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, λk(A) (k=1,2,…) are the eigenvalues with their multiplicities, A* is the adjoint operator, ∥A∥ is the operator norm, and Rλ(A)=(A-λI)-1 is the resolvent.

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

Let Ak and A~k (k=1,…,m<∞) be linear bounded operators in H. Consider the pencils
(1)Pm(λ)=∑k=0mAkλm-k,P~m(λ)=∑k=0mA~kλm-k(A~0=A0=I).
A point λ∈ℂ is called a regular value of Pm, if Pm(λ) is boundedly invertible. The complement of all regular points of Pm to the closed complex plane is called the spectrum of Pm and is denoted by Σ(Pm). Our main problem is as follows: if Ak and A~k (k=1,…,m) are close, how close are the spectra of Pm(λ) and P~m(λ)?

For an integer j≥1, we will say that
(2)zsvPm(j)(P~m)∶=supμ∈Σ(P~m)infs∈Σ(Pm)|μj-sj|
is the j-spectral variation of P~m with respect to Pm. Let T=T(Pm) and T~=T(P~m) be the operator m×m-matrices defined on the the orthogonal sum Hm of m exemplars of H by
(3)T=(-A1-A2⋯-Am-1-AmI0⋯000I⋯00··⋯··00⋯I0),T~=(-A~1-A~2⋯-A~m-1-A~mI0⋯000I⋯00··⋯··00⋯I0).

Lemma 1.

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

Proof.

For an integer μ<m, consider the operator m×m matrix
(4)Bm,μ=(b11b12⋯b1,m-μ⋯b1,m-1b1,mb21b22⋯b2,m-μ⋯b2,m-1b2,m··⋯·⋯··bμ1bμ,2⋯bμ,m-μ⋯bμ,m-1bμ,mI0⋯0⋯000I⋯0⋯00··⋯·⋯··00⋯I⋯00)
with some operators bjk. Direct calculations show that the operator matrix Dm,μ+1∶=Bm,μT has the form(5)Dm,μ+1=(d11d12⋯d1,m-μ-1⋯d1,m-1d1,md21d22⋯d2,m-μ-1⋯d2,m-1d2,m··⋯·⋯··dμ1dμ2⋯dμ,m-μ-1⋯dmu,m-1dμ,mdμ+1,1dμ+1,2⋯dμ+1,m-μ-1⋯dμ+1,m-1dμ+1,mI0⋯0⋯000I⋯0⋯00··⋯·⋯··00⋯I⋯00),where
(6)djk=-bj1Ak+bj,k+1(1≤j≤μ,k<m),djm=-bj1Am(1≤j≤μ),dμ+1,k=-Ak(1≤k≤μ).
But T has the form Bm,1, T2 has the form Bm,2, and so forth. Take Bm,μ=Tμ. Then, Dm,μ+1=Tμ+1. Denote by cjk(μ) the entries of Tμ. Then according to (6),
(7)cjk(μ+1)=-cj1(μ)Ak+cj,k+1(μ)(k<m),cjμ(μ+1)=-cj1(μ)Am(1≤j≤μ),cμ+1,k(μ+1)=-Ak.
Thus, taking μ=1,2,…,m-1, we can assert that cjk(m) are linear combinations of operators Ak and their products. This proves the required result.

For linear operators A and A~, we will say that svA(A~)∶=supμ∈σ(A~)infs∈σ(A)|μ-s| is the spectral variation of A~ with respect to A.

Lemma 2.

One has zsvPm(j)(P~m)=svTj(T~j) (j=1,2,…,m).

Proof.

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

2. The Main Results

The norm ∥·∥Hm in Hm is defined by the following way: let H=∑k=1m⊕Hk, where Hk≡H (k=1,…,m). Then,
(8)∥f∥Hm2=∑k=1m∥fk∥2(f=(fk∈Hk)k=1m∈Hm).
Put qm=∥Tm-T~m∥Hm and assume that, for an integer p≥1,
(9)Ak∈SN2p(k=1,…,m).

