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Perturbation Results on Semi-Fredholm Operators and Applications
Journal of Inequalities and Applications volume 2009, Article number: 284526 (2009)
Abstract
We give some results concerning stability in the Fredholm operators and Browder operators set, via the concept of measure of noncompactness. Moreover, we prove some localization results on the essential spectra of bounded operators on Banach space. As application, we describe the essential spectra of weighted shift operators. Finally, we describe the spectra of polynomially compact operators, and we use the obtained results to study the solvability for operator equations in Banach spaces.
1. Introduction
Throughout this paper, denotes an infinite dimensional complex Banach space. We denote by
the space of all bounded linear operators on
The subspace of all compact operators of
is denoted by
. We write
for the null space and
for the range of
. The nullity,
of
is defined as the dimension of
and the deficiency,
of
is defined as the codimension of
in
. The set of upper (lower) semi-Fredholm operators are defined, respectively by
and
and, respectively,
We use
for the set of Fredholm operators in
, and
for the set of semi-Fredholm operators in
. If
then
is called the index of
. It is well known that the index is a continuous function on the set of semi-Fredholm operators.
Various notions of essential spectrum appear in the applications of spectral theory (see, e.g., [1, 2]). We use for the spectrum of
for Wolf essential spectrum,
for Schechter essential spectrum, and
for approximate point spectrum.
Recall that (resp.,
), the ascent (resp., the descent) of
, is the smallest nonnegative integer
such that
(resp.
). If no such
exists, then
(resp.
). The sets of upper and lower semi-Browder operators are defined, respectively by
The set of Browder operators on
is
The corresponding spectrum is defined by
We are interested in this paper (Section 2) to the study of the stability problem in Fredholm operators set and semi-Fredholm operators set. In the past few years, a lot of work has been done along these lines, [3–5] and others. A well-known fact is that is an open set. An important question is to characterize, for a given
, the class of bounded operators
on
, such that
still belongs to
. Recall that if
then
(see [2, Theorem 16.9] ). More generally, this fact holds true also for
a strictly singular operator (see [6, Proposition
.c.
]). Noncompactness measures provide advanced techniques to obtain current precise results along this line; see for example [7, 8]. By means of the Kuratowski measure, for a given
, we describe in Theorem 2.2 a class of bounded operators
on
, for which
. We should notice that, in general, the size of the perturbation depends upon
. This key-result permits to prove in Corollary 2.3 some localization results about the essential spectra
and
of bounded operators on
. Next, we investigate the stability in the semi-Browder operators set. In [9], Grabiner proves that
and
are closed under commuting perturbation. In [4], Rakočević extends this result to the perturbation classes associated with the sets of semi-Fredholm operators. In Theorem 2.4, by means of the Kuratowski measure, we characteriz for a given
, a class of bounded operators
on
, that commute with
, such that
As the corollary of this theorem we obtain the main result of Grabiner. As the application of the obtained results, we describe the essential spectra of weighted shift operators.
In Section 3, we are interested in the study of polynomially compact operators. Consider For
there exists a unique unitary polynomial
of least degree such that
is compact. This polynomial will be called the minimal polynomial of
In this section, we describe
for
with compact commutator such that
Next, we show that if there exists an analytic function
in a neighborhood of
such that
is compact, then
As application, we use the obtained results to investigate the solvability for operator equations in Banach spaces,
For
we give affirmative answer under several sufficient conditions on
This result extends the analysis started in [10, 11] and generalizes the result obtained, in case
, in [12, Theorem
].
2. Some New Properties in Fredholm Theory by Means of the Kuratowski Measure of Noncompactness
In this section, we give some results concerning the classes of Fredholm operators and Browder operators via the concept of measures of noncompactness. General definition can be found in [13]. We write for the family of all nonempty and bounded subset of
. We deal with a specific measure: the Kuratowski measure of noncompactness defined on
as follows (see [14]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ1_HTML.gif)
For , we define the two nonnegative quantities (see [15]) associated with
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ2_HTML.gif)
Let be an infinite dimensional subspace of
and let
be the natural embedding of
into
. The disc (resp., circle) with center
and radius
is denoted by
(resp.,
). We write
for the closure of
and we use
for
We start this section by some fundamental properties satisfied by and
which will be useful in the remainder of the text. For more detail, we refer to [15].
