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

Authors: Maksim V. Kukushkin
Journal: Abstract and Applied Analysis
 (2020)

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## 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.


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## Body

## 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 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 assumptions 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 [23–27]. 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

LetC,Ci,i∈ℕ0 be positive real constants. We assume that the values of C can be different in various formulas but the values of Ci,i∈ℕ0 are certain. Everywhere further, we consider linear densely defined operators acting on a separable complex Hilbert space H. Denote by BH, the set of linear bounded operators acting in H. Denote by DL,RL,NL, the domain of definition, the range, and the inverse image of zero of the operator L accordingly. The deficiency (codimension) of RL is denoted by defL. Let PL be a resolvent set of the operator L. Denote by RLζ,ζ∈PL,RL≔RL0, the resolvent of the operator L. Let λiL,i∈ℕ, denote the eigenvalues of the operator L. Suppose L is a compact operator and ∣L∣≔L∗L1/2,r∣L∣≔dimR∣L∣, then the eigenvalues of the operator ∣L∣ are called singular numbers (s-numbers) of the operator L and are denoted by siL,i=1,2,⋯,,r∣L∣. If r∣L∣<∞, then we put by definition si=0,i=r∣L∣+1,2,⋯. According to the terminology of the monograph [33], the dimension of the root vector subspace corresponding to a certain eigenvalue λk is called an algebraic multiplicity of the eigenvalue λk. Let νL denote the sum of all algebraic multiplicities of the operator L. Denote by SpH,0<p<∞ the Schatten-von Neumann class, and let S∞H denote the set of compact operators. By definition, put
(1)SpH≔L:H⟶H,∑i=1∞sipL<∞,0<p<∞.SupposeL is an operator that has a compact resolvent and snRL≤Cn−μ,n∈ℕ,0≤μ<∞; then, we denote by μL the order of the operator L in accordance with the definition given in the paper [19]. Denote by LR≔L+L∗/2,LI≔L−L∗/2i the real and the imaginary component of the operator L accordingly, and let L~ denote the closure of the operator L. In accordance with the terminology of the monograph [34], the set ΘL≔z∈ℂ:z=Lf,fH,f∈DL,fH=1 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 Lγθ≔ζ:argζ−γ≤θ<π/2, where γ is the vertex and θ is the semiangle of the sector Lγθ. We shall say that the operator L has a positive sector if Imγ=0,γ>0. According to the terminology of the monograph [34], an operator L is called strictly accretive if the following relation holds ReLf,fH≥CfH2,f∈DL. In accordance with the definition [34], p.279, an operator L is called m-accretive if the next relation holds A+ζ−1∈BH,A+ζ−1≤Reζ−1,Reζ>0. An operator L is called m-sectorial if L is sectorial and L+β is m-accretive for some constant β. An operator L is called symmetric if one is densely defined and the next equality holds Lf,gH=f,LgH,f,g∈DL. A symmetric operator is called positive if the values of its quadratic form are nonnegative. Denote by HL,·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 t·,· defined on a linear manifold of the Hilbert space H (further we use the term form). Denote by t· the quadratic form corresponding to the sesquilinear form t·,·. Let Ret=t+t∗/2,Imt=t−t∗/2i be the real and imaginary component of the form t, respectively, where t∗u,v=tv,u¯,Dt∗=Dt. According to these definitions, we have Ret·=Ret·,Imt·=Imt·. Denote by t~ the closure of the form t. The range of a quadratic form tf,f∈Dt,fH=1 is called range of the sesquilinear form t and is denoted by Θt. 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′⊂Dl 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 Ll,LRel, associated with the closed sectorial forms l,Rel, respectively. The operator LRel is called a real part of the operator Ll and is denoted by ReLl. Suppose L is a sectorial densely defined operator and ku,v≔Lu,vH,Dk=DL; 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 form k~. 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 [32–34]. Consider a pair of complex Hilbert spaces H,H+ such that
(2)H+⊂⊂H.This denotation implies that we have a bounded embedding provided by the inequality(3)fH≤fH+,f∈H+,moreover any bounded set in the space H+ is a compact set in the space H. We also assume that H+ is a dense set in H. We consider nonselfadjoint operators that can be represented by a sum W=T+A, where the operators T and A act on H. We assume that: there exists a linear manifold M⊂H+ that is dense in H+, the operators T,A and their adjoint operators are defined on M. Further, we assume that DW=M. These give us the opportunity to claim that DW⊂DW∗ thus, by virtue of this fact, the real component of W, is defined on M. Suppose the operator W+ is the restriction of W∗ to DW; then, the operator W+ is called a formal adjoint operator with respect to W. Denote by W~+ the closure of the operator W+. Further, we assume that the following conditions are fulfilled:
(4)iReTf,fH≥C0fH+2,iiTf,gH≤C1fH+gH+,iiiReAf,fH≥C2fH2,iv∣Af,gH∣≤C3fH+gH,f,g∈M.Due to these conditions, it is easy to prove that the operatorsW,WR are closeable (see Theorem 4 [34], p.268). Denote by W~R the closure of the operator WR. To make some formulas readable, we also use the following form of notation:
(5)V≔RW~R,H≔W~R.


