#### Abstract

In this paper, we explore a certain class of Non-selfadjoint operators acting on a complex separable Hilbert space. We consider a perturbation of a nonselfadjoint operator by an operator that is also nonselfadjoint. Our consideration is based on known spectral properties of the real component of a nonselfadjoint compact operator. Using a technique of the sesquilinear forms theory, we establish the compactness property of the resolvent and obtain the asymptotic equivalence between the real component of the resolvent and the resolvent of the real component for some class of nonselfadjoint operators. We obtain a classification of nonselfadjoint operators in accordance with belonging their resolvent to the Schatten-von Neumann class and formulate a sufficient condition of completeness of the root vector system. Finally, we obtain an asymptotic formula for the eigenvalues.

#### 1. Introduction

It is remarkable that initially, the perturbation theory of selfadjoint operators was born in the works of Keldysh [1–3] and had been motivated by the works of famous scientists such as Carleman [4] and Tamarkin [5]. Many papers were published within the framework of this theory over time, for instance Browder [6], Livshits [7], Mukminov [8], Glazman [9], Krein [10], Lidsky [11], Marcus [12, 13], Matsaev [14, 15], Agmon [16], Katznelson [17], and Okazawa [18]. Nowadays, there exists a huge amount of theoretical results formulated in the work of Shkalikov [19]. However, for applying these results for a concrete operator , we must have a representation of it by a sum of operators . It is essential that must be an operator of a special type either a selfadjoint or normal operator. If we consider a case where in the representation the operator is neither selfadjoint nor normal and we cannot approach the required representation in an obvious way, then it is possible to use another technique based on the properties of the real component of the initial operator. Note that in this case, the made assumptions related to the initial operator allow us to consider a -accretive operator class which was thoroughly studied by mathematicians such as Kato [20] and Okazawa [21, 22]. This is a subject to consider in the second section. In the third section, we demonstrate the significance of the obtained abstract results and consider concrete operators. Note that the relevance of such consideration is based on the following. The eigenvalue problem is still relevant for the second-order fractional differential operators. Many papers were devoted to this question, for instance the papers [23–27]. The singular number problem for the resolvent of the second-order differential operator with the Riemann-Liouville fractional derivative in the final term is considered in the paper [23]. It is proved that the resolvent belongs to the Hilbert-Schmidt class. The problem of root functions system completeness is researched in the paper [24], also a similar problem is considered in the paper [25]. We would like to study the spectral properties of some class of nonselfadjoint operators in the abstract case. Via obtained results, we study a multidimensional case corresponding to the second-order fractional differential operator; this case can be reduced to the cases considered in the papers listed above. We consider a Kipriyanov fractional differential operator, considered in detail in the papers [28–30], which presents itself as a fractional derivative in a weaker sense with respect to the approach classically known with the name of the Riemann-Liouville derivative. More precisely, in the one-dimensional case, the Kipriaynov operator coincides with the Marchaud operator, in which relationship with the Weyl and Riemann-Liouville operators is well known [31, 32].

#### 2. Preliminaries

Let be positive real constants. We assume that the values of can be different in various formulas but the values of are certain. Everywhere further, we consider linear densely defined operators acting on a separable complex Hilbert space . Denote by , the set of linear bounded operators acting in Denote by , the domain of definition, the range, and the inverse image of zero of the operator accordingly. The deficiency (codimension) of is denoted by Let be a resolvent set of the operator Denote by , the resolvent of the operator Let , denote the eigenvalues of the operator Suppose is a compact operator and , then the eigenvalues of the operator are called *singular numbers* (*s-numbers*) of the operator and are denoted by If then we put by definition . According to the terminology of the monograph [33], the dimension of the root vector subspace corresponding to a certain eigenvalue is called an *algebraic multiplicity* of the eigenvalue . Let denote the sum of all algebraic multiplicities of the operator . Denote by the Schatten-von Neumann class, and let denote the set of compact operators. By definition, put

