Perturbation Results on Semi-Fredholm Operators and Applications

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]):

(2.1)

For , we define the two nonnegative quantities (see [15]) associated with by

(2.2)

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

(2.3)

Since then there exist compact operators and such that

(2.4)

Integrating along , we get

(2.5)

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

(2.6)

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

(2.7)

Hence, is a locally constant function of on the interval Since every locally constant function on a connected set is constant, then

(2.8)

Now, since then from [5, Proposition 1.6(i)]

(2.9)

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

(2.10)

We have . Since is finite dimensional subspace, then

(2.11)

Otherwise, by Proposition 2.1,

(2.12)

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

(2.13)

Proof.

For consider and We can write For define the operator by

(2.14)

Since, and then

(2.15)

This yields

(2.16)

Observe that is finite dimensional and then is finite rank. Hence, It remains to prove that, Consider the operator defined by

(2.17)

where is the corresponding un-weighted shift operator. We have , with being the sequence defined by

(2.18)

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

(2.19)

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

(2.20)

Since and for all then

(2.21)

Observe that is a finite set and Hence

(2.22)

Now, consider the sequence defined by

(2.23)

Clearly, and Hence, by Proposition 2.8, Since and then

(2.24)

Hence, we get, for

(2.25)

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

(2.26)

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

(2.27)

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),

(2.28)

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

(31)

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

(32)

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.

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

(33)

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.