Perturbations of Polynomials with Operator Coefficients
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.  and references therein. Recently, the spectral theory of operator pencils attracts the attention of many mathematicians. In particular, in the paper , spectral properties of the quadratic operator pencil of Schrődinger operators on the whole real axis are studied. The author of the paper  establishes sufficient conditions for the finiteness of the discrete spectrum of linear pencils. The paper  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  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  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 . A functional calculus generated by a quadratic operator pencil is investigated in . A quadratic pencil of differential operators with periodic generalized potential is considered in . The fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces is explored in . 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. . 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 be a separable complex Hilbert space with a scalar product and the norm . By the identity operator in is denoted. For a linear operator in , is the inverse operator, is the spectrum, () are the eigenvalues with their multiplicities, is the adjoint operator, is the operator norm, and is the resolvent.
() is the Schatten-von Neumann ideal of compact operators in having the finite norm .
Let and () be linear bounded operators in . Consider the pencils A point is called a regular value of , if is boundedly invertible. The complement of all regular points of to the closed complex plane is called the spectrum of and is denoted by . Our main problem is as follows: if and () are close, how close are the spectra of and ?
For an integer , we will say that is the -spectral variation of with respect to . Let and be the operator -matrices defined on the the orthogonal sum of exemplars of by
Lemma 1. Let all the operators () belong to some ideal of compact operators. Then, all the entries of the operator matrix also belong to .
Proof. For an integer , consider the operator matrix with some operators . Direct calculations show that the operator matrix has the formwhere But has the form , has the form , and so forth. Take . Then, . Denote by the entries of . Then according to (6), Thus, taking , we can assert that are linear combinations of operators and their products. This proves the required result.
For linear operators and , we will say that is the spectral variation of with respect to .
Lemma 2. One has ().
Proof. As it is well known, cf. , the spectra of and coincide. This implies the required result.
2. The Main Results
The norm in is defined by the following way: let , where (). Then, Put and assume that, for an integer ,
Theorem 3. Let condition (9) hold. Then, , where is the unique positive root of equation
This result is due to Theorem of  and Lemma 2.
Due to Corollary of , we have where
Now consider perturbations of pencils with almost commuting coefficients.
To this end, put and
Theorem 4. Let condition (9) hold. Then, , where is the unique positive root of equation
This theorem is proved in the next section. Replacing in (9) by , we obtain where It is clear that Theorem 4 is sharper than Theorem 3, provided .
Remark 5. Put . Then according to Theorem of , in Theorems 3 and 4, one can replace by , where
3. Proof of Theorem 4
Let and be bounded linear operators in , and . We begin with the following result.
Lemma 6. Let a be regular for both and . Then, the following equality holds:
Proof. We have as claimed.
Corollary 7. Let a be regular for and Then, is regular also for .
Indeed, put . Since the regular sets of operators are open, is regular for , provided is small enough. By Lemma 6, Hence, Thus, Taking , we obtain the required result.
Put . It is clear that . Now Corollary 7 implies the following.
Corollary 8. If is regular for and , then, is regular also for .
Furthermore, assume that Then due to Theorem of , we have the estimate where —the distance between and the spectrum of . Now Corollary 8 implies the following.
Corollary 9. If condition (24) holds, is regular for and then, is regular also for .
Lemma 10. Let condition (24) hold. Then, , where is the unique positive root of equation
Proof. For any , due to the previous corollary, we have Hence, it follows that . But . We thus arrive at the required result.
The assertion of Theorem 4 follows from Lemmas 10 and 2.
4. Quadratic Pencils
In this section, . So, , Now we can directly apply Theorems 3 and 4.
To derive bounds for the spectrum of , take an operator commuting with . For example, with a constant . If it is desirable to choose in such a way that the norm of is small enough. Put and Since and commute, one can enumerate their eigenvalues in such a way that the eigenvalues of for a fixed are . So, , where are the roots of the polynomial : We have where . So . Now inequality (11) implies the following.
Corollary 11. Let and let commute with . Then for any nonzero , there is a (), such that , where
Let be the spectral radius of : . From the previous corollary, it follows that . Besides,
Corollary 12. Under condition (33), let Then, .
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 with bounded operator coefficients . This equation is said to be asymptotically stable, if any of its solution tends to zero as . It is not hard to check that (37) is asymptotically stable, provided cf. . Now one can use the perturbation results due to Theorems 3 and 4. For example, let . So, Take an operator commuting with as in the previous section. Recall that it is desirable to choose such that the norm of 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 the differential equation where Numerous integrodifferential equations can be written in the form of (40). Impose the periodic conditions We seek a solution of problem (40), (42) in the form where () should be found. Substituting this expression into (40), we obtain This equation has a solution provided the spectrum of does not contain the numbers . Now one can apply Corollary 11.
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