Suppose is an operator that has a compact resolvent and then, we denote by the order of the operator in accordance with the definition given in the paper [19]. Denote by the real and the imaginary component of the operator accordingly, and let denote the closure of the operator . In accordance with the terminology of the monograph [34], the set is called *numerical range* of the operator . We use the definition of the sectorial property given in [34], p.280. An operator is called *sectorial*, if its numerical range belongs to a closed sector , where is the vertex and is the semiangle of the sector . We shall say that the operator has a positive sector if . According to the terminology of the monograph [34], an operator is called *strictly accretive* if the following relation holds . In accordance with the definition [34], p.279, an operator is called *-accretive* if the next relation holds . An operator is called *-sectorial* if is sectorial and is -accretive for some constant . An operator is called *symmetric* if one is densely defined and the next equality holds . A symmetric operator is called *positive* if the values of its quadratic form are nonnegative. Denote by the energetic space generated by the operator and the norm on this space, respectively (see [35, 36]). In accordance with the denotation of the paper [34], we consider a sesquilinear form defined on a linear manifold of the Hilbert space (further we use the term *form*). Denote by the quadratic form corresponding to the sesquilinear form . Let be the real and imaginary component of the form , respectively, where . According to these definitions, we have . Denote by the closure of the form . The range of a quadratic form is called *range* of the sesquilinear form and is denoted by . A form is called *sectorial* if its range belongs to a sector having the vertex situated at the real axis and the semiangle . Suppose is a closed sectorial form; then, a linear manifold is called *core* of if the restriction of to has the closure . Due to Theorem 2.7 [34], p.323, there exist unique -sectorial operators , associated with the closed sectorial forms , respectively. The operator is called a *real part* of the operator and is denoted by . Suppose is a sectorial densely defined operator and then, due to Theorem 1.27 [34], p.318, the form is closable, due to Theorem 2.7 [34], p.323 there exists a unique -sectorial operator associated with the form In accordance with the definition [34], p.325 the operator is called a *Friedrichs extension* of the operator

Further, if it is not stated otherwise we use the notations of the monographs [32–34]. Consider a pair of complex Hilbert spaces such that

This denotation implies that we have a bounded embedding provided by the inequality
moreover any bounded set in the space is a compact set in the space We also assume that is a dense set in We consider nonselfadjoint operators that can be represented by a sum where the operators and act on We assume that: there exists a linear manifold that is dense in the operators and their adjoint operators are defined on Further, we assume that These give us the opportunity to claim that thus, by virtue of this fact, the real component of , is defined on . Suppose the operator is the restriction of to then, the operator is called a *formal adjoint* operator with respect to . Denote by the closure of the operator . Further, we assume that the following conditions are fulfilled:

Due to these conditions, it is easy to prove that the operators are closeable (see Theorem 4 [34], p.268). Denote by the closure of the operator . To make some formulas readable, we also use the following form of notation:

#### 3. Main Results

In this section, we formulate abstract theorems that are generalizations of some particular results obtained by the author. First, we generalize Theorem 4.2 [37] establishing the sectorial property of the second-order fractional differential operator.

Lemma 1. *The operators have a positive sector .*

*Proof. *Due to inequalities (3) and (4), we conclude that the operator is strictly accretive, i.e.
Let us prove that the operator is canonical sectorial. Combining (4, ii) and (4, iii), we get
Obviously, we can extend the previous inequality to
By virtue of (8), we obtain . Note that we have the estimate
Using inequality (4, ii), the Young inequality, we get
where . Consider . Applying the Cauchy Schwartz inequality and inequality (4, iv), we obtain for arbitrary positive :
Hence
Finally, we have the following estimate
Thus, we conclude that the next inequality holds for arbitrary :
Using the continuity property of the inner product, we can extend the previous inequality to the set . It follows easily that
The previous inequality implies that the numerical range of the operator belongs to the sector with the vertex situated at the point and the semiangle . Solving system of equation (15) relative to , we obtain the positive root corresponding to the value and the following description for the coordinates of the sector vertex :
It follows that the operator has a positive sector. The proof corresponding to the operator follows from the reasoning given above if we note that is formal adjoint with respect to

Lemma 2. *The operators are -accretive; their resolvent sets contain the half-plane *

