Perturbations of Polynomials with Operator Coefficients

Abstract

We consider polynomials whose coefficients are operators belonging to the Schatten-von Neumann ideals of compact operators in a Hilbert space. Bounds for the spectra of perturbed pencils are established. Applications to differential and difference equations are also discussed.

1. Introduction and Preliminaries

Numerous mathematical and physical problems lead to polynomial operator pencils (polynomials with operator coefficients); cf. [1] and references therein. Recently, the spectral theory of operator pencils attracts the attention of many mathematicians. In particular, in the paper [2], spectral properties of the quadratic operator pencil of Schrődinger operators on the whole real axis are studied. The author of the paper [3] establishes sufficient conditions for the finiteness of the discrete spectrum of linear pencils. The paper [4] deals with the spectral analysis of a class of second-order indefinite nonself-adjoint differential operator pencils. In that paper, a method for solving the inverse spectral problem for the Schrődinger operator with complex periodic potentials is proposed. In [5, 6], certain classes of analytic operator valued functions in a Hilbert space are studied, and bounds for the spectra of these functions are suggested. The results of papers [5, 6] are applied to second-order differential operators and functional differential equations. The paper [7] considers polynomial pencils whose coefficients are compact operators. Besides, inequalities for the sums of absolute values and real and imaginary parts of characteristic values are derived. The paper [8] is devoted to the variational theory of the spectra of operator pencils with self-adjoint operators. A Banach algebra associated with a linear operator pencil is explored in [9]. A functional calculus generated by a quadratic operator pencil is investigated in [10]. A quadratic pencil of differential operators with periodic generalized potential is considered in [11]. The fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces is explored in [12]. Certainly, we could not survey the whole subject here and refer the reader to the pointed papers and references cited therein. Note that perturbations of pencils with nonself-adjoint operator coefficients, to the best of our knowledge, were not investigated in the available literature, although in many applications, for example, in numerical mathematics and stability analysis, bounds for the spectra of perturbed pencils are very important; cf. [13]. In the present paper, we derive such bounds in the case of polynomials whose coefficients are operators belonging to the Schatten-von Neumann ideals of compact operators in a Hilbert space.

Introduce the notations. Let 𝐻 be a separable complex Hilbert space with a scalar product (⋅,⋅) and the norm ‖⋅‖=(⋅,⋅). By 𝐼 the identity operator in 𝐻 is denoted. For a linear operator 𝐴 in 𝐻, 𝐴⁻¹ is the inverse operator, 𝜎(𝐴) is the spectrum, 𝜆_𝑘(𝐴) (𝑘=1,2,…) are the eigenvalues with their multiplicities, 𝐴^∗ is the adjoint operator, ‖𝐴‖ is the operator norm, and 𝑅_𝜆(𝐴)=(𝐴−𝜆𝐼)⁻¹ is the resolvent.

SN_𝜈 (1≤𝜈<∞) is the Schatten-von Neumann ideal of compact operators 𝐾 in 𝐻 having the finite norm 𝑁_𝜈(𝐾)=[Trace(𝐾𝐾^∗)^𝜈/2]^1/𝜈.

Let 𝐴_𝑘 and 𝐴^∼_𝑘 (𝑘=1,…,𝑚<∞) be linear bounded operators in 𝐻. Consider the pencils 𝑃_𝑚(𝜆)=^𝑚_𝑘=0𝐴_𝑘𝜆^𝑚−𝑘,𝑃^∼_𝑚(𝜆)=^𝑚∼_𝑘=0𝐴_𝑘𝜆^𝑚−𝑘_∼𝐴₀=𝐴₀=𝐼. (1) A point 𝜆∈ℂ is called a regular value of 𝑃_𝑚, if 𝑃_𝑚(𝜆) is boundedly invertible. The complement of all regular points of 𝑃_𝑚 to the closed complex plane is called the spectrum of 𝑃_𝑚 and is denoted by Σ(𝑃_𝑚). Our main problem is as follows: if 𝐴_𝑘 and 𝐴^∼_𝑘 (𝑘=1,…,𝑚) are close, how close are the spectra of 𝑃_𝑚(𝜆) and 𝑃^∼_𝑚(𝜆)?

