ON A CLASS OF NON-SELF-ADJOINT QUADRATIC MATRIX OPERATOR PENCILS ARISING IN ELASTICITY THEORY

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1 J. OPERATOR THEORY 4700, c Copyright by Theta, 00 ON A CLASS OF NON-SELF-ADJOINT QUADRATIC MATRIX OPERATOR PENCILS ARISING IN ELASTICITY THEORY VADIM ADAMJAN, VJACHESLAV PIVOVARCHIK and CHRISTIANE TRETTER Communicated by Florian-Horia Vasilescu Abstract. This paper deals with a class of non-self-adjoint quadratic pencils of block operator matrices. The main results concern the structure and location of the spectrum and theorems about the minimality, completeness and basis properties of the eigenvectors and associated vectors corresponding to certain parts of the spectrum. Finally, an application to the problem of vibrations of a rotating beam is given. Keywords: quadratic operator pencil, completeness of basis of eigenvectors. MSC 000: Primary 47A56; Secondary 47N0. 1. INTRODUCTION The spectral theory of quadratic operator pencils is a classical subject with many applications in elasticity theory. A fundamental contribution to the theory of self-adjoint operator pencils is due to Krein and Langer [7]. Also, operators which have a certain block matrix representation occur frequently in mathematical physics. Recent contributions to this area may be found e.g. in [], [1], [8], and [9]. In this paper we are going to study a class of damped non-self-adjoint quadratic operator pencils the coefficients of which are unbounded block operator matrices. Our aim is to investigate the spectrum of such pencils, to study the properties of the eigenvectors and associated vectors corresponding to certain parts of the spectrum, and to apply the results to the problem of vibrations of a rotating beam with inner and outer damping in a possibly inhomogeneous outer medium. A fundamental tool here are factorization theorems by Markus, Matsaev and Russu [11], [1], [13], [10]. Let H be a separable infinite dimensional Hilbert space. We consider a quadratic operator pencil L acting in the product space H H and given by the matrix representation 1.1 Lλ = λ I 0 A + K1 0 A βa +λ +, λ C. 0 I 0 A + K βa A

2 36 Vadim Adamjan, Vjacheslav Pivovarchik and Christiane Tretter Here A is an unbounded self-adjoint operator in H with domain DA having compact resolvent, A δi with some δ > 0, and β are positive constants, and K 1, K are bounded operators, 0 K 1, K γi with some > γ 0. The domain of Lλ is independent of λ and given by DLλ = DA DA. An example for such a quadratic operator pencil arises in elasticity theory: The system of differential equations EI 4 u z 4 + ωκei 4 v z 4 + κei 5 u z 4 t + ε u 1 t + u m t = 0, EI 4 v z 4 ωκei 4 u z 4 + κei 5 v z 4 t + ε v t + v m t = 0 on the finite interval [0, l] describes the vibrations of a rotating beam of length l and mass density m per unit length. Here EI > 0 is the constant bending stiffness of the beam sections, ω > 0 is the angular frequency of the rotation, κ > 0 is the coefficient of inner damping Voigt material, and ε 1, ε are nonnegative continuous functions on [0, l] describing the outer viscous damping. In the general case when the outer medium is inhomogeneous, one has ε 1 ε see e.g. [3]. The boundary conditions to be imposed e.g. in the case of hinged ends are u0, t = ul, t = u 1.4 z 0, t = u l, t = 0, z v0, t = vl, t = v z 0, t = v l, t = 0. z For simplicity, we assume that m 1. Then separation of variables 1.5 uz, t, vz, t t = e λt y 1 z, y z t, z [0, l], t 0, leads to a spectral problem of the form Lλy = 0, λ C, for y = y 1, y t in the Hilbert space L 0, l L 0, l where the operators A and K in L 0, l are given by 1.6 Ay := EI y 4, DA := {y L 0, l : y0 = yl = y 0 = y l = 0}, 1.7 K i y := ε i y, DK i := L 0, l, i = 1,, and the constants and β are given by := κ, β := ωκ. In Section we first determine the structure of the spectrum of the quadratic operator pencil 1.1. We show that its essential spectrum consists of the points 1+iβ, 1 iβ, and that outside the essential spectrum L has 3 branches of eigenvalues accumulating at the points of the essential spectrum and at. Secondly, we prove a criterion for the stability of the pencil L, that is, a criterion guaranteeing that the spectrum of L lies in the open left half plane. In Section 3 we consider the case K 1 = K where L in fact decomposes into two quadratic pencils in H. In this case the eigenvalue branch accumulating at splits again into two branches. We derive theorems about the minimality, completeness and basis properties of the eigenvectors and associated vectors corresponding to the 4 branches of eigenvalues of L.