Theorem 3.

Let condition (9) hold. Then, zsvPm(m)(P~m)≤ym,p, where ym,p is the unique positive root of equation
(10)qm∑s=0p-1(2N2p(Tm))szs+1exp[12+(2N2p(Tm))2p2z2p]=1.

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

Due to Corollary 8.5.5 of [14], we have
(11)zsvPm(m)(P~m)≤Δp,m,
where
(12)Δp,m:={peqmif2N2p(Tm)≤epqm,2N2p(Tm)[ln(2N2p(Tm)qmp)]-1/2pif2N2p(Tm)>epqm.

Now consider perturbations of pencils with almost commuting coefficients.

To this end, put Ξm=T~m-Tm and
(13)ζ^m(T,Ξm)∶=∥TmΞm-ΞmTm∥+∥Ξm2∥.

Theorem 4.

Let condition (9) hold. Then, zsvPm(m)(P~m)≤xm,p, where xm,p is the unique positive root of equation
(14)ζ^m(T,Ξm)∑s=0p-1(2N2p(Tm))szs+1exp[12+(2N2p(Tm))2p2z2p]=1.

This theorem is proved in the next section. Replacing in (9) qm by ζm∶=ζ^m(T,Ξm), we obtain
(15)zsvPm(m)(P~m)≤δp,m,
where
(16)δp,m∶={peζmif2N2p(Tm)≤epζm,2N2p(Tm)[ln(2N2p(Tm)ζmp)]-1/2pif2N2p(Tm)>epζm.
It is clear that Theorem 4 is sharper than Theorem 3, provided ζ^m(T,Ξm)<qm.

Remark 5.

Put ImTm=(Tm-(T*)m)/2i. Then according to Theorem 7.9.1 of [14], in Theorems 3 and 4, one can replace 2N2p(Tm) by βpN2p(ImTm), where
(17)βp∶=2(1+2pexp(2/3)ln2).

3. Proof of Theorem <xref ref-type="statement" rid="thm2.2">4</xref>

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:
(18)Rλ(A~)-Rλ(A)=Rλ(A~)ZRλ2(A)-ERλ2(A).

Proof.

We have
(19)Rλ(A~)(A~E-EA)Rλ2(A)-ERλ2(A)=(Rλ(A~)(A~E-EA)-E)Rλ2(A)=Rλ(A~)(A~E-EA-(A~-Iλ)E)Rλ2(A)=-Rλ(A~)(-Eλ+EA)Rλ2(A)=-Rλ(A~)ERλ(A)=-Rλ(A~)(A~-Iλ-(A-Iλ))Rλ(A)=-(I-Rλ(A~)(A-Iλ))Rλ(A)=Rλ(A~)-Rλ(A),
as claimed.

Denote η(A,E,λ)=sup0≤t≤1t∥(AE-EA+tE2)Rλ2(A)∥.

Corollary 7.

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

Indeed, put At=A+tE(t∈[0,1]). Since the regular sets of operators are open, λ is regular for At, provided t is small enough. By Lemma 6,(21)Rλ(At)-Rλ(A)=Rλ(At)(t(A+tE)E-tEA)Rλ2(A)-tERλ2(A).
Hence,
(22)∥Rλ(At)∥-∥Rλ(A)-tERλ2(A)∥≤∥Rλ(At)∥∥[t(EA-AE)+t2E2]Rλ2(A)∥≤∥Rλ(At)∥η(A,E,λ).
Thus,
(23)∥Rλ(At)∥≤∥Rλ(A)-tERλ2(A)∥1-η(A,E,λ).
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)∥Rλ2(A)∥. Now Corollary 7 implies the following.

Corollary 8.

If λ is regular for A and ζ(A,E)∥Rλ(A)∥<1, then, λ is regular also for A~.