## 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 operatorsW~,W~+ have a positive sector.Proof.
Due to inequalities (3) and (4), we conclude that the operator W is strictly accretive, i.e.
(6)ReWf,fH≥C0fH2,f∈DW.
Let us prove that the operatorW~ is canonical sectorial. Combining (4, ii) and (4, iii), we get
(7)ReWf,fH=ReTf,fH+ReAf,fH≥C0fH++C2fH,f∈DW.
Obviously, we can extend the previous inequality to(8)ReW~f,fH≥C0fH++C2fH,f∈DW~.
By virtue of (8), we obtain DW~⊂H+. Note that we have the estimate
(9)ImWf,fH≤ImTf,fH+ImAf,fH=I1+I2,f∈DW.
Using inequality (4, ii), the Young inequality, we get(10)I1=Tv,uH−Tu,vH≤Tv,uH+Tu,vH≤2C1uH+vH+≤C1fH+2,where f=u+iv. Consider I2. Applying the Cauchy Schwartz inequality and inequality (4, iv), we obtain for arbitrary positive ε:
(11)Av,uH≤C3vH+uH≤C321εuH2+εvH+2;Au,vH≤C321εvH2+εuH+2.
Hence(12)I2=Av,uH−Au,vH≤Av,uH∣+∣Au,vH≤C321εfH2+εfH+2.
Finally, we have the following estimate(13)ImWf,fH≤C32ε−1fH2+C32ε+C1fH+2,f∈DW.
Thus, we conclude that the next inequality holds for arbitraryk>0:
(14)ReWf,fH-kImWf,fH≥≥C0-kC32ε+C1fH+2+C2-kC32ε-1fH2,f∈DW.
Using the continuity property of the inner product, we can extend the previous inequality to the setDW~. It follows easily that
(15)ImW~−γεf,fH≤1kεReW~−γεf,fH,f∈DW~,kε=C0C32ε+C1−1,γε=C2−kεC32ε−1.
The previous inequality implies that the numerical range of the operatorW~ belongs to the sector Lγθ with the vertex situated at the point γ and the semiangle θ=arctan1/k. 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:
(16)γ≔γ<0,ε∈0,ξ,γ≥0,ε∈ξ,∞,ξ=C1C32+C0C2−C1C3.
It follows that the operatorW~ has a positive sector. The proof corresponding to the operator W~+ follows from the reasoning given above if we note that W+ is formal adjoint with respect to W.Lemma 2.
The operatorsW~,W~+ are m-accretive; their resolvent sets contain the half-plane ζ:ζ∈ℂ,Reζ<C0.Proof.
Due to Lemma1, we know that the operator W~ has a positive sector, i.e., the numerical range of W~ belongs to the sector Lγθ,γ>0. In consequence to Theorem 3.2 [34], p.268, we have ∀ζ∈ℂ\Lγθ, the set RW~−ζ is a closed space, and the next relation holds:
(17)defW~−ζ=η,η=const.
Due to Theorem 3.2 [34], p.268, the inverse operator W~+ζ−1 is defined on the subspace RW~+ζ,Reζ>0. In accordance with the definition of m-accretive operator given in the monograph [34], p.279, we need to show that
(18)defW~+ζ=0,W~+ζ−1≤Reζ−1,Reζ>0.
For this purpose, assume thatζ0∈ℂ\Lγθ,Reζ0<0. Using (6), we get
(19)Ref,W~−ζ0fH≥C0−Reζ0fH2,f∈DW~.
Since the operatorW~−ζ0 has the closed range RW~−ζ0, it follows that
(20)H=RW~−ζ0⊕RW~−ζ0⊥.
Note that the intersection of the setsM and RW~−ζ0⊥ is zero. If we assume otherwise, then applying inequality (19) for any element u∈M∩RW~−ζ0⊥, we get
(21)C0−Reζ0uH2≤Reu,W~−ζ0uH=0.
