- Research Article
- Open Access
Perturbation Results on Semi-Fredholm Operators and Applications
© B. Abdelmoumen and H. Baklouti. 2009
- Received: 14 July 2009
- Accepted: 26 September 2009
- Published: 11 October 2009
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.
- Banach Space
- Operator Equation
- Compact Operator
- Essential Spectrum
- Fredholm Operator
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 , Grabiner proves that and are closed under commuting perturbation. In , 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 ].
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 . 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 ):
For , we define the two nonnegative quantities (see ) associated with by
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 .
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 .
For define (resp., ) to be the limit of the sequence (resp., ). For the existence of these limits see [2, Lemma ].
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.
Thus, and again by [5, Proposition ], it follows that
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
Since then By Theorem 2.2, On the other hand, (i) yields According to [17, Theorem ], we get
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 .
The following statements hold true.
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.
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
Now, we prove the following result.
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
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.
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.
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
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,
Notice that in general, the converse inclusion in (i) does not hold.
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.
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
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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