Furthermore, assume that
(24)A∈SN2p(p=1,2,…).
Then due to Theorem 7.9.1 of [14], we have the estimate
(25)∥Rλ(A)∥≤∑s=0p-1(2N2p(A))sρs+1(A,λ)exp[12+(2N2p(A))2p2ρ2p(A,λ)],
where ρ(A,λ)=infs∈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
(26)ζ(A,E)∑s=0p-1(2N2p(A))sρs+1(A,λ)exp[12+(2N2p(A))2p2ρ2p(A,λ)]<1,
then, λ is regular also for A~.

Lemma 10.

Let condition (24) hold. Then, svA(A~)≤zp, where zp is the unique positive root of equation
(27)ζ(A,E)∑s=0p-1(2N2p(A))szs+1exp[12+(2N2p(A))2p2z2p]=1.

Proof.

For any μ∈σ(A), due to the previous corollary, we have
(28)ζ(A,E)∑s=0p-1(2N2p(A))sρs+1(A,μ)exp[12+(2N2p(A))2p2ρ2p(A,μ)]≥1.
Hence, it follows that ρ(A,μ)≤zp. But svA(A~)=supμ∈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,
(29)T=T(P2)=(-A1-A2I0),T2=(A12-A2A1A2-A1-A2),T~2=(A~12-A~2A~1A~2-A~1-A~2).
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=cA12 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
(30)S=(-A1-BI0).Then,S2=(A12-BA1B-A1-B).
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+λk(A1)z+λk(B). So, Σ(Q)={zk1(Q),zk2(Q)}k=1∞, where zk1(Q),zk2(Q) are the roots of the polynomial z2+λk(A1)z+λk(B):
(31)zk1(Q)=-λk(A1)2+λk2(A1)4-λk(B),zk2(Q)=-λk(A1)2-λk2(A1)4-λk(B).
We have
(32)T2(P2)-S2=(-CA1C0-C),
where C=A2-B. So ∥T2(P2)-S2∥H2≤qS:=∥C∥(1+∥A1∥). Now inequality (11) implies the following.

Corollary 11.

Let
(33)A1,B∈SN2p,
and let B commute with A1. Then for any nonzero s∈Σ(P2), there is a zkl(Q) (l=1,2;k=1,2,…), such that |s2-zk,l2|≤δ(S), where
(34)δ(S):={peqSif2N2p(S2)≤epqS,2N2p(S2)[ln(2N2p(S2)qSp)]-1/2pif2N2p(S2)>epqS.

Let rs(P2) be the spectral radius of P2: rs(P2):=sups∈Σ(P2)|s|. From the previous corollary, it follows that rs2(P2)≤rs2(Q)+δ(S). Besides,
(35)rs(Q)=supl=1,2;k=1,2,…|zkl(Q)|.

Corollary 12.

Under condition (33), let
(36)rs2(Q)+δ(S)<1.
Then, rs(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
(37)∑k=0mAm-kv(j+k)=0(A0=I;j=m,m+1,…),
with bounded operator coefficients Ak. 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
(38)rs(Pm)<1;
cf. [13]. Now one can use the perturbation results due to Theorems 3 and 4. For example, let m=2. So,
(39)v(j+2)+A1v(j+1)+A2v(j)=0(j=2,3,…).
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
(40)uxx(x)+A1ux(x)+A2u(x)=f(x)(0<x<1),
where
(41)f(x)=∑k=-∞∞fke2πikx∈L2([0,1],H)(fk∈H).
Numerous integrodifferential equations can be written in the form of (40). Impose the periodic conditions
(42)u(0)=u(1),u′(0)=u′(1).
We seek a solution of problem (40), (42) in the form
(43)u(x)=∑k=-∞∞uke2πikx,
where uk (k=0,±1,…) should be found. Substituting this expression into (40), we obtain
(44)(-4π2k2I+2πikA1+A2)uk=fk.
This equation has a solution provided the spectrum of P2(z) does not contain the numbers 2πik. Now one can apply Corollary 11.

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