For an integer 𝑗≥1, we will say that 𝑧𝑠𝑣𝑃_𝑃^(𝑗)_𝑚^∼_𝑚∶=sup_{𝜇∈Σ(∼𝑃𝑚)}inf_{𝑠∈Σ(𝑃𝑚)}𝜇^𝑗−𝑠^𝑗 (2) is the 𝑗-spectral variation of 𝑃^∼_𝑚 with respect to 𝑃_𝑚. Let 𝑇=𝑇(𝑃_𝑚) and 𝑇^∼=𝑇(𝑃^∼_𝑚) be the operator 𝑚×𝑚-matrices defined on the the orthogonal sum 𝐻^𝑚 of 𝑚 exemplars of 𝐻 by 𝑇=−𝐴₁−𝐴₂⋯−𝐴_𝑚−1−𝐴_𝑚,𝐼0⋯000𝐼⋯00⋅⋅⋯⋅⋅00⋯𝐼0_∼𝑇=−𝐴^∼₁−𝐴^∼₂⋯−𝐴^∼_𝑚−1−𝐴^∼_𝑚.𝐼0⋯000𝐼⋯00⋅⋅⋯⋅⋅00⋯𝐼0 (3)

Lemma 1. Let all the operators 𝐴_𝑘 (𝑘=1,…,𝑚) belong to some ideal 𝐽 of compact operators. Then, all the entries of the operator matrix 𝑇^𝑚 also belong to 𝐽.

Proof. For an integer 𝜇<𝑚, consider the operator 𝑚×𝑚 matrix 𝐵_𝑚,𝜇=𝑏₁₁𝑏₁₂⋯𝑏_{1,𝑚−𝜇}⋯𝑏_1,𝑚−1𝑏_1,𝑚𝑏₂₁𝑏₂₂⋯𝑏_{2,𝑚−𝜇}⋯𝑏_2,𝑚−1𝑏_2,𝑚𝑏⋅⋅⋯⋅⋯⋅⋅_𝜇1𝑏_𝜇,2⋯𝑏_{𝜇,𝑚−𝜇}⋯𝑏_{𝜇,𝑚−1}𝑏_𝜇,𝑚𝐼0⋯0⋯000𝐼⋯0⋯00⋅⋅⋯⋅⋯⋅⋅00⋯𝐼⋯00 (4) with some operators 𝑏_𝑗𝑘. Direct calculations show that the operator matrix 𝐷_𝑚,𝜇+1∶=𝐵_𝑚,𝜇𝑇 has the form𝐷_𝑚,𝜇+1=𝑑₁₁𝑑₁₂⋯𝑑_{1,𝑚−𝜇−1}⋯𝑑_1,𝑚−1𝑑_1,𝑚𝑑₂₁𝑑₂₂⋯𝑑_{2,𝑚−𝜇−1}⋯𝑑_2,𝑚−1𝑑_2,𝑚𝑑⋅⋅⋯⋅⋯⋅⋅_𝜇1𝑑_𝜇2⋯𝑑_{𝜇,𝑚−𝜇−1}⋯𝑑_{𝑚𝑢,𝑚−1}𝑑_𝜇,𝑚𝑑_𝜇+1,1𝑑_𝜇+1,2⋯𝑑_{𝜇+1,𝑚−𝜇−1}⋯𝑑_{𝜇+1,𝑚−1}𝑑_𝜇+1,𝑚𝐼0⋯0⋯000𝐼⋯0⋯00⋅⋅⋯⋅⋯⋅⋅00⋯𝐼⋯00, (5)where 𝑑_𝑗𝑘=−𝑏_𝑗1𝐴_𝑘+𝑏_𝑗,𝑘+1,𝑑1≤𝑗≤𝜇,𝑘<𝑚_𝑗𝑚=−𝑏_𝑗1𝐴_𝑚,𝑑1≤𝑗≤𝜇_𝜇+1,𝑘=−𝐴_𝑘1≤𝑘≤𝜇. (6) But 𝑇 has the form 𝐵_𝑚,1, 𝑇² has the form 𝐵_𝑚,2, and so forth. Take 𝐵_𝑚,𝜇=𝑇^𝜇. Then, 𝐷_𝑚,𝜇+1=𝑇^𝜇+1. Denote by 𝑐^(𝜇)_𝑗𝑘 the entries of 𝑇^𝜇. Then according to (6), 𝑐^(𝜇+1)_𝑗𝑘=−𝑐^(𝜇)_𝑗1𝐴_𝑘+𝑐^(𝜇)_𝑗,𝑘+1(),𝑐𝑘<𝑚_{(𝜇+1)𝑗𝜇}=−𝑐^(𝜇)_𝑗1𝐴_𝑚,𝑐1≤𝑗≤𝜇_{(𝜇+1)𝜇+1,𝑘}=−𝐴_𝑘. (7) Thus, taking 𝜇=1,2,…,𝑚−1, we can assert that 𝑐^(𝑚)_𝑗𝑘 are linear combinations of operators 𝐴_𝑘 and their products. This proves the required result.

