On One Method of Studying Spectral Properties of Non-selfadjoint Operators

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.


Introduction
It is remarkable that initially, the perturbation theory of selfadjoint operators was born in the works of Keldysh [1][2][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 W, we must have a representation of it by a sum of operators W = T + A. It is essential that T 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 T 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 assump-tions related to the initial operator W allow us to consider a m-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][24][25][26][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][29][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].

Preliminaries
Let C, C i , i ∈ ℕ 0 be positive real constants. We assume that the values of C can be different in various formulas but the values of C i , i ∈ ℕ 0 are certain. Everywhere further, we consider linear densely defined operators acting on a separable complex Hilbert space H. Denote by BðHÞ, the set of linear bounded operators acting in H: Denote by DðLÞ, RðLÞ, NðLÞ, the domain of definition, the range, and the inverse image of zero of the operator L accordingly. The deficiency (codimension) of RðLÞ is denoted by def L: Let PðLÞ be a resolvent set of the operator L: Denote by R L ðζÞ, ζ ∈ PðLÞ, ½R L ≔ R L ð0Þ, the resolvent of the operator L: Let λ i ðLÞ, i ∈ ℕ, denote the eigenvalues of the operator L: Suppose L is a compact operator and |L | ≔ðL * LÞ 1/2 , rð|L | Þ ≔ dim Rð|L | Þ, then the eigenvalues of the operator |L | are called singular numbers (s-numbers) of the operator L and are denoted by s i ðLÞ, i = 1, 2, ⋯; ;rð|L | Þ: If rð|L | Þ < ∞, then we put by definition s i = 0, i = rð|L | Þ + 1, 2, ⋯. According to the terminology of the monograph [33], the dimension of the root vector subspace corresponding to a certain eigenvalue λ k is called an algebraic multiplicity of the eigenvalue λ k . Let νðLÞ denote the sum of all algebraic multiplicities of the operator L. Denote by S p ðHÞ, 0 < p < ∞ the Schatten-von Neumann class, and let S ∞ ðHÞ denote the set of compact operators. By definition, put Suppose L is an operator that has a compact resolvent and s n ðR L Þ ≤ C n −μ , n ∈ ℕ, 0 ≤ μ < ∞ ; then, we denote by μðLÞ the order of the operator L in accordance with the definition given in the paper [19]. Denote by L R ≔ ðL + L * Þ/2, L I ≔ ðL − L * Þ/2i the real and the imaginary component of the operator L accordingly, and letL denote the closure of the operator L. In accordance with the terminology of the monograph [34], the set ΘðLÞ ≔ fz ∈ ℂ : z = ðLf, f Þ H , f ∈ D ðLÞ,k f k H = 1g is called numerical range of the operator L. We use the definition of the sectorial property given in [34], p.280. An operator L is called sectorial, if its numerical range belongs to a closed sector L γ ðθÞ ≔ fζ : jarg ðζ − γÞj ≤ θ < π/2g, where γ is the vertex and θ is the semiangle of the sector L γ ðθÞ. We shall say that the operator L has a positive sector if Im γ = 0, γ > 0. According to the terminology of the monograph [34], an operator L is called strictly accretive if the following relation holds A symmetric operator is called positive if the values of its quadratic form are nonnegative. Denote by H L , k·k L the energetic space generated by the operator L and the norm on this space, respectively (see [35,36]). In accordance with the denotation of the paper [34], we consider a sesquilinear form t½·, · defined on a linear manifold of the Hilbert space H (further we use the term form). Denote by t½· the quadratic form corresponding to the sesquilinear form t½·, · . Let Re t = ðt + t * Þ/2, Im t = ðt − t * Þ/ 2i be the real and imaginary component of the form t, respectively, where t * ½u, v = t ½v, u, Dðt * Þ = DðtÞ. According to these definitions, we have Re t½· = Re t½·, Im t½· = Im t½·. Denote byt the closure of the form t. The range of a quadratic form t½ f , f ∈ DðtÞ, k f k H = 1 is called range of the sesquilinear form t and is denoted by ΘðtÞ. A form t is called sectorial if its range belongs to a sector having the vertex γ situated at the real axis and the semiangle 0 ≤ θ < π/2. Suppose l is a closed sectorial form; then, a linear manifold D ′ ⊂ DðlÞ is called core of l if the restriction of l to D ′ has the closure l. Due to Theorem 2.7 [34], p.323, there exist unique m-sectorial operators L l , L Re l , associated with the closed sectorial forms l, Re l, respectively. The operator L Re l is called a real part of the operator L l and is denoted by Re L l . Suppose L is a sectorial densely defined operator and k½u, v ≔ ðLu, vÞ H , DðkÞ = DðLÞ ; then, due to Theorem 1.27 [34], p.318, the form k is closable, due to Theorem 2.7 [34], p.323 there exists a unique m-sectorial operator Tk associated with the formk: In accordance with the definition [34], p.325 the operator Tk is called a Friedrichs extension of the operator L: Further, if it is not stated otherwise we use the notations of the monographs [32][33][34]. Consider a pair of complex Hilbert spaces H, H + such that This denotation implies that we have a bounded embedding provided by the inequality Due to these conditions, it is easy to prove that the operators W, W R are closeable (see Theorem 4 [34], p.268). Denote byW R the closure of the operator W R . To make some formulas readable, we also use the following form of notation:

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 operatorsW,W + have a positive sector.