Proposition 2.1.
Let be in
Then one has the following.
(i) and
, for all
(ii)
(iii) and
(iv)
(v)If is an isomorphism, then
(vi)if and only if
(vii) and
In the following theorem we establish a stability property in the upper semi-Fredholm operators set. This result provides, in particular, an extension of Theorem in [8].
Theorem 2.2.
Let be two bounded operators on
and let
be an analytic function in a neighborhood
of
not vanishing on a connected component of
.
(i)If , then
and
Suppose moreover that the commutator and
, then one has the following.
(ii)
(iii) implies that
.
(iv) for some
implies that
Proof.
By Proposition 2.1, we have for all then
for all
in particular,
By the continuity of the index on
, we get
and this proves (i).
Now, assume that applying (i), we get
and
Let
be an open set with closure
and whose boundary
consists of finite number of simple closed curves that do not intersect, and such that
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ3_HTML.gif)
Since then there exist compact operators
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ4_HTML.gif)
Integrating along , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ5_HTML.gif)
where It is easily checked that
and
This leads to
If
then
By (2.5), we conclude that
Now, if then
yields
. Therefore, by (ii)
By the continuity of the index function on
we get
For define
(resp.,
) to be the limit of the sequence
(resp.,
). For the existence of these limits see [2, Lemma
].
Corollary 2.3.
Let be a bounded operator on
, the one has the following.
(i)
(ii)If then
(iii)If then
(iv)If then
(v)If then
Proof.
Let and suppose that
then, by Theorem 2.2(iv), we have
and
Hence, if
then
and this proves (i).
Notice that if then
and the results are all trivial. Suppose that
. For
, there exists
such that
Then, by Theorem 2.2(iv), we have
and
Hence, we get easily (ii)–(v).
2.1. Stability in the Browder and the Semi-Browder Operators
The following theorem uses the measure of noncompactness to establish stability in the semi-Browder operators set. More precisely, we have the following.
Theorem 2.4.
Suppose that and
are commuting bounded linear operators on the Banach space
Assume that
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ6_HTML.gif)
Proof.
For we have
and then, by Theorem 2.2(i),
Set
and
Since
and
are commuting, then according to [16, Theorem
], for all
there exists
such that, for all
in the disk
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ7_HTML.gif)
Hence, is a locally constant function of
on the interval
Since every locally constant function on a connected set is constant, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ8_HTML.gif)
Now, since then from [5, Proposition 1.6(i)]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ9_HTML.gif)
Thus, and again by [5, Proposition
], it follows that
Remark 2.5.
Theorem 2.4 extends the results of Grabiner [9, Theorem ]. Indeed, if
is compact, we obtain
Hence, Theorem 2.4 yields
if and only if
This proves that
is closed under commuting compact perturbation. By duality argument, we prove the closeness of
Corollary 2.6.
Let be commuting bounded operators on
Suppose that there exists
such that
(i)If then
(ii)If then
Proof.
-
(i)
Let
Since
then from Theorem 2.2,
Arguing as in the proof of Theorem 2.4, we get the result.
-
(ii)
Since
then
By Theorem 2.2,
On the other hand, (i) yields
According to [17, Theorem
], we get
Corollary 2.7.
Let be a bounded operator on
, thene one has the following.
(i)
(ii)If then
Proof.
-
(i)
For
, there exists
such that
By Corollary 2.6, we have
. The result follows since we can choose
arbitrary large.
-
(ii)
Since
then
and hence
For
, there exists
such that
. Corollary 2.6 implies that
since
.
2.2. Application: Weighted Shift Operators
Let be a bounded complex sequence. Consider the unilateral backward weighted shift operator
defined on
by
In [18, Proposition
], the authors give a localization results for the spectrum and the approximate point spectrum of unilateral backward weighted shift operator. In this section, we investigate the Wolf essential spectrum of
.
Proposition 2.8.
The following statements hold true.
(i) and
(ii)
Proof.
For the set
is finite. Consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ10_HTML.gif)
We have . Since
is finite dimensional subspace, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ11_HTML.gif)
Otherwise, by Proposition 2.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ12_HTML.gif)
Since we can choose arbitrary small, then we get (i).