Hence,u=0. Thus, the intersection of the sets M and RW~−ζ0⊥ is zero. It implies that
(22)g,vH=0,∀g∈RW~−ζ0⊥,∀v∈M.
SinceM is a dense set in H+, then taking into account (3), we obtain that M is a dense set in H. Hence, RW~−ζ0⊥=0,defW~−ζ0=0. Combining this fact with Theorem 3.2 [34], p.268, we get defW~−ζ=0,ζ∈ℂ\Lγθ. It is clear that defW~+ζ=0,∀ζ,Reζ>0. Let us prove that W~+ζ−1≤Reζ−1,∀ζ,Reζ>0. We must notice that
(23)C0+ReζfH2≤Ref,W~+ζfH≤fHW~+ζfH,f∈DW~,Reζ>0.
By virtue of the factdefW~+ζ=0,∀ζ,Reζ>0, we know that the resolvent is defined. Therefore
(24)W~+ζ−1fH≤C0+Reζ−1fH≤Reζ−1fH,f∈H.
This implies that(25)W~+ζ−1≤Reζ−1,∀ζ,Reζ>0.
If we combine inequality (8) with Theorem 3.2 [34], p.268, we get PW~⊃ζ:ζ∈ℂ,Reζ<C0. The proof corresponding to the operator W~+ is absolutely analogous.Lemma 3.
The operatorW~R is strictly accretive, m-accretive, and selfadjoint.Proof.
It is obvious thatWR is a symmetric operator. Due to the continuity property of the inner product, we can conclude that W~R is symmetric, too. Hence, ΘW~R⊂ℝ. By virtue of (7), we have
(26)WRf,fH≥C0fH+2,f∈DW.
Using inequality (3) and the continuity property of the inner product, we obtain
(27)W~Rf,fH≥C0fH+2≥C0fH2,f∈DW~R.
It implies thatW~R is strictly accretive. In the same way as in the proof of Lemma 2, we come to conclusion that W~R is m-accretive. Moreover, we obtain the relation defW~R−ζ=0,Imζ≠0. Hence, by virtue of Theorem 3.16 [34], p.271, the operator W~R is selfadjoint.Theorem 4.
The operatorsW~R,W~,W~+ have compact resolvents.Proof.
First, note that due to Lemma3, the operator W~R is selfadjoint. Using (27), we obtain the estimates
(28)fH≥C0fH+≥C0fH,f∈HH,where H≔W~R. Since H+⊂⊂H, then we conclude that each set bounded with respect to the energetic norm generated by the operator W~R is compact with respect to the norm ·H. Hence, in accordance with the theorem in [35], p.216, we conclude that W~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 that W~R has a compact resolvent.
Further, we need the technique of the sesquilinear forms theory stated in [34]. Consider the sesquilinear forms:
(29)tf,g=W~f,gH,f,g∈DW~,hf,g=W~Rf,gH,f,g∈DW~R.
Recall that due to inequality (8), we came to the conclusion that DW~⊂H+. In the same way, we can deduce that DW~R⊂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:
(30)t~f,g=Wf,gH,t~g,f¯=W+f,gH,f,g∈M.
On the other hand(31)h~f,g=W~Rf,gH=WRf,gH,f,g∈M.
Hence(32)Ret~f,g=h~f,g,f,g∈M.
Using (4), we get
(33)C0fH+2≤Ret~f≤C4fH+2,C0fH+2≤h~f≤C4fH+2,f∈M,where C4=C1+C3. Since Ret~f=Ret~f,f∈M, the sesquilinear forms Ret~,h~ are closed forms; then, using (33), it is easy to prove that DRet~=Dh~=H+. Using estimates (33), it is not hard to prove that M is a core of the forms Ret~,h~. Hence, using (32), we obtain Ret~f=h~f,f∈H+. In accordance with the polarization principle (see (1.1) [34], p.309), we have Ret~=h~. Now, recall that the forms t~,h~ are generated by the operators W~,W~R, respectively. Note that in consequence of Lemmas 1–3, these operators are m-sectorial. Hence, by virtue of Theorem 2.9 [34], p.326, we get Tt~=W~,Th~=W~R. Since we have proved that Ret~=h~, then TRet~=W~R. Therefore, by definition, we conclude that the operator W~R is the real part of the m-sectorial operator W~, by symbol W~R=ReW~. Since we proved above that W~R has a compact resolvent, then using Theorem 3.3 [34], p.337, we conclude that the operator W~ has a compact resolvent. The proof corresponding to the operator W~+ is absolutely analogous.Theorem 5.