For linear operators 𝐴 and 𝐴^∼, we will say that 𝑠𝑣_𝐴(𝐴^∼)∶=sup_𝜇∈𝜎(^∼_𝐴)inf_{𝑠∈𝜎(𝐴)}|𝜇−𝑠| is the spectral variation of 𝐴^∼ with respect to 𝐴.

Lemma 2. One has 𝑧𝑠𝑣_𝑃^(𝑗)_𝑚(𝑃^∼_𝑚)=𝑠𝑣_𝑇^𝑗(𝑇^∼𝑗) (𝑗=1,2,…,𝑚).

Proof. As it is well known, cf. [1], the spectra of 𝑇 and 𝑃_𝑚(⋅) coincide. This implies the required result.

2. The Main Results

The norm ‖⋅‖_𝐻^𝑚 in 𝐻^𝑚 is defined by the following way: let 𝐻=∑^𝑚_𝑘=1⊕𝐻_𝑘, where 𝐻_𝑘≡𝐻 (𝑘=1,…,𝑚). Then, ‖𝑓‖²_𝐻𝑚=^𝑚_𝑘=1‖𝑓_𝑘‖²𝑓=(𝑓_𝑘∈𝐻_𝑘)^𝑚_𝑘=1∈𝐻^𝑚. (8) Put 𝑞_𝑚=‖𝑇^𝑚−𝑇^∼𝑚‖_𝐻^𝑚 and assume that, for an integer 𝑝≥1, 𝐴_𝑘∈SN_2𝑝(𝑘=1,…,𝑚). (9)

Theorem 3. Let condition (9) hold. Then, 𝑧𝑠𝑣_𝑃^(𝑚)_𝑚(𝑃^∼_𝑚)≤𝑦_𝑚,𝑝, where 𝑦_𝑚,𝑝 is the unique positive root of equation 𝑞_𝑚^𝑝−1_𝑠=0(2𝑁_2𝑝(𝑇^𝑚))^𝑠𝑧^𝑠+11exp2+(2𝑁_2𝑝(𝑇^𝑚))^2𝑝2𝑧^2𝑝=1. (10)

This result is due to Theorem 8.5.4 of [14] and Lemma 2.

Due to Corollary 8.5.5 of [14], we have 𝑧𝑠𝑣𝑃_𝑃^(𝑚)_𝑚^∼_𝑚≤Δ_𝑝,𝑚, (11) where Δ_𝑝,𝑚∶=𝑝𝑒𝑞_𝑚if2𝑁_2𝑝𝑇^𝑚≤𝑒𝑝𝑞_𝑚,2𝑁_2𝑝𝑇^𝑚ln2𝑁_2𝑝𝑇^𝑚𝑞_𝑚𝑝^−1/2𝑝if2𝑁_2𝑝𝑇^𝑚>𝑒𝑝𝑞_𝑚. (12)

Now consider perturbations of pencils with almost commuting coefficients.