3 On a class of non-self-adjoint quadratic matrix operator pencils 37 In Section 4 we consider the case K 1 K. We prove a theorem about the minimality, completeness and basis properties of the eigenvectors and associated vectors corresponding to the branch of eigenvalues of L accumulating at. Finally, in Section 5, we apply all results to the problem of vibrations of a rotating beam.. THE SPECTRUM OF L We define the resolvent set ρl of the quadratic operator pencil L as ρl := {λ C : Lλ : DA DA H H is bijective, Lλ 1 is bounded} and its spectrum σl as σl := C \ ρl. For λ C, the operator Lλ is called Fredholm if Lλ is closed, its kernel is finite dimensional and its range is finite codimensional see e.g. [6], Chapter IV, Section 5.1. A point λ 0 C is said to be an eigenvalue of L if Lλ is not injective. An eigenvalue λ 0 C of L is called normal or of finite type if λ 0 is isolated and Lλ 0 is Fredholm. The essential spectrum of L is defined as σ ess L := {λ C : Lλ is not Fredholm}. In order to determine the essential spectrum of the operator pencil L, we consider the transformed pencil L d given by L d λ := S 1 LλS on DA DA for λ C where the operator matrix S in H H is of the form I ii S :=. ii I Then, for λ C, L d λ := λ I 0 0 I A λ K i 1 + K K 1 K i K 1 K A + 1 K 1 + K iβa iβa and L d λ is closed Fredholm if and only if Lλ is closed Fredholm. Theorem.1. The essential spectrum of L consists of the two points 1 + iβ iβ, 1. The other points of the spectrum of L are normal eigenvalues which accumulate at most at the points 1+iβ, 1 iβ, and at. Proof. Let λ C. If we write L d λ in the form λ+1+iβa+λ I +λ 1 K 1+K λ i K 1 K λ i K 1 K λ+1 iβa+λ I +λ 1 K, 1+K,

4 38 Vadim Adamjan, Vjacheslav Pivovarchik and Christiane Tretter it is not difficult to see that L d λ with domain DA DA is closed if and only if λ 1±iβ 1±iβ. Now let λ. Since A 1 0 L d λ 0 A 1 =λ A 1 0 I A 1 +λ K 1 + K A 1 i K 1 K A 1 i.1 K 1 K A 1 I + 1 K 1 + K A iβi 0 + = 0 1 iβi 1 + iβ + λi iβ + λi + Kλ, where Kλ is a compact operator in H H, the operator on the left hand side of.1 is Fredholm see e.g. [4], Chapter XI, Theorem 4.. Hence the same is true for L d λ. On the other hand, since H is infinite dimensional, it follows from.1 that L d 1±iβ is not Fredholm. Moreover, the operator in.1 is bijective for λ = 0. Now the theorem follows e.g. from a theorem about analytic Fredholm operator valued functions see [4], Chapter XI, Corollary 8.4. Remark.. From the minimality and completeness results which will be proved in the next two sections more precise statements about the accumulation of the eigenvalues will follow if the resolvent of A belongs to a certain von Neumann Schatten class: In the next section we are going to show that in the case K 1 = K the normal eigenvalues of L split into 4 branches of eigenvalues, two branches accumulating at the points 1+iβ, 1 iβ, and two branches at. In Section 4, for the case K 1 K, it will turn out that again is an accumulation point of eigenvalues, but the branches of eigenvalues accumulating at can, in general, not be separated as in the case K 1 = K. Theorem.3. Assume that there exists a µ > 0 with K 1 µi, K µi, and such that. µ δ and β 4µ < 1, or µ < δ and β δ δ + µ < 1. Then the spectrum of L lies in the open left half plane. Proof. If λ 0 σ ess L, then obviously Re λ 0 < 0 by the above theorem. Otherwise, if λ 0 σl \ σ ess L, then λ 0 is an eigenvalue of L. Let Y = y, z t DA DA, y + z = 1, be a corresponding eigenvector. Then Lλ 0 Y, Y = 0 implies that.3.4 Re λ 0 Im λ 0 + Re λ 0 Ay, y + Az, z + K 1 y, y + K z, z + Ay, y + Az, z = 0, Re λ 0 Im λ 0 + Im λ 0 Ay, y + Az, z + K 1 y, y + K z, z + β Im Az, y = 0.