The following relation holds:(34)λiRH≍λiV.Proof.
It was shown in the proof of Theorem4 that H=ReW~. Hence, in consequence to Lemmas 1, 2, and Theorem 3.2 [34], p.337, there exist the selfadjoint operators Bi≔Bi∈BH,Bi≤tanθ,i=1,2 (where θ is the semiangle of the sector L0θ⊃ΘW~) such that
(35)W~=H1/2I+iB1H1/2,W~+=H1/2I+iB2H1/2.
Since the set of linear operators generates ring, it follows that(36)Hf=12H1/2I+iB1+H1/2I+iB2H1/2==12H1/2I+iB1+I+iB2H1/2==Hf+i2H1/2B1+B2H1/2f,f∈M.
Consequently(37)H1/2B1+B2H1/2f=0,f∈M.
Let us show thatB1=−B2. In accordance with Lemma 3, the operator H is m-accretive; hence, we have H+ζ−1∈BH,Reζ>0. Using this fact, we get
(38)ReH+ζ-1Hf,fH=ReH+ζ-1H+ζf,fH-ReζH+ζ-1f,fH≥≥fH2-ζ·H+ζ-1·fH2=fH21-ζ·H+ζ-1,Reζ>0,f∈DH.
Applying inequality (27), we obtain
(39)fHH+ζ−1fH≥f,H+ζ−1f≥Reζ+C0H+ζ−1fH2,f∈H.
It implies that(40)H+ζ−1≤Reζ+C0−1,Reζ>0.
Combining this estimate with (38), we have
(41)ReH+ζ−1Hf,fH≥fH21−ζReζ+C0,Reζ>0,f∈DH.
Applying formula (3.45) [34], p.282, and taking into account that H1/2 is selfadjoint, we get
(42)H1/2f,fH=1π∫0∞ζ−1/2ReH+ζ−1Hf,fHdζ≥≥fH2·C0π∫0∞ζ−1/2ζ+C0dζ=C0fH2,f∈DH.
Since in accordance with Theorem 3.35 [34], p.281, the set DH is the core of the operator H1/2, then we can extend (42) to
(43)H1/2f,fH≥C0fH2,f∈DH1/2.
Hence,NH1/2=0. Combining this fact and (37), we obtain
(44)B1+B2H1/2f=0,f∈M.
Let us show that the setM is a core of the operator H1/2. Note that due to Theorem 3.35 [34], p.281, the operator H1/2 is selfadjoint, and DH is a core of the operator H1/2. Hence, we have the representation
(45)H1/2fH2=Hf,fH,f∈DH.
To achieve our aim, it is sufficient to show the following:(46)∀f0∈DH1/2,∃fn1∞⊂M:fn→Hf0,H1/2fn→HH1/2f0.
Since in accordance with the definition the setM is a core of H, then we can extend second relation (33) to C0fH+≤Hf,fH≤C4fH+,f∈DH. Applying (45), we can write
(47)C0fH+≤H1/2fH≤C4fH+,f∈DH.
Using lower estimate (47) and the fact that DH is a core of H1/2, it is not hard to prove that DH1/2⊂H+. Taking into account this fact and using upper estimate (47), we obtain (46). It implies that M is a core of H1/2. Note that in accordance with Theorem 3.35 [34], p.281, the operator H1/2 is m-accretive. Hence, combining Theorem 3.2 [34], p.268, with (43), we obtain RH1/2=H. Taking into account that M is a core of the operator H1/2, we conclude that RH˘1/2 is dense in H, where H˘1/2 is the restriction of the operator H1/2 to M. Finally, by virtue of (44), we conclude that the sum B1+B2 equal to zero on the dense subset of H. Since these operators are defined on H and bounded, then B1=−B2. Further, we use the denotation B1≔B.
Note that due to Lemma2, there exist the operators RW~,RW~+. Using the properties of the operator B, we get I±iBfHfH≥ReI±iBf,fH=fH2,f∈H. Hence
(48)I±iBfH≥fH,f∈H.
It implies that the operatorsI±iB are invertible. Since it was proved above that RH1/2=H, NH1/2=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
(49)RW~=H−1/2I+iB−1H−1/2,RW~+=H−1/2I−iB−1H−1/2.
Note that the following equality can be proved easilyRW~∗=RW~+. Hence, we have
(50)V=12RW~+RW~+.