To this end, put Ξ_𝑚=𝑇^∼𝑚−𝑇^𝑚 and 𝜁̂_𝑚𝑇,Ξ_𝑚∶=𝑇^𝑚Ξ_𝑚−Ξ_𝑚𝑇^𝑚+Ξ²_𝑚. (13)

Theorem 4. Let condition (9) hold. Then, 𝑧𝑠𝑣_𝑃^(𝑚)_𝑚(𝑃^∼_𝑚)≤𝑥_𝑚,𝑝, where 𝑥_𝑚,𝑝 is the unique positive root of equation 𝜁̂_𝑚𝑇,Ξ_𝑚^𝑝−1_𝑠=02𝑁_2𝑝𝑇^𝑚𝑠𝑧^𝑠+11exp2+2𝑁_2𝑝𝑇^𝑚2𝑝2𝑧^2𝑝=1. (14)

This theorem is proved in the next section. Replacing in (9) 𝑞_𝑚 by 𝜁_𝑚∶=𝜁̂_𝑚(𝑇,Ξ_𝑚), we obtain 𝑧𝑠𝑣𝑃_𝑃^(𝑚)_𝑚^∼_𝑚≤𝛿_𝑝,𝑚, (15) where 𝛿_𝑝,𝑚∶=𝑝𝑒𝜁_𝑚if2𝑁_2𝑝𝑇^𝑚≤𝑒𝑝𝜁_𝑚,2𝑁_2𝑝𝑇^𝑚ln2𝑁_2𝑝𝑇^𝑚𝜁_𝑚𝑝^−1/2𝑝if2𝑁_2𝑝𝑇^𝑚>𝑒𝑝𝜁_𝑚. (16) It is clear that Theorem 4 is sharper than Theorem 3, provided 𝜁̂_𝑚(𝑇,Ξ_𝑚)<𝑞_𝑚.

Remark 5. Put Im𝑇^𝑚=(𝑇^𝑚−(𝑇^∗)^𝑚)/2𝑖. Then according to Theorem 7.9.1 of [14], in Theorems 3 and 4, one can replace 2𝑁_2𝑝(𝑇^𝑚) by 𝛽_𝑝𝑁_2𝑝(Im𝑇^𝑚), where 𝛽_𝑝∶=21+2𝑝().exp2/3ln2 (17)

3. Proof of Theorem 4

Let 𝐴 and 𝐴^∼ be bounded linear operators in 𝐻, 𝐸=𝐴^∼−𝐴 and 𝑍=𝐴^∼𝐸−𝐸𝐴. We begin with the following result.

Lemma 6. Let a 𝜆∈ℂ be regular for both 𝐴 and 𝐴^∼. Then, the following equality holds: 𝑅𝐴_𝜆^∼−𝑅_𝜆(𝐴)=𝑅𝐴_𝜆^∼𝑍𝑅²_𝜆(𝐴)−𝐸𝑅²_𝜆(𝐴). (18)

Proof. We have 𝑅𝐴_𝜆^∼𝐴^∼𝐸−𝐸𝐴𝑅²_𝜆(𝐴)−𝐸𝑅²_𝜆(𝐴)=𝑅𝐴_𝜆^∼𝐴^∼𝐸−𝐸𝐴−𝐸𝑅²_𝜆(𝐴)=𝑅𝐴_𝜆^∼𝐴^∼𝐸−𝐸𝐴−𝐴^∼−𝐼𝜆𝐸𝑅²_𝜆(𝐴)=−𝑅𝐴_𝜆^∼(−𝐸𝜆+𝐸𝐴)𝑅²_𝜆(𝐴)=−𝑅𝐴_𝜆^∼𝐸𝑅_𝜆(𝐴)=−𝑅𝐴_𝜆^∼𝐴^∼−𝐼𝜆−(𝐴−𝐼𝜆)𝑅_𝜆(𝐴)=−𝐼−𝑅𝐴_𝜆^∼(𝐴−𝐼𝜆)𝑅_𝜆(𝐴)=𝑅𝐴_𝜆^∼−𝑅_𝜆(𝐴), (19) as claimed.