We should notice that if is a cluster point for the sequence
, then
and (ii) follows from Corollary 2.3(i). If not, then
is a finite set and
is a Fredholm operator with index
More precisely,
and
, here
denotes the cardinal of
Now, by Corollary 2.3(iii), we get
, which proves the proposition.
Remark 2.9.
Notice that if converges to
, then according to Proposition 2.8, we get
and
Since
then by the continuity of the index function on
, we obtain
This is a well-known fact (see, e.g., [19, Proposition 27.7, page 139]).
In what follows, we investigate more precisely the essential spectrum of . For this end define
to be the limit set of
, that is, the set of all cluster points of the sequence
, and
to be the limit set of
for
Proposition 2.10.
Suppose that is finite, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ13_HTML.gif)
Proof.
For consider
and
We can write
For
define the operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ14_HTML.gif)
Since, and
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ15_HTML.gif)
This yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ16_HTML.gif)
Observe that is finite dimensional and then
is finite rank. Hence,
It remains to prove that,
Consider the operator
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ17_HTML.gif)
where is the corresponding un-weighted shift operator. We have
, with
being the sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ18_HTML.gif)
Observe that converges and
then
Since
then, as above,
Hence,
and this completes the proof.
Now, we prove the following result.
Theorem 2.11.
Suppose that there exists such that
is a finite set, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ19_HTML.gif)
Proof.
(by induction). For the result follows by Proposition 2.10. Let
be an integer and suppose that if
is a finite set, then (2.19) holds true. Suppose now that
is a finite set. For
and
, we consider
and
Define the sequence
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ20_HTML.gif)
Since and
for all
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ21_HTML.gif)
Observe that is a finite set and
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ22_HTML.gif)
Now, consider the sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ23_HTML.gif)
Clearly, and
Hence, by Proposition 2.8,
Since
and
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ24_HTML.gif)
Hence, we get, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ25_HTML.gif)
Since we can choose arbitrary small, then by (2.21), (2.22), and (2.25), we get (2.19).
Finally, consider the superposition of two weighted shift operators Suppose that
then, by Proposition 2.8,
By Theorem 2.2,
and
To close this section, we define a special class of bounded operators on a Banach space that presents some interesting properties. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ26_HTML.gif)
First, we observe that and
for all
Also, we notice that if
is a complex sequence that converges, then the weighted shift operator
is a nontrivial element of
Now, we prove the following result.
Proposition 2.12.
(i)For all one has
and
(ii)If is invertible, then
Proof.
We observe, by Proposition 2.1, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ27_HTML.gif)
This proves the statement (i). Again, by Proposition 2.1, for invertible, we have
This proves (ii).
As an immediate result we get, for all being in
In the following proposition we describe the essential spectra for a given
Proposition 2.13.
Let be in
and suppose that
then one has the following.
(i)
(ii)If then
(iii)If then
Proof.
According to Corollary 2.3, we have By Corollary 2.7, we get
. Since
, then we get (i). The assertion (ii) follows from Corollary 2.3(i)–(ii). For (iii), on one hand, by Corollary 2.3(iii), we have
on the other hand, the boundary
Notice that if is a complex sequence that converges to
, then by Proposition 2.13 (i),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ28_HTML.gif)
3. Fredholm Theory for Polynomially Compact Operators
In this section, we present a spectral analysis for polynomially compact operators. We begin by proving an important result about perturbation by polynomially compact operators in the general context of normed spaces. First, we make the following definition.
Definition 3.1.
Let be a normed space, let
be the minimal polynomial of
and let
We say that
and
communicate if There exists a continuous map
and
such that, for all
zero of
Theorem 3.2.
Let be two bounded operators on a normed space
with compact commutator. Suppose that
and
for all
Then
If moreover, and
communicate, then
Proof.
Since for all
then we can write
with
This yields
On the other hand
is compact, then
Writing
with
and
we conclude that
Now, consider then
Thus,
is compact and, for all
This yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ29_HTML.gif)
By the continuity of the index function on we get
constant for all
. In particular,
Remark 3.3.
Theorem 3.2 is an improvement of [12, Theorem ]. Indeed, if
is a discrete set of
then
and
communicate. In the particular case where
we have
. Therefore,
is a Fredholm operator of index zero.