*Proof. *Due to Lemma 1, we know that the operator has a positive sector, i.e., the numerical range of belongs to the sector . In consequence to Theorem 3.2 [34], p.268, we have , the set is a closed space, and the next relation holds:
Due to Theorem 3.2 [34], p.268, the inverse operator is defined on the subspace . In accordance with the definition of -accretive operator given in the monograph [34], p.279, we need to show that
For this purpose, assume that . Using (6), we get
Since the operator has the closed range it follows that
Note that the intersection of the sets and is zero. If we assume otherwise, then applying inequality (19) for any element , we get
Hence, Thus, the intersection of the sets and is zero. It implies that
Since is a dense set in , then taking into account (3), we obtain that is a dense set in . Hence, . Combining this fact with Theorem 3.2 [34], p.268, we get . It is clear that . Let us prove that . We must notice that
By virtue of the fact , we know that the resolvent is defined. Therefore
This implies that
If we combine inequality (8) with Theorem 3.2 [34], p.268, we get . The proof corresponding to the operator is absolutely analogous.

Lemma 3. *The operator is strictly accretive, -accretive, and selfadjoint.*

*Proof. *It is obvious that is a symmetric operator. Due to the continuity property of the inner product, we can conclude that is symmetric, too. Hence, . By virtue of (7), we have
Using inequality (3) and the continuity property of the inner product, we obtain
It implies that is strictly accretive. In the same way as in the proof of Lemma 2, we come to conclusion that is -accretive. Moreover, we obtain the relation . Hence, by virtue of Theorem 3.16 [34], p.271, the operator is selfadjoint.

Theorem 4. *The operators have compact resolvents.*

*Proof. *First, note that due to Lemma 3, the operator is selfadjoint. Using (27), we obtain the estimates
where . Since , then we conclude that each set bounded with respect to the energetic norm generated by the operator is compact with respect to the norm . Hence, in accordance with the theorem in [35], p.216, we conclude that has a discrete spectrum. Note that in consequence to Theorem 5 [35], p.222, we conclude that a selfadjoint strictly accretive operator with discrete spectrum has a compact inverse operator. Thus, using Theorem 6.29 [34], p.187, we obtain that has a compact resolvent.

Further, we need the technique of the sesquilinear forms theory stated in [34]. Consider the sesquilinear forms:
Recall that due to inequality (8), we came to the conclusion that . In the same way, we can deduce that . By virtue of Lemmas 1 and 3, it is easy to prove that the sesquilinear forms are sectorial. Applying Theorem 1.27 [34], p.318, we conclude that these forms are closable. Now, note that is a sum of two closed sectorial forms. Hence, in consequence to Theorem 1.31 [34], p.319, we conclude that is a closed form. Let us show that . First, note that this equality is true on the elements of the linear manifold . This fact can be obtained from the following obvious relations:
On the other hand
Hence
Using (4), we get
where . Since , the sesquilinear forms are closed forms; then, using (33), it is easy to prove that . Using estimates (33), it is not hard to prove that is a core of the forms . Hence, using (32), we obtain . In accordance with the polarization principle (see (1.1) [34], p.309), we have . Now, recall that the forms are generated by the operators , respectively. Note that in consequence of Lemmas 1–3, these operators are -sectorial. Hence, by virtue of Theorem 2.9 [34], p.326, we get . Since we have proved that , then . Therefore, by definition, we conclude that the operator is the real part of the -sectorial operator , by symbol . Since we proved above that has a compact resolvent, then using Theorem 3.3 [34], p.337, we conclude that the operator has a compact resolvent. The proof corresponding to the operator is absolutely analogous.