Proof. Due to inequalities (3) and (4), we conclude that the operator W is strictly accretive, i.e.
Let us prove that the operatorW 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 DðWÞ ⊂ H + . Note that we have the estimate Using inequality (4, ii), the Young inequality, we get where f = u + i v. Consider I 2 . Applying the Cauchy Schwartz inequality and inequality (4, iv), we obtain for arbitrary positive ε: Hence Finally, we have the following estimate Thus, we conclude that the next inequality holds for arbitrary k > 0: Using the continuity property of the inner product, we can extend the previous inequality to the set DðWÞ. It follows easily that The previous inequality implies that the numerical range of the operatorW belongs to the sector L γ ðθÞ with the vertex situated at the point γ and the semiangle θ = arctan ð1/kÞ. Solving system of equation (15) relative to ε, we obtain the positive root ξ corresponding to the value γ = 0 and the following description for the coordinates of the sector vertex γ: 3 Abstract and Applied Analysis It follows that the operatorW has a positive sector. The proof corresponding to the operatorW + follows from the reasoning given above if we note that W + is formal adjoint with respect to W: The operatorsW,W + are m-accretive; their resolvent sets contain the half-plane fζ : ζ ∈ ℂ, Re ζ < C 0 g: Proof. Due to Lemma 1, we know that the operatorW has a positive sector, i.e., the numerical range ofW belongs to the sector L γ ðθÞ, γ > 0. In consequence to Theorem 3.2 [34], p.268, we have ∀ζ ∈ ℂ \ L γ ðθÞ, the set RðW − ζÞ is a closed space, and the next relation holds: Due to Theorem 3.2 [34], p.268, the inverse operator ðW + ζÞ −1 is defined on the subspace RðW + ζÞ, Reζ > 0.
Proof. It is obvious that W R is a symmetric operator. Due to the continuity property of the inner product, we can conclude thatW R is symmetric, too. Hence, ΘðW R Þ ⊂ ℝ. By virtue of (7), we have Using inequality (3) and the continuity property of the inner product, we obtaiñ It implies thatW R is strictly accretive. In the same way as in the proof of Lemma 2, we come to conclusion thatW R is m-accretive. Moreover, we obtain the relation def ðW R − ζÞ = 0, Im ζ ≠ 0. Hence, by virtue of Theorem 3.16 [34], p.271, the operatorW R is selfadjoint. Proof. First, note that due to Lemma 3, the operatorW R is selfadjoint. Using (27), we obtain the estimates where H ≔W R . Since H + ⊂ ⊂H, then we conclude that each set bounded with respect to the energetic norm generated by the operatorW R is compact with respect to the norm k·k H . Hence, in accordance with the theorem in [35], p.216, we 4 Abstract and Applied Analysis conclude thatW R 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 thatW R 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 DðWÞ ⊂ H + . In the same way, we can deduce that DðW R Þ ⊂ H + . By virtue of Lemmas 1 and 3, it is easy to prove that the sesquilinear forms t, h are sectorial. Applying Theorem 1.27 [34], p.318, we conclude that these forms are closable. Now, note that Ret is a sum of two closed sectorial forms. Hence, in consequence to Theorem 1.31 [34], p.319, we conclude that Ret is a closed form. Let us show that Ret =h. First, note that this equality is true on the elements of the linear manifold M ⊂ H + . This fact can be obtained from the following obvious relations: On the other hand Hence Using (4), we get where Proof. It was shown in the proof of Theorem 4 that H = Rẽ W. Hence, in consequence to Lemmas 1, 2, and Theorem 3.2 [34], p.337, there exist the