5 On a class of non-self-adjoint quadratic matrix operator pencils 39 Calculating Im λ 0 from.4, using the estimate Im Az, y Az, y = A 1/ z, A 1/ y A 1/ z + A 1/ y = Ay, y + Az, z and substituting it into.3, we arrive at Re λ 0 β Ay, y + Az, z.5 Re λ 0 + Ay, y + Az, z + K 1 y, y + K y, y +Re λ 0 Ay, y+az, z+k 1 y, y + K z, z+ay, y+az, z 0. The left hand side is monotonically increasing for Re λ 0 [0,. But the condition. and A δ > 0 imply that β Ay, y + Az, z + Ay, y + Az, z > 0, Ay, y + Az, z + K 1 y, y + K z, z a contradiction to.5. Hence Re λ 0 < THE CASE K 1 = K = K In this case the operator pencil L d in H H is the orthogonal sum of two quadratic operator pencils L ± in H, L+ λ 0 L d λ =, λ C, 0 L λ where L ± λ := λ I + λa + K + 1 ± iβa, λ C. In the following we are going to prove minimality, completeness and basis results for the eigenvectors and associated vectors corresponding to various branches of eigenvalues of L. With regard to the eigenvalues which will prove to accumulate at, we introduce the auxiliary operator pencils L 1 ± given by L 1 ±λ := λ A 1/ L ± = λ 1 ± iβ I + λ 1 λ A 1/ I + 1 A 1/ KA 1/ + 1 A 1, λ C. Lemma 3.1. The spectrum and the essential spectrum of L 1 ± are given by: i σl 1 ± = { 1 λ : λ σl ± } {0}; ii σ ess L 1 ± = { } 0, 1±iβ. Proof. The assertions are immediate from Theorem.1 and from the definition of L 1 ±.

6 330 Vadim Adamjan, Vjacheslav Pivovarchik and Christiane Tretter The numerical range or root domain of a quadratic operator polynomial T in H is the set of all roots of all possible polynomials T y, y, y H, y 0 see [10], Section 6.3. It consists of at most two components. If the numerical range of T consists of two disjoint components called the root zones of T, then, clearly, for any y H, y 0, the polynomial T y, y has exactly one root in each component. Lemma 3.. The numerical range of L 1 + consists of two components 1 +, 1 + which are bounded and separated by the strip { 3.1 λ C : 0 < Im λ < β } 1 + β, say 1 + below and 1 + above this strip. There exists a simple contour Γ 1 + surrounding 1 + {0} and separating 1 + {0} from 1 +, and such that inf L 1 +λy, y > 0. λ Γ 1 +, y =1 Proof. The first assertion follows from the fact that L 1 + is a pencil of bounded operators. For the proof of the second statement, fix an element y H, y = 1, consider the function ϕλ, η := λ 1 + iβ + λ 1 + η 1 KA 1/ y, A 1/ y + η 1 A 1/ y, A 1/ y for λ C, η [0, 1], and assume that there exists a point λ 0 in the strip 3.1 and an η [0, 1] such that ϕλ 0, η = 0. The point λ 0 has a representation λ 0 = t + id with t R and 0 < d < β 1+β. Hence t +itd d 1 + iβ + t+id 1 + η 1 KA 1/ y, A 1/ y +η 1 A 1/ y = 0. Taking the imaginary part and multiplying by β yields t + td β d + d 1 + η 1 KA β 1/ y, A 1/ y = 0. This is equivalent to t + d β = d β which implies, since K 0, that d1 + β β 1 + η 1 KA 1/ y, A 1/ y, d β 1 + β, are separated by the strip 3.1. Then, according to what was proved above, so are λ 1 η, λ η for all η [0, 1]. In particular, the zeros of L 1 + y, y = y ϕ, 1 are separated by the strip 3.1. This proves the second statement. The remaining assertions are immediate. a contradiction. Thus ϕλ, η 0, η [0, 1], and in particular L 1 +λy, y = y ϕλ, 1 0 for all λ in the strip 3.1. The zeros λ 1 η, λ η of ϕ, η = 0 depend continuously on the parameter η, and λ 1 0 = 0, λ 0 = 1+iβ