Theorem 5. *The following relation holds:
*

*Proof. *It was shown in the proof of Theorem 4 that . Hence, in consequence to Lemmas 1, 2, and Theorem 3.2 [34], p.337, there exist the selfadjoint operators (where is the semiangle of the sector ) such that
Since the set of linear operators generates ring, it follows that
Consequently
Let us show that . In accordance with Lemma 3, the operator is -accretive; hence, we have . Using this fact, we get
Applying inequality (27), we obtain
It implies that
Combining this estimate with (38), we have
Applying formula (3.45) [34], p.282, and taking into account that is selfadjoint, we get
Since in accordance with Theorem 3.35 [34], p.281, the set is the core of the operator , then we can extend (42) to
Hence, . Combining this fact and (37), we obtain
Let us show that the set is a core of the operator . Note that due to Theorem 3.35 [34], p.281, the operator is selfadjoint, and is a core of the operator . Hence, we have the representation
To achieve our aim, it is sufficient to show the following:
Since in accordance with the definition the set is a core of , then we can extend second relation (33) to . Applying (45), we can write
Using lower estimate (47) and the fact that is a core of , it is not hard to prove that . Taking into account this fact and using upper estimate (47), we obtain (46). It implies that is a core of . Note that in accordance with Theorem 3.35 [34], p.281, the operator is -accretive. Hence, combining Theorem 3.2 [34], p.268, with (43), we obtain . Taking into account that is a core of the operator , we conclude that is dense in , where is the restriction of the operator to . Finally, by virtue of (44), we conclude that the sum equal to zero on the dense subset of . Since these operators are defined on and bounded, then . Further, we use the denotation .

Note that due to Lemma 2, there exist the operators . Using the properties of the operator , we get . Hence
It implies that the operators are invertible. Since it was proved above that , , then there exists an operator defined on . Using representation (35) and taking into account the reasonings given above, we obtain
Note that the following equality can be proved easily . Hence, we have
Combining (49) and (50), we get
Using the obvious identity , by direct calculation, we get
Combining (51) and (52), we obtain
Let us evaluate the form . Note that there exists the operator (see Lemma 3). Since is selfadjoint (see Lemma 3), then due to Theorem 3 [38], p.136, is selfadjoint. It is clear that is positive because is positive. Hence, by virtue of the well-known theorem (see [39], p.174) there exists a unique square root of the operator , the selfadjoint operator such that . Using the decomposition , we get . Hence, , but . It implies that . Using the uniqueness property of the square root, we obtain . Let us use the shorthand notation . Note that due to the obvious inequality , the operator is bounded on the set . Taking into account the reasoning given above, we get
On the other hand, it is easy to see that . At the same time, it is obvious that is bounded, and we have . Using these estimates, we have
Note that due to Theorem 4, the operator is compact. Combining (50) with Theorem 4, we conclude that the operator is compact. Taking into account these facts and using Lemma 1.1 [33], p.45, we obtain (34).

*Remark 6. *Since it was proved above that is selfadjoint and positive, then we have . Note that in accordance with the facts established above, the operator has a discrete spectrum and a compact resolvent. Due to results represented in [40–42], we have an opportunity to obtain the order of the operator in an easy way in most particular cases*.*

The following theorem is formulated in terms of order and devoted to the Schatten-von Neumann classification of the operator .

Theorem 7. *We have the following classification:
**Moreover, under the assumption , we have
where *

*Proof. *Consider the case . Since we already know that , then it can easily be checked that the operator is a selfadjoint positive compact operator. Due to the well-known fact [39], p.174, there exists the operator . By virtue of Theorem 9.2 [39], p.178, the operator is compact. Since , it follows that , Hence applying Theorem [38], p.189, we conclude that the operator has an infinite set of the eigenvalues. Using condition (4, iii), we get
Hence
Since we already know that the operators are compact, then using Lemma 1.1 [33], p.45, and Theorem 5, we get
Recall that by definition, we have . Note that the operators have the same eigenvectors. This fact can be easily proved if we note the obvious relation and the spectral representation for the square root of a selfadjoint positive compact operator
where are the eigenvectors of the operators , respectively (see (10.25) [39], p.201). Hence, . Combining this fact with (60), we get
This completes the proof for the case

Consider the case . It follows from (50) that the operator is positive and bounded. Hence, by virtue of Lemma 8.1 [33], p.126, we conclude that for any orthonormal basis , the following equalities hold
where is the orthonormal basis of the eigenvectors of the operator . Due to Theorem 5, we get
By virtue of Lemma 1, we get . Combining this fact with (63), we conclude that the following series is convergent
Hence, by definition [33], p.125, the operator has a finite matrix trace. Using Theorem 8.1 [33], p.127, we get . This completes the proof for the case .