Combining (49) and (50), we get
(51)V=12H−1/2I+iB−1+I−iB−1H−1/2.
Using the obvious identityI+B2=I+iBI−iB=I−iBI+iB, by direct calculation, we get
(52)I+iB−1+I−iB−1=I+B2−1.
Combining (51) and (52), we obtain
(53)V=12H−1/2I+B2−1H−1/2.
Let us evaluate the formVf,fH. Note that there exists the operator RH (see Lemma 3). Since H is selfadjoint (see Lemma 3), then due to Theorem 3 [38], p.136, RH is selfadjoint. It is clear that RH is positive because H is positive. Hence, by virtue of the well-known theorem (see [39], p.174) there exists a unique square root of the operator RH, the selfadjoint operator R^ such that R^R^=RH. Using the decomposition H=H1/2H1/2, we get H−1/2H−1/2H=I. Hence, RH⊂H−1/2H−1/2, but DRH=H. It implies that RH=H−1/2H−1/2. Using the uniqueness property of the square root, we obtain H−1/2=R^. Let us use the shorthand notation S≔I+B2. Note that due to the obvious inequality SfH≥fH,f∈H, the operator S−1 is bounded on the set RS. Taking into account the reasoning given above, we get
(54)Vf,fH=H−1/2S−1H−1/2f,fH=S−1H−1/2f,H−1/2fH≤≤S−1H−1/2fHH−1/2fH≤S−1·H−1/2fH2=S−1·RHf,fH,f∈H.
On the other hand, it is easy to see thatS−1f,fH≥S−1fH2,f∈RS. At the same time, it is obvious that S is bounded, and we have S−1fH≥S−1fH,f∈RS. Using these estimates, we have
(55)Vf,fH=S−1H−1/2f,H−1/2fH≥S−1H−1/2fH2≥≥S−2·H−1/2fH2=S−2·RHf,fH,f∈H.
Note that due to Theorem4, the operator RH 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 thatRH is selfadjoint and positive, then we have λiRH=siRH,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 [40–42], 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 orderμ≔μH and devoted to the Schatten-von Neumann classification of the operator RW~.Theorem 7.
We have the following classification:(56)RW~∈Sp,p=l,l>2/μ,μ≤1,1,μ>1.
Moreover, under the assumptionλnRH≥Cn−μ,n∈ℕ, we have
(57)RW~∈Sp⇒μp>1,1≤p<∞,where μ≔μH.Proof.
Consider the caseμ≤1. Since we already know that RW~∗=RW~+, then it can easily be checked that the operator RW~∗RW~ is a selfadjoint positive compact operator. Due to the well-known fact [39], p.174, there exists the operator RW~. By virtue of Theorem 9.2 [39], p.178, the operator RW~ is compact. Since NRW~2=0, it follows that NRW~=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
(58)ReRW~f,fH≥C0RW~fH2,f∈H.
Hence(59)RW~2f,fH=RW~fH2≤C0−1ReRW~f,fH=C0−1Vf,fH,V≔RW~R.
Since we already know that the operatorsRW~2,V are compact, then using Lemma 1.1 [33], p.45, and Theorem 5, we get
(60)λiRW~2≤C0−1λiV≤Ci−μ,i∈ℕ.
Recall that by definition, we havesiRW~=λi∣RW~∣. Note that the operators RW~,RW~2 have the same eigenvectors. This fact can be easily proved if we note the obvious relation RW~2fi=λiRW~2fi,i∈ℕ and the spectral representation for the square root of a selfadjoint positive compact operator
(61)∣RW~∣f=∑i=1∞λiRW~2f,φiφi,f∈H,where fi,φi are the eigenvectors of the operators RW~,RW~2, respectively (see (10.25) [39], p.201). Hence, λi∣RW~∣=λiRW~2,i∈ℕ. Combining this fact with (60), we get
(62)∑i=1∞sipRW~=∑i=1∞λip/2RW~2≤C∑i=1∞i−μp/2.
This completes the proof for the caseμ≤1.
Consider the caseμ>1. 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 ψi1∞⊂H, the following equalities hold
(63)∑i=1∞ReRW~ψi,ψiH=∑i=1∞Vψi,ψiH=∑i=1∞Vφi,φiH,where φi1∞ is the orthonormal basis of the eigenvectors of the operator V. Due to Theorem 5, we get
(64)∑i=1∞Vφi,φiH=∑i=1∞siV≤C∑i=1∞i−μ.
