Perturbations of Polynomials with Operator Coefficients

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 manymathematicians. 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 inmany 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 Schattenvon 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 inH is denoted. For a linear operator A in H, A−1 is the inverse operator, σ(A) is the spectrum, λ


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 Schattenvon 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 ,  −1 is the inverse operator, () is the spectrum,   () ( = 1, 2, . ..) are the eigenvalues with their multiplicities,  * is the adjoint operator, ‖‖ is the operator norm, and Let   and Ã ( = 1, . . .,  < ∞) be linear bounded operators in .Consider the pencils is the -spectral variation of P with respect to   .Let  = (  ) and T = ( P ) be the operator  × -matrices defined on the the orthogonal sum   of  exemplars of  by ) . ( Lemma 1.Let all the operators   ( = 1, . . ., ) 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  ,+1 :=  ,  has the form where But  has the form  ,1 ,  2 has the form  ,2 , and so forth.Take  , =   .Then,  ,+1 =  +1 .Denote by the entries of   .Then according to (6), Thus, taking  = 1, 2, . . .,  − 1, we can assert that  ()  are linear combinations of operators   and their products.This proves the required result.

Lemma 2. One has
Proof.As it is well known, cf.[1], the spectra of  and   (⋅) coincide.This implies the required result.
Due to Corollary 8.5.5 of [14], we have where Now consider perturbations of pencils with almost commuting coefficients.
To this end, put Theorem 4. Let condition (9) hold.Then, V ()   ( P ) ≤  , , where  , is the unique positive root of equation This theorem is proved in the next section.Replacing in (9)   by   := ζ (, Ξ  ), we obtain where It is clear that Theorem 4 is sharper than Theorem 3, provided ζ (, Ξ  ) <   .

Proof of Theorem 4
Let  and Ã be bounded linear operators in ,  = Ã −  and  = Ã − .We begin with the following result.
Lemma 6.Let a  ∈ C be regular for both  and Ã.Then, the following equality holds: Proof.We have as claimed.
Corollary 9.If condition (24) holds,  is regular for  and  (, ) then,  is regular also for Ã.
The assertion of Theorem 4 follows from Lemmas 10 and 2.
To derive bounds for the spectrum of  2 , take an operator  commuting with  1 .For example,  =  2  1 with a constant .If it is desirable to choose  in such a way that the norm of  2 −  is small enough.Put () = We have where Then,   ( 2 ) < 1.

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. [13].Now one can use the perturbation results due to Theorems 3 and 4. For example, let  = 2. So, V ( + 2) +  1 V ( + 1) +  2 V () = 0 ( = 2, 3, . ..) .(39) Take an operator  commuting with  1 as in the previous section.Recall that it is desirable to choose  such that the norm of  2 −  is small enough.Now Corollary 12 implies the following.
A point  ∈ C 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 Σ(  ).Our main problem is as follows: if   and Ã ( = 1, . . ., ) are close, how close are the spectra of   () and P ()?