Denote 𝜂(𝐴,𝐸,𝜆)=sup_0≤𝑡≤1𝑡‖(𝐴𝐸−𝐸𝐴+𝑡𝐸²)𝑅²_𝜆(𝐴)‖.

Corollary 7. Let a 𝜆∈ℂ be regular for 𝐴 and 𝜂(𝐴,𝐸,𝜆)<1. (20) Then, 𝜆 is regular also for 𝐴^∼.

Indeed, put 𝐴_𝑡=𝐴+𝑡𝐸(𝑡∈[0,1]). Since the regular sets of operators are open, 𝜆 is regular for 𝐴_𝑡, provided 𝑡 is small enough. By Lemma 6,𝑅_𝜆𝐴_𝑡−𝑅_𝜆(𝐴)=𝑅_𝜆𝐴_𝑡(𝑡(𝐴+𝑡𝐸)𝐸−𝑡𝐸𝐴)𝑅²_𝜆(𝐴)−𝑡𝐸𝑅²_𝜆(𝐴). (21) Hence, 𝑅_𝜆𝐴_𝑡−𝑅_𝜆(𝐴)−𝑡𝐸𝑅²_𝜆(𝐴≤𝑅_𝜆)𝐴_𝑡𝑡(𝐸𝐴−𝐴𝐸)+𝑡²𝐸²𝑅²_𝜆(𝐴≤𝑅_𝜆)𝐴_𝑡𝜂(𝐴,𝐸,𝜆). (22) Thus, 𝑅_𝜆𝐴_𝑡≤𝑅_𝜆(𝐴)−𝑡𝐸𝑅²_𝜆(𝐴)().1−𝜂𝐴,𝐸,𝜆 (23) Taking 𝑡=1, we obtain the required result.

Put 𝜁(𝐴,𝐸)∶=‖𝐴𝐸−𝐸𝐴‖+‖𝐸²‖. It is clear that 𝜂(𝐴,𝐸,𝜆)≤𝜁²(𝐴,𝐸)‖𝑅²_𝜆(𝐴)‖. Now Corollary 7 implies the following.

Corollary 8. If 𝜆 is regular for 𝐴 and 𝜁(𝐴,𝐸)‖𝑅_𝜆(𝐴)‖<1, then, 𝜆 is regular also for 𝐴^∼.

Furthermore, assume that 𝐴∈SN_2𝑝𝑝=1,2,…. (24) Then due to Theorem 7.9.1 of [14], we have the estimate 𝑅_𝜆(𝐴)≤^𝑝−1_𝑠=02𝑁_2𝑝(𝐴)^𝑠𝜌^𝑠+1()1𝐴,𝜆exp2+2𝑁_2𝑝(𝐴)^2𝑝2𝜌^2𝑝(),𝐴,𝜆 (25) where 𝜌(𝐴,𝜆)=inf_{𝑠∈𝜎(𝐴)}|𝑠−𝜆|—the distance between 𝜆 and the spectrum of 𝐴. Now Corollary 8 implies the following.

Corollary 9. If condition (24) holds, 𝜆 is regular for 𝐴 and 𝜁(𝐴,𝐸)^𝑝−1_𝑠=02𝑁_2𝑝(𝐴)^𝑠𝜌^𝑠+1()1𝐴,𝜆exp2+2𝑁_2𝑝(𝐴)^2𝑝2𝜌^2𝑝(𝐴,𝜆)<1, (26) then, 𝜆 is regular also for 𝐴^∼.