7 On a class of non-self-adjoint quadratic matrix operator pencils 331 The subsequent corollary about the existence of a spectral root of L 1 + or, equivalently, of a canonical factorization of λ 1 L 1 +λ follows from the above lemma by some general results of Markus and Matsaev [11], [1], see also [10], Theorems 6.19 and 6.1. Corollary 3.3. The operator pencil L 1 + has a spectral root Z 1 + such that σz 1 + = σl In the following let S p, 1 p, denote the von Neumann Schatten classes of compact operators see [5], Chapter III, Section 7, or [10], Section.4. Further, for an operator T S we denote by nτ, T the sum of the algebraic multiplicities of the eigenvalues of T in {λ C : λ > τ 1 }. For a quadratic operator pencil T the nonzero spectrum of which in some bounded domain G containing 0 consists of a sequence of eigenvalues of finite algebraic multiplicity converging to 0, we denote by nτ, G, T the sum of the algebraic multiplicities of the eigenvalues of T in {λ C : λ > τ 1 } G see [10], Section.4. Using a theorem of Markus, Matsaev and Russu [13], we obtain: Theorem 3.4. i The set of eigenvectors and associated vectors corresponding to the eigenvalues of L 1 + in 1 + is minimal in H. ii If A 1 S p for some p <, then the set of eigenvectors and associated vectors corresponding to the eigenvalues of L 1 + in 1 + is complete in H. If, in addition, n τ, 1 A 1 c 1 τ c as τ with some 0 < c 1, c <, then nτ, G 1 +, L 1 + c 1 τ c, where G 1 + is the interior of the curve Γ 1 +. iii If nτ, A 1 = Oτ γ for some γ 0, 1 ], then the set of eigenvectors and associated vectors corresponding to the eigenvalues of L 1 + in 1 + is a Riesz basis with parentheses in H. If, in addition, n τ, 1 A 1 = c 1 τ c +Oτ β for some 0 < c 1, c <, 0 β < β + γ, then also nτ, G 1 +, L 1 + = c 1 τ c + Oτ β. Proof. The theorem follows from a general result of Markus, Matsaev and Russu see [13] which is contained in [10], Theorem.13 and Corollary 6.0. To apply the statements therein, we choose H = 1 A 1, T = 0 for ii and H = 1 A 1, D 0 = 0, D 1 = 1 A 1/ KA 1/+γ where γ 0, 1 ] for iii. Remark 3.5. Analogous assertions hold for the pencil L 1. With regard to the branches of eigenvalues possibly accumulating at 1±iβ, we first consider the pencils L ± themselves. Lemma 3.6. The numerical range of L + consists of two components +, + which are separated by the strip where {λ C : δ < Im λ < δ } δ := β + K + 4 δ 1 + K δ + 16β δ 1/4.