selfadjoint operators B i ≔ fB i ∈ BðHÞ,kB i k≤ tan θg, i = 1, 2 (where θ is the semiangle of the sector L 0 ðθÞ ⊃ ΘðWÞ) such that Since the set of linear operators generates ring, it follows that ð36Þ Consequently Let us show that B 1 = −B 2 . In accordance with Lemma 3, the operator H is m-accretive; hence, we have ðH + ζÞ −1 ∈ BðHÞ, Re ζ > 0. Using this fact, we get Applying inequality (27), we obtain It implies that Applying formula (3.45) [34], p.282, and taking into account that H 1/2 is selfadjoint, we get Since in accordance with Theorem 3.35 [34], p.281, the set DðHÞ is the core of the operator H 1/2 , then we can extend (42) to Hence, NðH 1/2 Þ = 0. Combining this fact and (37), we obtain Let us show that the set M is a core of the operator H 1/2 . Note that due to Theorem 3.35 [34], p.281, the operator H 1/2 is selfadjoint, and DðHÞ is a core of the operator H 1/2 . Hence, we have the representation To achieve our aim, it is sufficient to show the following: Since in accordance with the definition the set M is a core of H, then we can extend second relation (33) to Using lower estimate (47) and the fact that DðHÞ is a core of H 1/2 , it is not hard to prove that DðH 1/2 Þ ⊂ H + . Taking into account this fact and using upper estimate (47), we obtain (46). It implies that M is a core of H 1/2 . Note that in accordance with Theorem 3.35 [34], p.281, the operator H 1/2 is m-accretive. Hence, combining Theorem 3.2 [34], p.268, with (43), we obtain RðH 1/2 Þ = H. Taking into account that M is a core of the operator H 1/2 , we conclude that RðH 1/2 Þ is dense in H, where H 1/2 is the restriction of the operator H 1/2 to M. Finally, by virtue of (44), we conclude that the sum B 1 + B 2 equal to zero on the dense subset of H. Since these operators are defined on H and bounded, then B 1 = − B 2 . Further, we use the denotation B 1 ≔ B.
Note that due to Lemma 2, there exist the operators RW, RW+. Using the properties of the operator B, we get It implies that the operators I ± iB are invertible. Since it was proved above that RðH 1/2 Þ = H, NðH 1/2 Þ = 0, then there exists an operator H −1/2 defined on H. Using representation (35) and taking into account the reasonings given above, we obtain Note that the following equality can be proved easily R * W = RW+. Hence, we have Combining (49) and (50), we get Using the obvious identity ðI + B 2 Þ = ðI + iBÞðI − iBÞ = ðI − iBÞðI + iBÞ, by direct calculation, we get Combining (51) and (52), we obtain . Using the uniqueness property of the square root, we obtain H −1/2 =R. Let us use the shorthand notation S ≔ I + B 2 . Note that due to the obvious inequality ðkSf k H ≥ k f k H , f ∈ HÞ, the operator S −1 is bounded on the set RðSÞ. Taking into account the reasoning given above, we get 6 Abstract and Applied Analysis On the other hand, it is easy to see that Using these estimates, we have Note that due to Theorem 4, the operator R H is compact. Combining (50) with Theorem 4, we conclude that the operator V 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 R H is selfadjoint and positive, then we have λ i ðR H Þ = s i ðR H Þ, i ∈ ℕ. Note that in accordance with the facts established above, the operator H ≔W R has a discrete spectrum and a compact resolvent. Due to results represented in [40][41][42], we have an opportunity to obtain the order of the operator H in an easy way in most particular cases.
The following theorem is formulated in terms of order μ ≔ μðHÞ and devoted to the Schatten-von Neumann classification of the operator RW.

Theorem 7.