Lemma 10. Let condition (24) hold. Then, 𝑠𝑣_𝐴(𝐴^∼)≤𝑧_𝑝, where 𝑧_𝑝 is the unique positive root of equation 𝜁(𝐴,𝐸)^𝑝−1_𝑠=02𝑁_2𝑝(𝐴)^𝑠𝑧^𝑠+11exp2+(2𝑁_2𝑝(𝐴))^2𝑝2𝑧^2𝑝=1. (27)

Proof. For any 𝜇∈𝜎(𝐴), due to the previous corollary, we have 𝜁(𝐴,𝐸)^𝑝−1_𝑠=02𝑁_2𝑝(𝐴)^𝑠𝜌^𝑠+11𝐴,𝜇exp2+(2𝑁_2𝑝(𝐴))^2𝑝2𝜌^2𝑝𝐴,𝜇≥1. (28) Hence, it follows that 𝜌(𝐴,𝜇)≤𝑧_𝑝. But 𝑠𝑣_𝐴(𝐴^∼)=sup_𝜇∈𝜎(^∼_𝐴)𝜌(𝐴,𝜇). We thus arrive at the required result.

The assertion of Theorem 4 follows from Lemmas 10 and 2.

4. Quadratic Pencils

In this section, 𝑚=2. So, 𝑃₂(𝑧)=𝑧²+𝐴₁𝑧+𝐴₂,𝑃^∼₂(𝑧)=𝑧²+𝐴^∼₁𝑧+𝐴^∼₂, 𝑇=𝑇𝑃₂=−𝐴₁−𝐴₂,𝑇𝐼0₂=𝐴²₁−𝐴₂𝐴₁𝐴₂−𝐴₁−𝐴_2∼,𝑇²=^∼𝐴²₁−𝐴^∼𝐴₂^∼𝐴₁^∼₂−𝐴^∼₁−𝐴^∼₂. (29) Now we can directly apply Theorems 3 and 4.

To derive bounds for the spectrum of 𝑃₂, take an operator 𝐵 commuting with 𝐴₁. For example, 𝐵=𝑐𝐴²₁ with a constant 𝑐. If it is desirable to choose 𝐵 in such a way that the norm of 𝐴₂−𝐵 is small enough. Put 𝑄(𝑧)=𝑧²+𝐴₁𝑧+𝐵 and 𝑆=−𝐴₁−𝐵𝐼0.Then,𝑆²=𝐴²₁−𝐵𝐴₁𝐵−𝐴₁−𝐵. (30) Since 𝐴₂ and 𝐵 commute, one can enumerate their eigenvalues in such a way that the eigenvalues of 𝑄(𝑧) for a fixed 𝑧 are 𝑧²+𝜆_𝑘(𝐴₁)𝑧+𝜆_𝑘(𝐵). So, Σ(𝑄)={𝑧_𝑘1(𝑄),𝑧_𝑘2(𝑄)}^∞_𝑘=1, where 𝑧_𝑘1(𝑄),𝑧_𝑘2(𝑄) are the roots of the polynomial 𝑧²+𝜆_𝑘(𝐴₁)𝑧+𝜆_𝑘(𝐵): 𝑧_𝑘1(𝑄)𝜆=−^𝑘𝐴₁2+𝜆²_𝑘𝐴₁4−𝜆_𝑘(𝐵),𝑧_𝑘2(𝑄)𝜆=−^𝑘𝐴₁2−𝜆²_𝑘𝐴₁4−𝜆_𝑘(𝐵). (31) We have 𝑇²𝑃₂−𝑆²=−𝐶𝐴₁𝐶0−𝐶, (32) where 𝐶=𝐴₂−𝐵. So ‖𝑇²(𝑃₂)−𝑆²‖_𝐻²≤𝑞_𝑆∶=‖𝐶‖(1+‖𝐴₁‖). Now inequality (11) implies the following.