Now, assume that . Let us show that the operator has the complete orthonormal system of the eigenvectors. Using formula (53), we get
Let us prove that . Note that the set consists of the elements , where . Using representation (35), it is easy to prove that . This gives the desired result. Taking into account the facts proven above, we get
where . Since is selfadjoint, then due to Theorem 3 [38], p.136, the operator is selfadjoint. Combining (67) with Lemma 3, we conclude that is strictly accretive. Using these facts, we can write
Since the operator has a discrete spectrum (see Theorem 5.3 [37]), then any set bounded with respect to the norm is a compact set with respect to the norm (see Theorem 4 [35], p.220). Combining this fact with (68) and Theorem 3 [35], p.216, we conclude that the operator has a discrete spectrum, i.e., it has the infinite set of the eigenvalues , and the complete orthonormal system of the eigenvectors. Now note that the operators have the same eigenvectors. Therefore the operator has the complete orthonormal system of the eigenvectors. Recall that any complete orthonormal system is a basis in separable Hilbert space. Hence, the complete orthonormal system of the eigenvectors of the operator is a basis in the space . Let be the complete orthonormal system of the eigenvectors of the operator , and suppose ; then, by virtue of inequalities (7.9) [33], p.123, and Theorem 5, we get
We claim that . Assuming the converse in the previous inequality, we come to the contradiction with the condition . This completes the proof.

The following theorem establishes the completeness property of the system of root vectors of the operator .

Theorem 8. *Suppose ; then, the system of root vectors of the operator is complete, where is the semiangle of the sector .*

*Proof. *Using Lemma 1, we have
Therefore, . Note that the map takes each eigenvalue of the operator to the eigenvalue of the operator . It is also clear that . Using the definition [33], p.302, let us consider the following set:
It is easy to see that coincides with a closed sector of the complex plane with the vertex situated at the point zero. Let us denote by the angle of this sector. It is obvious that . Therefore, . Let us prove that , i.e., the strict inequality holds. If we assume that , then we get , where is a constant independent on . In consequence to this fact, we have . Hence, the operator is symmetric (see Problem 3.9 [34], p.269), and by virtue of the fact , one is selfadjoint. On the other hand, taking into account the equality (see the proof of Theorem 5), we have . Hence, . In the particular case, we have . It implies that . This contradiction concludes the proof of the fact . Let us use Theorem 6.2 [33], p.305, according to which we have the following. If the following two conditions (a) and (b) are fulfilled, then the system of root vectors of the operator is complete.
(a), where (b)For some , the operator .Let us show that conditions (a) and (b) are fulfilled. Note that due to Lemma 1, we have . Hence, . It implies that there exists such that . Thus, condition (a) is fulfilled. Let us choose the certain value in condition (b) and notice that . Since the operator is selfadjoint, then we have . In consequence to Theorem 5, we obtain
Hence, to achieve condition (b), it is sufficient to show that . By virtue of the conditions , we have . Hence, we obtain . Since both conditions (a) and (b) are fulfilled, then using Theorem 6.2 [33], p.305, we complete the proof.

Theorem 7 is devoted to the description of -number behavior, but questions related with asymptotic of the eigenvalues , are still relevant in our work. It is a well-known fact that for any bounded operator with the compact imaginary component, there is a relationship between the -numbers of the imaginary component and the eigenvalues (see [33]). Similarly, using the information on -numbers of the real component, we can obtain an asymptotic formula for the eigenvalues . This idea is realized in the following theorem.

Theorem 9. *The following inequality holds:
**Moreover, if and the order , then the following asymptotic formula holds:
*

*Proof. *Let be a bounded operator with a compact imaginary component. Note that according to Theorem 6.1 [33], p.81, we have
where is the sum of all algebraic multiplicities corresponding to the not real eigenvalues of the operator