By virtue of Lemma1, we get ImRW~ψi,ψiH≤k−1ξReRW~ψi,ψiH. Combining this fact with (63), we conclude that the following series is convergent
(65)∑i=1∞RW~ψi,ψiH<∞.
Hence, by definition [33], p.125, the operator RW~ has a finite matrix trace. Using Theorem 8.1 [33], p.127, we get RW~∈S1. This completes the proof for the case μ>1.
Now, assume thatλnRH≥Cn−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
(66)V−1=2H1/2I+B2H1/2,DV−1=RV.
Let us prove thatDV−1⊂DH. Note that the set DV−1consists of the elements f+g, where f∈DW~,g∈DW~+. Using representation (35), it is easy to prove that DW~⊂DH,DW~+⊂DH. This gives the desired result. Taking into account the facts proven above, we get
(67)V−1f,fH=2SH1/2f,H1/2fH≥2H1/2fH2=2Hf,fH,f∈DV−1,where S=I+B2. 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
(68)fV−1≥CfH,f∈HV−1.
Since the operatorH has a discrete spectrum (see Theorem 5.3 [37]), then any set bounded with respect to the norm HH 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 λ1≤λ2≤⋯≤λi≤⋯,λi⟶∞,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 φi1∞ be the complete orthonormal system of the eigenvectors of the operator V, and suppose RW~∈Sp; then, by virtue of inequalities (7.9) [33], p.123, and Theorem 5, we get
(69)∑i=1∞siRW~p≥∑i=1∞RW~φi,φiHp≥∑i=1∞ReRW~φi,φiHp==∑i=1∞Vφi,φiHp=∑i=1∞λiVp≥C∑i=1∞i−μp.
We claim thatμp>1. Assuming the converse in the previous inequality, we come to the contradiction with the condition RW~∈Sp. This completes the proof.The following theorem establishes the completeness property of the system of root vectors of the operatorRW~.Theorem 8.
Supposeθ<πμ/2; then, the system of root vectors of the operator RW~ is complete, where θ is the semiangle of the sector L0θ⊃ΘW~,μ≔μH.Proof.
Using Lemma1, we have
(70)∣ImRW~f,fH∣≤k−1ξReRW~f,fH,f∈H.
Therefore,ΘRW~¯⊂L0θ. Note that the map z:ℂ⟶ℂ,z=1/ζ takes each eigenvalue of the operator RW~ to the eigenvalue of the operator W~. It is also clear that z:L0θ⟶L0θ. Using the definition [33], p.302, let us consider the following set:
(71)P≔z:z=tξ,ξ∈ΘRW~¯,0≤t<∞.
It is easy to see thatP coincides with a closed sector of the complex plane with the vertex situated at the point zero. Let us denote by ϑRW~ the angle of this sector. It is obvious that P⊂L0θ. Therefore, 0≤ϑRW~≤2θ. Let us prove that 0<ϑRW~, i.e., the strict inequality holds. If we assume that ϑRW~=0, then we get e−iargz=ς,∀z∈P\0, where ς is a constant independent on z. In consequence to this fact, we have ImΘςRW~=0. Hence, the operator ςRW~ is symmetric (see Problem 3.9 [34], p.269), and by virtue of the fact DςRW~=H, one is selfadjoint. On the other hand, taking into account the equality RW~∗=RW~+ (see the proof of Theorem 5), we have ςRW~f,gH=f,ς¯RW~+gH,f,g∈H. Hence, ςRW~=ς¯RW~+. In the particular case, we have ∀f∈H,Imf=0:ReςRW~f=ReςRW~+f,ImςRW~f=−ImςRW~+f. It implies that NRW~≠0. This contradiction concludes the proof of the fact ϑRW~>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.
(a)
ϑRW~=π/d, where d>1(b)
For someβ, the operator B≔eiβRW~I:siB=oi−1/d,i⟶∞.
Let us show that conditions (a) and (b) are fulfilled. Note that due to Lemma1, we have 0≤θ<π/2. Hence, 0<ϑRW~<π. It implies that there exists 1<d<∞ such that ϑRW~=π/d. Thus, condition (a) is fulfilled. Let us choose the certain value β=π/2 in condition (b) and notice that eiπ/2RW~I=RW~R. Since the operator V≔RW~R is selfadjoint, then we have siV=λiV,i∈ℕ. In consequence to Theorem 5, we obtain
(72)siVi1/d=siViμ·i1/d−μ≤C·i1/d−μ,i∈ℕ.