Corollary 11. Let 𝐴₁,𝐵∈𝑆𝑁_2𝑝, (33) and let 𝐵 commute with 𝐴₁. Then for any nonzero 𝑠∈Σ(𝑃₂), there is a 𝑧_𝑘𝑙(𝑄) (𝑙=1,2;𝑘=1,2,…), such that |𝑠²−𝑧²_𝑘,𝑙|≤𝛿(𝑆), where 𝛿(𝑆)∶=𝑝𝑒𝑞_𝑆𝑖𝑓2𝑁_2𝑝𝑆²≤𝑒𝑝𝑞_𝑆,2𝑁_2𝑝𝑆²ln2𝑁_2𝑝𝑆²𝑞_𝑆𝑝^−1/2𝑝𝑖𝑓2𝑁_2𝑝𝑆²>𝑒𝑝𝑞_𝑆. (34)

Let 𝑟_𝑠(𝑃₂) be the spectral radius of 𝑃₂: 𝑟_𝑠(𝑃₂)∶=sup_{𝑠∈Σ(𝑃2)}|𝑠|. From the previous corollary, it follows that 𝑟²_𝑠(𝑃₂)≤𝑟²_𝑠(𝑄)+𝛿(𝑆). Besides, 𝑟_𝑠(𝑄)=sup_{𝑙=1,2;𝑘=1,2,…}𝑧_𝑘𝑙(𝑄). (35)

Corollary 12. Under condition (33), let 𝑟²_𝑠(𝑄)+𝛿(𝑆)<1. (36) Then, 𝑟_𝑠(𝑃₂)<1.

5. Difference and Differential Equations

In the present section, we briefly discuss applications of our results to difference and differential equations.

Consider the difference equation ^𝑚_𝑘=0𝐴_𝑚−𝑘𝑣𝑗+𝑘=0𝐴₀=𝐼;𝑗=𝑚,𝑚+1,…, (37) with bounded operator coefficients 𝐴_𝑘. This equation is said to be asymptotically stable, if any of its solution tends to zero as 𝑡→∞. It is not hard to check that (37) is asymptotically stable, provided 𝑟_𝑠𝑃_𝑚<1; (38) cf. [13]. Now one can use the perturbation results due to Theorems 3 and 4. For example, let 𝑚=2. So, 𝑣𝑗+2+𝐴₁𝑣𝑗+1+𝐴₂𝑣𝑗=0𝑗=2,3,…. (39) Take an operator 𝐵 commuting with 𝐴₁ as in the previous section. Recall that it is desirable to choose𝐵 such that the norm of 𝐴₂−𝐵 is small enough. Now Corollary 12 implies the following.

Corollary 13. Under conditions (33) and (36), (39) is asymptotically stable.

Furthermore, let us consider in 𝐻 the differential equation 𝑢_𝑥𝑥(𝑥)+𝐴₁𝑢_𝑥(𝑥)+𝐴₂𝑢(𝑥)=𝑓(𝑥)(0<𝑥<1), (40) where 𝑓(𝑥)=^∞_𝑘=−∞𝑓_𝑘𝑒^{2𝜋𝑖𝑘𝑥}∈𝐿²([0,1],𝐻)𝑓_𝑘∈𝐻. (41) Numerous integrodifferential equations can be written in the form of (40). Impose the periodic conditions 𝑢(0)=𝑢(1),𝑢^′(0)=𝑢^′(1). (42) We seek a solution of problem (40), (42) in the form 𝑢(𝑥)=^∞_𝑘=−∞𝑢_𝑘𝑒^{2𝜋𝑖𝑘𝑥}, (43) where 𝑢_𝑘 (𝑘=0,±1,…) should be found. Substituting this expression into (40), we obtain −4𝜋²𝑘²𝐼+2𝜋𝑖𝑘𝐴₁+𝐴₂𝑢_𝑘=𝑓_𝑘. (44) This equation has a solution provided the spectrum of 𝑃₂(𝑧) does not contain the numbers 2𝜋𝑖𝑘. Now one can apply Corollary 11.