8 33 Vadim Adamjan, Vjacheslav Pivovarchik and Christiane Tretter The component located in the half plane {λ C : Im λ δ }, say +, is bounded, and there exists a simple closed curve Γ + surrounding + and separating it from +, and such that inf λ Γ + y DA, y =1 L + λy, y > 0. Proof. Let y DA, y = 1. Then the solutions λ ± y of L + λy, y = 0 are given by Ay, y + Ky, y Ay, y + Ky, y 3. λ ± y = ± 1 + iβay, y. 4 Hence Im λ ± y = 1 Im Ay, y + Ky, y 41 + iβay, y 1/ = 1 4Ay, y Ay, y + Ky, y + Ay, y + Ky, y 4Ay, y + 16β Ay, y 1/ 1/ Ay, y + Ky, y 1/4 4Ay, y β Ay, y + 16β Ay, y δ. This proves the first assertion. The second statement follows from the fact that the root λ + y lying in the half plane {λ C : Im λ δ } tends to 1+iβ when Ay, y tends to infinity. The remaining assertions are then immediate. In order to apply the results of Markus, Matsaev and Russu used before we need to consider pencils of bounded operators. Therefore we introduce L ±λ := 1 A 1/ L ± λ 1 ± iβ A 1/ where = λ 1 ± iβ A 1 + λ I A A 1/ KA 1/ B ± = 1 ± iβ 3 A 1 1 ± iβ A 1/ KA 1/. + B ± Corollary 3.7. The numerical range of L + consists of two components +, + which are bounded and separated by the strip { λ C : β δ < Im λ < β } + δ, say + below and + above this strip. There exists a simple closed curve Γ + surrounding + {0} and separating it from +, and such that inf L + λy, y > 0. λ Γ +, y =1 Proof. All assertions follow immediately from Lemma 3.6 by definition of L + note that δ < β.

9 On a class of non-self-adjoint quadratic matrix operator pencils 333 Theorem 3.8. i The set of eigenvectors and associated vectors corresponding to the eigenvalues of L + in + is minimal in H. ii If A 1 S p for some p <, then the set of eigenvectors and associated vectors corresponding to the eigenvalues of L + in + is complete in H. If, in addition, nτ, B + c 1 τ c as τ with some 0 < c 1, c <, then nτ, G +, L + c 1 τ c, where G + is the interior of the curve Γ +. Proof. As in the proof of Theorem 3.4, we use [10], Theorem.13 and Corollary 6.0. To apply the statements therein, we now choose H = B +, T = 0 for ii, and we note that if A 1 S p for some p <, then also B + S p see e.g. [5]. Remark 3.9. Analogous assertions hold for the pencil L. In order to formulate statements for the original pencil L, we first note that σl = σl d = σl + σl. We denote by λ 1 k and λ k, k = 1,,..., the eigenvalues of the pencil L + located in the upper and lower half plane, respectively counted according to their algebraic multiplicities. It is not difficult to see that then the complex conjugates λ 1 k and λ k, k = 1,,..., are the eigenvalues of the pencil L located in the lower and upper half plane, respectively. We denote the corresponding eigenvectors and associated vectors of L + by yk 1 and y k, and those of L by yk 1 and y k. Then the eigenvalues of L d and hence those of L can be separated into the 4 branches {λ 1 k }, {λ k } and {λ1 k }, {λ k }, which lie symmetrically to the real axis. The corresponding eigenvectors and associated vectors of L d are of the form y 1 k, 0 t, y k, 0 t, 0, y 1 k t, 0, y k t. The respective eigenvectors of L are given by y 1 k, iy 1 k t, y k, iy k t, iy 1 k, y 1 k t, iy k, y k t, and there are analogous formulas for the associated vectors of L. Theorem i The set of eigenvectors and associated vectors of L corresponding to the eigenvalues λ 1 k and λ1 k, k = 1,,..., is minimal in the space H A 1 H A 1 where H A 1 = H, A 1, A 1, complete in H A 1 H A 1 if A 1 S p for some p <, and a Riesz basis with parentheses in H A 1 H A 1 if nτ, A 1 = Oτ γ for some γ 0, 1 ]. ii The set of eigenvectors and associated vectors of L corresponding to the eigenvalues λ k and λ k, k = 1,,..., is minimal in the space H A 1 H A 1 and complete in H A 1 H A 1 if A 1 S p for some p <. In particular, λ 1 k, λ1 k, λ k 1+iβ, λ k 1 iβ for k if A 1 S p for some p <. Proof. The assertions in i and ii follow from Theorems 3.4 and 3.8. The statement about the accumulation at the points of the essential spectrum follows from Theorem.1 together with the minimality and completeness from i and ii.