Hence, to achieve condition (b), it is sufficient to show thatd>μ−1. By virtue of the conditions ϑRW~≤2θ,θ<πμ/2, we have d=π/ϑRW~≥π/2θ>μ−1. Hence, we obtain siV=oi−1/d. Since both conditions (a) and (b) are fulfilled, then using Theorem 6.2 [33], p.305, we complete the proof.Theorem7 is devoted to the description of s-number behavior, but questions related with asymptotic of the eigenvalues λiRW~,i∈ℕ, 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 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 λiRW~,i∈ℕ. This idea is realized in the following theorem.Theorem 9.
The following inequality holds:(73)∑i=1nλiRW~p≤secpθS-1∑i=1nλipRH,(74)n=1,2,⋯,νRW~,1≤p<∞.
Moreover, ifνRW~=∞ and the order μH≠0, then the following asymptotic formula holds:
(75)λiRW~=oi−μ+ε,i⟶∞,∀ε>0.Proof.
LetL be a bounded operator with a compact imaginary component. Note that according to Theorem 6.1 [33], p.81, we have
(76)∑m=1kImλmLp≤∑m=1ksmLIp,k=1,2,⋯,νIL,1≤p<∞,where νIL≤∞ 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
(77)iLI=LR,ImλmiL=ReλmL,m∈ℕ.
By virtue of (70), we have ReλmRW~>0,m=1,2,⋯,νRW~. Combining this fact with (77), we get νIiRW~=νRW~. Taking into account the previous equality and combining (76) and (77), we obtain
(78)∑m=1kReλmRW~p≤∑m=1ksmVp,k=1,2,⋯,,νRW~,V≔RW~R.
Note that by virtue of (70), we have
(79)ImλmRW~≤tanθReλmRW~,m∈ℕ.
Hence(80)λmRW~=ImλmRW~2+ReλmRW~2≤≤tan2θ+1ReλmRW~=secθReλmRW~,m∈ℕ.
Combining (78), (80), we get
(81)∑m=1kλmRW~p≤secpθ∑m=1ksmVp,k=1,2,⋯,,νRW~.
Using (34), we complete the proof of inequality (73).
SupposeνRW~=∞,μH≠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
(82)∣λiRW~∣i1/p⟶0,i⟶∞.
It is obvious that(83)∣λiRW~∣iμ−ε<∣λiRW~∣i1/p,i∈ℕ.
Taking into account (82), we obtain (75).


## 4. Applications

We begin with Definitions. SupposeΩ is a convex domain of the n-dimensional Euclidian space with the sufficient smooth boundary, L2Ω is a complex Lebesgue space of summable with square functions, H2Ω,H1Ω are complex Sobolev spaces, Dif≔∂f/∂xi,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]):
(84)Lu≔-DjaijDif+D0+αf,DL=H2Ω∩H01Ω,with the following assumptions relative to the real-valued coefficients
(85)aijQ∈C1Ω¯,aijξiξj≥aξ2,a>0.It was proved in the paper [37] that the operator L+f≔−DiaijDjf+Dd−αf,DL+=DL is formal adjoint with respect to L. Note that in accordance with Theorem 2 [43], we have RL=RL+=L2Ω, 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 LR. Having made the absolutely analogous reasonings as in the previous case, we conclude that the operator LR is closed. Applying the reasonings of Theorem 4.3 [37], we obtain that the operator LR 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≔af is called a T-subordinated form if the following condition holds:
(86)af≤btf+MfH2,Da⊃Dt,b<1,M>0,where tf=T1/2H2,f∈DT1/2. The form a is called a completely T-subordinated form if, besides of (86), we have the following additional condition ∀ε>0∃b,M>0:b<ε.Let us consider the trivial decomposition of the operatorL on the sum L=2LR−L+ and let us use the notation T≔2LR,A≔−L+. Then, we have L=T+A. Due to the sectorial property proven in Theorem 4.2 [37], we have