We have the following classification: Moreover, under the assumption λ n ðR H Þ ≥ C n −μ , n ∈ ℕ, we have where μ ≔ μðHÞ: Proof. Consider the case ðμ ≤ 1Þ. Since we already know that R * W = RW + , then it can easily be checked that the operator R * W RW is a selfadjoint positive compact operator. Due to the well-known fact [39], p.174, there exists the operator jRWj. By virtue of Theorem 9.2 [39], p.178, the operator jRWj is compact. Since NðjRWj 2 Þ = 0, it follows that NðjRWjÞ = 0, Hence applying Theorem [38], p.189, we conclude that the operator |RW | has an infinite set of the eigenvalues. Using condition (4, iii), we get Since we already know that the operators jRWj 2 , V are compact, then using Lemma 1.1 [33], p.45, and Theorem 5, we get Recall that by definition, we have s i ðRWÞ = λ i ð|RW | Þ. Note that the operators jRWj, jRWj 2 have the same eigenvectors. This fact can be easily proved if we note the obvious relation jRWj 2 f i = jλ i ðjRWjÞj 2 f i , i ∈ ℕ and the spectral representation for the square root of a selfadjoint positive compact operator where f i , φ i are the eigenvectors of the operators jRWj, jRWj 2 , respectively (see (10.25) [39], p.201). Hence, Combining this fact with (60), we get This completes the proof for the case ðμ ≤ 1Þ: Consider the case ðμ > 1Þ. It follows from (50) that the operator V is positive and bounded. Hence, by virtue of Lemma 8.1 [33], p.126, we conclude that for any orthonormal basis fψ i g ∞ 1 ⊂ H, the following equalities hold where fφ i g ∞ 1 is the orthonormal basis of the eigenvectors of the operator V. Due to Theorem 5, we get By virtue of Lemma 1, we get jIm ðRWψ i , ψ i Þ H j ≤ k −1 ðξÞ Re ðRWψ i , ψ i Þ H . Combining this fact with (63), we conclude that the following series is convergent Hence, by definition [33], p.125, the operator RW has a finite matrix trace. Using Theorem 8.1 [33], p.127, we get RW ∈ S 1 . This completes the proof for the case ðμ > 1Þ.

Abstract and Applied Analysis
Now, assume that λ n ðR H Þ ≥ C n −μ , n ∈ ℕ, 0 ≤ μ < ∞. Let us show that the operator V has the complete orthonormal system of the eigenvectors. Using formula (53), we get Let us prove that DðV −1 Þ ⊂ DðHÞ. Note that the set DðV −1 Þconsists of the elements f + g, where f ∈ DðWÞ, g ∈ DðW + Þ. Using representation (35), it is easy to prove that DðWÞ ⊂ DðHÞ, DðW + Þ ⊂ DðHÞ. This gives the desired result.
Taking into account the facts proven above, we get where S = I + B 2 . Since V is selfadjoint, then due to Theorem 3 [38], p.136, the operator V −1 is selfadjoint. Combining (67) with Lemma 3, we conclude that V −1 is strictly accretive.
Using these facts, we can write Since the operator H has a discrete spectrum (see Theorem 5.3 [37]), then any set bounded with respect to the norm H H is a compact set with respect to the norm H (see Theorem 4 [35], p.220). Combining this fact with (68) and Theorem 3 [35], p.216, we conclude that the operator V −1 has a discrete spectrum, i.e., it has the infinite set of the eigenvalues λ 1 ≤ λ 2 ≤ ⋯ ≤ λ i ≤ ⋯, λ i ⟶ ∞, i ⟶ ∞, and the complete orthonormal system of the eigenvectors. Now note that the operators V, V −1 have the same eigenvectors. Therefore the operator V 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 V is a basis in the space H. Let fφ i g ∞ 1 be the complete orthonormal system of the eigenvectors of the operator V, and suppose RW ∈ S p ; then, by virtue of inequalities (7.9) [33], p.123, and Theorem 5, we get We claim that μp > 1. Assuming the converse in the previous inequality, we come to the contradiction with the condition RW ∈ S p . This completes the proof.
The following theorem establishes the completeness property of the system of root vectors of the operator RW. Theorem 8. Suppose θ < πμ/2; then, the system of root vectors of the operator RW is complete, where θ is the semiangle of the sector L 0 ðθÞ ⊃ ΘðWÞ, μ ≔ μðHÞ.
Proof. Using Lemma 1, we have Therefore, ΘðRWÞ ⊂ L 0 ðθÞ. Note that the map z : ℂ ⟶ ℂ, z = 1/ζ takes each eigenvalue of the operator RW to the eigenvalue of the operatorW. It is also clear that z : L 0 ðθÞ ⟶ L 0 ðθÞ. Using the definition [33], p.302, let us consider the following set: It is easy to see that P coincides with a closed sector of the complex plane with the vertex situated at the point zero. Let us denote by ϑðRWÞ the angle of this sector. It is obvious that P ⊂ L 0 ðθÞ. Therefore, 0 ≤ ϑðRWÞ ≤ 2θ. Let us prove that 0 < ϑðRWÞ, i.e., the strict inequality holds. If we assume that ϑðRWÞ = 0, then we get e −i arg z = ς, ∀z ∈ P \ 0, where ς is a constant independent on z. In consequence to this fact, we have Im ΘðςRWÞ = 0. Hence, the operator ςRW is symmetric (see Problem 3.9 [34], p.269), and by virtue of the fact DðςRWÞ = H, one is selfadjoint. On the other hand, taking into account the equality R * W = RW + (see the proof of Theorem 5), we have ðςRW f , gÞ H = ðf , ςRW + gÞ H , f , g ∈ H. Hence, ςRW = ςRW+. In the particular case, we have ∀f ∈ H, Im f = 0 : Re ς RW f = Re ς RW + f , Im ς RW f = −Im ς RW + f . It implies that NðRWÞ ≠ 0. This contradiction concludes the proof of the fact ϑðRWÞ > 0. 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 RW is complete.
