
the necessary conditions for the equality
to hold and further, for the tensor product of two operators satisfying the property
, the Fredholm spectrum, the Weyl spectrum, the Browder spectrum, the upper Fredholm spectrum, the upper Weyl spectrum and the upper Browder spectrum to coincide with each other and the same for Berkani type spectra are investigated.
2. On Spectra and the Property (Bv)
In this section, we give partitions of the spectrum of for operators that satisfy the property The approximate point spectrum equals the spectrum, and the upper semi-Fredholm spectrum equals the essential spectrum if is a-polaroid.
Aiena [
1] has shown that the spectrum of
is the union of
and surjective spectrum, i.e.,
where
Thus, we obtain the following Theorem.
Theorem 2. If then .
Proof. Let If then This implies that has at , i.e., Hence, and thus □
By Theorems 1 and 2, it results that if the interior of is empty, then the approximate point spectrum coincides with the union of the upper semi Browder spectrum and the surjective spectrum.
Corollary 1. Let . If , then .
Next, we find a condition through Riesz operators for an operator to satisfy property .
Theorem 3. Let If , for some Riesz operator such that , then .
Proof. It is known that, see [
1],
By hypothesis we have that , whereby if then . Hence, , i.e., verifies a-Browder’s theorem, or equivalently . □
For an a-polaroid operator that satisfies property , we obtain that the approximate point spectrum becomes the spectrum, and the upper semi-Fredholm spectrum becomes the essential spectrum. Indeed,
Theorem 4. Let be an a-polaroid operator. If then
- 1.
.
- 2.
Proof. 1. By ([
26], Theorem 3.1), we have
, or equivalently
So, as
and
, we deduce that
. By Theorem 1, we have that
Hence,
.
2. As in part 1., we have that . Since verifies property , it results that On the other hand, let .
If , then and hence
If , then Therefore, is both upper semi-Fredholm and lower semi-Fredholm; hence
So, in both cases and hence □
3. Property (Bv) and Norm Equivalent Operators
For
, we define
. Clearly,
We recall that an operator
is
Drazin invertible if there exists an operator
(called the
Drazin inverse of
) and an integer
such that
Of conclusions in [
27], we deduce that for
is Drazin invertible with Drazin inverse
, it results that:
In [
28], the concept of equivalent norm operator was introduced and studied under linear operators with closed range. Two operators
and
in
are said to be norm equivalent if there exist two positive real numbers
and
such that
In particular, the condition that for all and with was studied in detail. This condition implies equivalent norm.
In this section, we show that if
are norm equivalent, then
In order to test the latter, in the following theorem we first show that a nonzero scalar multiple of an operator that verifies the property also verifies the property
Theorem 5. Let and . Then,
- 1.
- 2.
- 3.
if and only if .
Proof. 1. Let , observe that:
- (a)
.
- (b)
.
By part (b), we have that and by part (a) it turns out .
Thus, if
, then
and
is closed, whereby
and by part (b) we obtain that
is closed. Hence,
and so
, we deduce that
Similarly, it results that . Therefore, .
- 2.
In similar way to 1., we have that
Therefore, and so
- 3.
If then , so By parts 1 and 2, We obtain that equivalently Hence . Similarly implies . □
Now, we can show that if and are norm equivalent, then iff .
Theorem 6. Let such that for all and with two positive real numbers. Then, the following holds:
- 1.
if and only if .
- 2.
⇒.
Proof. 1. By ([
28], Theorem 2.7), for some
result that
. Then, the result follows by Theorem 5 part 3.
2. Since implies , so by Theorem 2, it turns out that □
By Theorem 3, we obtain the following corollary.
Corollary 2. Let If , for some Riesz operator such that , then , for each .
Remark 2. It is already known that there are several classes of operators that satisfy a-Browder’s theorem. Then, by the results of the present section, we have that: there exist infinite such that , or the same , or verify a-Browder’s theorem.
4. Property (Bv) for Tensor Product
Throughout this section, we assume that and . Thus, we focus on obtaining the conditions that . This will make it possible to see the equality between the different spectra of . In addition, with an example of one of our main results, we show an important stability result with respect to the property under tensor product. Thus, we can test the stability of the various spectra for the tensor product.
Note that the set is not equal to in general. However, we do have an inclusion.
Theorem 7. .
Proof. Let the factorization of be such that and This implies that Hence it follows from the tensor product of upper semi-Browder spectrum identity that Therefore □
Next, we show that if the spectrum coincides with the approximate point spectrum for and then the equality holds in the above Theorem.
Theorem 8. If and , then .
Proof. By Theorem 7, it is sufficient to prove . Let Then thus for any factorization it results that and As and we have that and This implies that and That is Therefore, and so . □
Hereafter in the remaining part of this section, we assume that the essential approximate point spectrum verifies the spectral identity for .
In the following theorem, we discuss the necessary conditions for so that the equality becomes .
Theorem 9. If such that and then .
Proof. Assume
, so
and
verify
a-Browder’s theorem, whereby
verifies a-Browder’s theorem, consequently
. Thus:
By Theorem 8 it results that
. Therefore,
. □
Theorem 4 allows us to obtain the same result for a-polaroid operators in the class of operators verifying property
Theorem 10. If are a-polaroid, then .
The following theorem establishes three equalities, whereby various spectral properties are satisfied for the tensor product , for example the property .
Theorem 11. If are two a-polaroid operators, then:
- 1.
.
- 2.
.
- 3.
.
Proof. 1. It is obtained by applying part (1) of Theorem 4.
2. Let By Theorem 4, we have Hence, Since the reverse inclusion is always true,
3. By hypothesis
, so
and
verify a-Browder’s Theorem; by Theorem 4, we have that
and
, thus by ([
25], Theorem 3.2), we obtain that
and
verify property
. We have assumed that
Therefore, by ([
23], Corollary 1) we obtain that
. □
The following Theorem is the main result of this section which establishes the equality between the different spectra of .
Theorem 12. If and , then:
- 1.
.
- 2.
. And
Proof. 1. By hypothesis, it results that , so by Theorem 1, we obtain that , and Hence, by Theorem 9 we obtain that and so ; by Theorem 1 we have that , or .
2. Let
So
and if
it results that
and
Since
and
, we have that
and
. This implies that
and
Hence,
. Therefore,
By hypothesis, we conclude that
. Hence,
and the results follow by ([
13],
Section 5). □
Corollary 3. If and , then .
Proof. As in the proof of part 2.-3. of Theorem 12, we obtain that , also By using Corollary 1, we obtain that . □
Example 1. If and are two left m-invertible contractions such that , then is a pole of if and only if is closed (see [29], for definition and details). On the other hand, for , it results that is closed; hence , whereby Hence, Theorem 12 applies to Example 2. If and are two quasi-nilpotent operators commuting with and , respectively. Then, by Corollaries and , of [30], we have that , , , . Hence, and . Thus, if , then Theorem 12 applies to