(87)∣Af,fL2∣=secθf∣ReAf,fL2∣=secθf12Tf,fL2,f∈DT,where 0≤θf≤θ,θf≔argL+f,fL2,L2≔L2Ω and θ is the semiangle corresponding to the sector L0θ. 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 DT is a core of the operator T1/2. It implies that we can extend relation (87) to
(88)12tf≤∣af∣≤secθ12tf,f∈Dt,where a is a quadratic form generated by A and tf=T1/2fH2. 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:
(89)af≤btf+MfL22,f∈Dt.Hence, the forma 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
(90)12tf≤εtf+MεfL22;tf≤2Mε1−2εfL22,f∈Dt,ε<1/2.Using the strictly accretive property of the operatorL (see inequality (4.9) [37]), we obtain
(91)fH012C≤tf,f∈Dt.On the other hand, using the results of the paper [37], it is easy to prove that H01Ω⊂Dt. Taking into account the facts considered above, we get
(92)fH01≤CfL2,f∈H01Ω,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≔L2Ω, H+≔H01Ω, Tf≔−DjaijDif, Af≔D0+αf, DT, DA=H2Ω∩H01Ω; 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 LR 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(93)Lf≔ckfkk+ck-1fk-1k-1+⋯+c0f,DL=H2kI∩H0kI,k∈N,where I≔a,b⊂ℝ, and the complex-valued coefficients cjx∈CjI¯ satisfy the condition signRecj=−1j,j=1,2,⋯,k. It is easy to see that
(94)ReLf,fL2I≥∑j=0kRecjfjL2I2≥CfjH0kI2,f∈DL.On the other hand(95)Lf,fL2I=∑j=0k−1jcjfj,gjL2I≤∑j=0kcjfj,gjL2I≤≤C∑j=0kfjL2IgjL2I≤fH0kIgH0kI,f∈DL.Consider the Riemann-Liouville operators of fractional differentiation of arbitrary nonnegative orderα (see [32], p.44) defined by the expressions
(96)Da+αf=ddxα+1Ia+1−αf;Db−αf=−ddxα+1Ib−1−αf,where the fractional integrals of arbitrary positive order α are defined by
(97)Ia+αfx=1Γα∫axftx−t1−αdt,Ib−αfx=1Γα∫xbftt−x1−αdt,f∈L1I.Suppose0<α<1,f∈ACl+1I¯,fja=fjb=0,j=0,1,⋯,l; then, the next formulas follow from Theorem 2.2 [32], p.46:
(98)Da+α+lf=Ia+1−αfl+1,Db−α+lf=−1l+1Ib−1−αfl+1.Further, we need the following inequalities (see [45]):
(99)ReDa+αf,fL2I≥CfL2I2,f∈Ia+αL2,ReDb−αf,fL2I≥CfL2I2,f∈Ib−αL2,where Ia+αL2,Ib−αL2 are the classes of the functions representable by the fractional integrals (see [32]). Consider the following operator with the constant real-valued coefficients:
(100)Df≔pnDa+αn+qnDb-βn+pn-1Da+αn-1+qn-1Db-βn-1+⋯+p0Da+α0+q0Db-β0,DD=H2kI∩H0kI,n∈N,where αj,βj≥0,0≤αj,βj<k,j=0,1,⋯,n,(101)qj≥0,signpj=−1αj+1/2,αj=2m−1,m∈ℕ,−1αj/2,αj=2m,m∈ℕ0.Using (98) and (99), we get
(102)pjDa+αjf,fL2I=pjddxmDa+m−1+αjf,fL2I=−1mpjIa+1−αjfm,fmL2I≥CIa+1−αjfmL2I2=CDa+αjfm−1L2I2≥Cfm−1L2I2,where f∈DD is a real-valued function and αj=2m−1,m∈ℕ. Similarly, we obtain for orders αj=2m, m∈ℕ0(103)pjDa+αjf,fL2I=pjDa+2m+αjf,fL2I=−1mpjDa+m+αjf,fmL2I=−1mpjDa+αjfm,fmL2I≥CfmL2I2.Thus, in both cases, we have(104)pjDa+αjf,fL2I≥CfsL2I2,s=αj/2.In the same way, we obtain the inequality(105)qjDb−αjf,fL2I≥CfsL2I2,s=αj/2.Hence, in the complex case, we have(106)ReDf,fL2I≥CfL2I2,f∈DD.Combining Theorem 2.6 [32], p.53, with (98), we get
(107)pjDa+αjfL2I=Ia+1−αjfαj+1L2I≤Cfαj+1L2I≤CfH0kI;qjDb−αjfL2I≤CfH0kI,f∈DD.Hence, we obtain(108)DfL2I≤CfH0kI,f∈DD.Now, we can formulate the main result. Consider the operator(109)G=L+D,DG=H2kI∩H0kI.SupposeH≔L2I,H+≔H0kI,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
(110)LRf=Reckfkk+Reck−1fk−1k−1+⋯+Rec0f,f∈DDand using the method described in the paper [46], we can prove that the operator G~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.



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