Theorem 7 is devoted to the description of s-number behavior, but questions related with asymptotic of the eigenvalues λ i ðRWÞ, i ∈ ℕ, are still relevant in our work. It is a wellknown fact that for any bounded operator with the compact imaginary component, there is a relationship between the s -numbers of the imaginary component and the eigenvalues (see [33]). Similarly, using the information on s-numbers of the real component, we can obtain an asymptotic formula for the eigenvalues λ i ðRWÞ, i ∈ ℕ. This idea is realized in the following theorem. Theorem 9. The following inequality holds: Moreover, if νðRWÞ = ∞ and the order μðHÞ ≠ 0, then the following asymptotic formula holds: Proof. Let L be a bounded operator with a compact imaginary component. Note that according to Theorem 6.1 [33], p.81, we have where ν I ðLÞ ≤ ∞ is the sum of all algebraic multiplicities corresponding to the not real eigenvalues of the operator L (see [33], p.79). It can easily be checked that By virtue of (70), we have Re λ m ðRWÞ > 0, m = 1, 2, ⋯, νðRWÞ. Combining this fact with (77), we get ν I ðiRWÞ = νðRWÞ. Taking into account the previous equality and combining (76) and (77), we obtain Note that by virtue of (70), we have Hence Combining (78), (80), we get Using (34), we complete the proof of inequality (73). Suppose νðRWÞ = ∞, μðHÞ ≠ 0, and let us prove (75). Note that for μ > 0 and for any ε > 0, we can choose p so that μp > 1, μ − ε < 1/p. Using the condition μp > 1, we obtain convergence of the series on the left side of (73). It implies that It is obvious that Taking into account (82), we obtain (75).

Applications
We begin with Definitions. Suppose Ω is a convex domain of the n-dimensional Euclidian space with the sufficient smooth boundary, L 2 ðΩÞ is a complex Lebesgue space of summable with square functions, H 2 ðΩÞ,H 1 ðΩÞ are complex Sobolev spaces, D i f ≔ ∂f /∂x i , 1 ≤ i ≤ n is the weak partial derivatives of the function f . Consider a sum of a uniformly elliptic operator and the extension of the Kipriyanov fractional differential operator of order 0 < α < 1 (see Lemma 2.5 [37]): with the following assumptions relative to the real-valued coefficients It was proved in the paper [37] that the operator L + f ≔ −D i ða ij D j f Þ + D α d− f ,DðL + Þ = DðLÞ is formal adjoint with respect to L. Note that in accordance with Theorem 2 [43], we have RðLÞ = RðL + Þ = L 2 ðΩÞ, due to the reasonings of Theorem 3.1 [44], the operators L, L + are strictly accretive. Taking into account these facts, we can conclude that the operators L, L + are closed (see Problem 5.15 [34], p.165). Consider the operator L R . Having made the absolutely analogous reasonings as in the previous case, we conclude that 9 Abstract and Applied Analysis the operator L R is closed. Applying the reasonings of Theorem 4.3 [37], we obtain that the operator L R is selfadjoint and strictly accretive. Recall that to apply the methods described in the paper [19], we must have some decomposition of the initial operator L on a sum L = T + A, where T must be an operator of a special type either a selfadjoint or a normal operator. Note that the uniformly elliptic operator of second-order is neither selfadjoint no normal in the general case. To demonstrate the significance of the method obtained in this paper, we would like to note that a search for a convenient decomposition of L on a sum of a selfadjoint operator and some operator does not seem to be a reasonable way. Now, to justify this claim, we consider one of possible decompositions of L on a sum. Consider a selfadjoint strictly accretive operator T : H ⟶ H.

Definition 10.