We notice that if, for some then, for all
and
communicate. Hence, we obtain the following.
Corollary 3.4.
Let be two bounded operators on a normed space
with compact commutator. Suppose that
for some
If
then
and
Corollary 3.5.
Let be two commuting bounded operators on the Banach space
. Suppose that
,
and assume that
and
communicate, then
Proof.
As in the proof of Theorem 3.2, (3.1) we obtain Arguing as in the proof of Theorem 2.4, we get
. Now, by Theorem 3.2, we have
Therefore, according to [17, Theorem
] we get
The following proposition is a well-know result, see [12, 20]. Here, we present a simple proof for this fact.
Proposition 3.6.
Let and let
be the minimal polynomial of
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ30_HTML.gif)
Proof.
Since is compact, then
By [3, Theorem
],
Hence,
Let
be such that
we can write
Since
is compact and, by the minimality of
,
is not compact, then
. Hence,
Proposition 3.7.
Let be two bounded operators on
with compact commutator.
(i)
(ii)If there exists such that
then
Proof.
-
(i)
If
then
On the other hand
and
is compact. According to Theorem 3.2, there exists
such that
where
is the minimal polynomial of
Hence
where
Finally, the result follows from Proposition 3.6
-
(ii)
By (i),
Since
then
and we obtain
Notice that in general, the converse inclusion in (i) does not hold.
Example 3.8.
Consider the unweighted shift operator According to Remark 2.9, we have
the unit circle. Let
be a bounded complex sequence and let
be defined by
Suppose that
then
Consider
then
Suppose that
then, applying Theorem 3.2, we get that
is a Fredholm operator. By Proposition 3.7, we get
The index of depends on the position of
with respect to
If
then
and
communicate and by Theorem 3.2,
If we suppose that
with
then
and
is invertible. In this case
Observe that in this case,
and
do not communicate.
Theorem 3.9.
Let be a bounded operator on
Suppose that there exists an analytic function
in a neighborhood of
which does not vanish on a connected component of
such that
then
Proof.
From [3, Theorem ] , we have
Since
then
Hence,
and therefore,
is a finite set
. Write
where
and
is an analytic function with
Since
does not vanish on
then
Thus,
and
3.1. Application: Solvability of Operator Equations
In the following theorem, we treat the question of the solvability of operator equations. We will prove, under several sufficient conditions, that if the homogeneous equation only has the trivial solution
then for all
the nonhomogeneous equation
has a unique solution
, and this solution depends continuously on
Theorem 3.10.
Let be a normed space and let
be two communicating commuting bounded operators on
. Suppose that
and let
be the minimal polynomial of
Assume that
If is injective, then the inverse operator
exists and is bounded.
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_IEq560_HTML.gif)
is injective, then thus
Applying Theorem 3.2, we get
It follows that
and therefore, the operator
is surjective. Hence, the inverse operator
exists. Since
is not necessary a Banach space, we have to prove that
is bounded. Suppose that it is not so, then there exists
with
and the sequence
satisfies:
as
Set
Then
as
and
Since
and
then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F284526/MediaObjects/13660_2009_Article_1930_Equ31_HTML.gif)
Since is compact, we can choose a subsequence
such that
as
Using (3.3), we observe that
as
On the one hand,
as
On the other hand,
Hence,
which implies that
This is in contradiction with
Theorem 3.11.
Let and
be communicating, commuting operators on the Banach space
. Suppose that
and set
. Then the projection
defined by the decomposition
is compact, and the operator
is bijective.
Proof.
First we notice that by Corollary 3.5, , then
and
. Thus,
which implies that
is finite dimensional. Hence, the projection
is continuous and compact. Now, we claim that
is bijective. Let
Since
then
which implies that
Thus
Since
then
We get by iteration
On the other hand, from Theorem 3.10 applied to the operator
we conclude that
is surjective.
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Abdelmoumen, B., Baklouti, H. Perturbation Results on Semi-Fredholm Operators and Applications. J Inequal Appl 2009, 284526 (2009). https://doi.org/10.1155/2009/284526
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DOI: https://doi.org/10.1155/2009/284526