In accordance with the definition of the paper [19], a quadratic form a ≔ a½ f is called a T -subordinated form if the following condition holds: where t½ f = kT 1/2 k 2 H , f ∈ DðT 1/2 Þ. The form a is called a completely T -subordinated form if, besides of (86), we have the following additional condition ∀ε > 0∃b, Let us consider the trivial decomposition of the operator L on the sum L = 2L R − L + and let us use the notation T ≔ 2L R , A ≔ −L + . Then, we have L = T + A. Due to the sectorial property proven in Theorem 4.2 [37], we have where 0 ≤ θ f ≤ θ, θ f ≔ jarg ðL + f , f Þ L 2 j, L 2 ≔ L 2 ðΩÞ and θ is the semiangle corresponding to the sector L 0 ðθÞ. Due to Theorem 4.3 [37], the operator T is m-accretive. Hence, in consequence to Theorem 3.35 [34], p.281, we conclude that DðT Þ is a core of the operator T 1/2 . It implies that we can extend relation (87) to where a is a quadratic form generated by A and t½ f = H . If we consider the case 0 < θ < π/3, then it is obvious that there exist constants b < 1 and M > 0 such that the following inequality holds: Hence, the form a is a T -subordinated form. In accordance with the definition given in the paper [19], it means T -subordination of the operator A in the sense of form.
Assume that ∀ε > 0∃b, M > 0 : b < ε. Using inequality (88), we get Using the strictly accretive property of the operator L (see inequality (4.9) [37]), we obtain On the other hand, using the results of the paper [37], it is easy to prove that H 1 0 ðΩÞ ⊂ DðtÞ. Taking into account the facts considered above, we get but as it is well known, this inequality is not true. This contradiction shows us that the form a is not a completely T -subordinated form. It implies that we cannot use Theorem 8.4 [19] which could give us an opportunity to describe the spectral properties of the operator L. Note that the reasonings corresponding to another trivial decomposition of L on a sum are analogous. This rather particular example does not aim at showing the inability of using remarkable methods considered in the paper [19] but only creates prerequisite for some value of another method based on using spectral properties of the real component of the initial operator L. Now, we would like to demonstrate the effectiveness of this method. Suppose H ≔ L 2 ðΩÞ, H + ≔ H 1 0 ðΩÞ, T f ≔ −D j ða ij D i f Þ, Af ≔ D α 0+ f , DðTÞ, DðAÞ = H 2 ðΩÞ ∩ H 1 0 ðΩÞ; then, due to the Rellich-Kondrachov theorem, we conclude that condition (2) is fulfilled. Due to the results obtained in the paper [37], we conclude that condition (4) is fulfilled. Applying the results obtained in the paper [37], we conclude that the operator L R has nonzero order. Hence, we can apply the abstract results of this paper to the operator L. In fact, Theorems 7-9 describe the spectral properties of the operator L: We deal with the differential operator acting in the complex Sobolev space and defined by the following expression where I ≔ ða, bÞ ⊂ ℝ, and the complex-valued coefficients c j ðxÞ ∈ C ðjÞ ð IÞ satisfy the condition sign ðRe c j Þ = ð−1Þ j , j = 1, 2, ⋯, k. It is easy to see that Re c j f j D α+l Further, we need the following inequalities (see [45]): where I α a+ ðL 2 Þ, I α b− ðL 2 Þ are the classes of the functions representable by the fractional integrals (see [32]). Consider the following operator with the constant real-valued coefficients: Df ≔ p n D α n a+ + q n D where α j , β j ≥ 0, 0 ≤ ½α j , ½β j < k, j = 0, 1, ⋯, n, Using (98) and (99), we get where f ∈ DðDÞ is a real-valued function and ½α j = 2m − 1, m ∈ ℕ. Similarly, we obtain for orders ½α j = 2m, m ∈ ℕ 0 p j D Thus, in both cases, we have In the same way, we obtain the inequality Hence, in the complex case, we have Hence, we obtain Now, we can formulate the main result. Consider the operator Suppose H ≔ L 2 ðIÞ, H + ≔ H k 0 ðIÞ, T ≔ L, A ≔ D; then due to the well-known fact of the Sobolev spaces theory, condition (2) is fulfilled, due to the reasonings given above, condition (4) is fulfilled. Taking into account the equality and using the method described in the paper [46], we can prove that the operatorG R has nonzero order. Hence, we can successfully apply the abstract results of this paper to the operator G. Now, it is easily seen that Theorems 7-9 describe the spectral properties of the operator G.

Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.