10 334 Vadim Adamjan, Vjacheslav Pivovarchik and Christiane Tretter 4. THE GENERAL CASE Now we consider the general case of a quadratic block operator matrix pencil 1.1 with possibly different K 1, K. In this case L cannot be written as the orthogonal sum of two pencils in H, and hence there exists no decomposition of the spectrum as it was used in the previous section. In order to guarantee a certain subdivision of the spectrum of L also here and to obtain minimality, completeness and basis results for the eigenvectors and associated vectors, we have to assume in addition that 4.1 δ > 4. We choose ρ > 0 such that 4. ρ > βδ δ 4. Lemma 4.1. On the segment Γ 1 := {λ C : Re λ =, Im λ ρ} we have the estimate inf LλY, Y > 0. λ Γ 1 Y DA DA, Y =1 Proof. Let λ = + iτ with τ R, Y = y, zt with y, z DA and y + z = 1. Then LλY, Y = + iτ + Ay, + iτ y + Az, z + K1 y, y + K z, z + Ay, y + Az, z + βaz, y βay, z 4 τ Ay, y Az, z K 1y, y K z, z δ 4 > 0 by assumption 4.1. Lemma 4.. On the rays Γ ± have the estimate := {λ C : Re λ, Im λ = ±ρ} we inf LλY, Y > 0. λ Γ ± Y DA DA, Y =1 Proof. Let λ = t + iρ with t, Y = y, zt with y + z = 1. Then y, z DA and

11 On a class of non-self-adjoint quadratic matrix operator pencils 335 LλY, Y Im LλY, Y = tρ + ρ Ay, y + Az, z + K1 y, y + K z, z + β Im Az, y = t + Ay, y + Az, z + K1 y, y + K z, z ρ + β Im Az, y t + Ay, y + Az, z + K 1 y, y + K z, z 4 + δ β Im Az, y ρ t + Ay, y + Az, z + K 1 y, y + K z, z δ 4 β Az, y ρ δ 4 δ 4 ρ β A1/ z A 1/ y δ 4 δ 4 ρ β A 1/ z + A 1/ y δ 4 δ 4 ρ βδ δ 4 > 0 by assumption 4.1 and the choice of ρ according to 4.. Lemma 4.3. On the segment Γ 3 := {λ C : Re λ = ρ 1, Im λ ρ} with ρ 1 > ρ we have inf LλY, Y > 0. λ Γ 3 Y DA DA, Y =1 Proof. Let λ = ρ 1 + iτ with τ ρ, Y = y, z t with y, z DA and y + z = 1. Then LλY, Y Re LλY, Y = ρ 1 τ + ρ 1 Ay, y + Az, z + K1 y, y + K z, z + Ay, y + Az, z > δρ 1 + δ > 0 since ρ 1 > ρ. By Γ we now denote the rectangle the sides of which are Γ 1, Γ 3 and parts of the rays Γ +, Γ. Further, we denote by G the interior and by G + the exterior of Γ without the boundary of Γ. Lemma 4.4. For any fixed Y DA DA, the polynomial LλY, Y has one root in G and one root in G +. Proof. Consider the auxiliary pencil L 1 λ = L λ, λ C,

12 336 Vadim Adamjan, Vjacheslav Pivovarchik and Christiane Tretter and the corresponding polynomial L 1 λy, Y for fixed Y = y, z t with y DA, z DA, y + z = 1. The roots of L 1 λy, Y = 0 are the roots of the equation λ + λ 4 + Ay, y + Az, z + K 1y, y + K z, z Ay, y Az, z K 1y, y K z, z+iβ Im Az, y = 0. Due to assumption 4.1 the quadratic equation 4.4 λ + λ δ δ = 0 possesses exactly one solution on the positive half axis and exactly one on the negative half axis excluding 0. Now we consider A and K i, i = 1,, as perturbations of δi and 0, respectively, i.e., we consider Aη := ηa δi + δi, K i η = ηk i, i = 1,, for η [0, 1] and the pencil L 1 η, λ which arises if we substitute A and K i, i = 1,, in L 1 λ by Aη and K i η, i = 1,, i.e., L 1 η, λ := λ I 0 Aη + K1 η 4 + λ 0 0 I 0 Aη + K η 4 Aη + K 1η + 4 βa βa Aη K η + 4, η [0, 1], λ C. Then L 1 1, λ = L 1 λ and the equation L 1 0, λy, Y = 0 is just the equation 4.4. The quadratic form L 1 η, λy, Y is analytic with respect to η and λ. The roots λ 1 η and λ η of L 1 η, λy, Y = 0 are piecewise analytic, they may fail to be analytic in [0, 1] only if for some η [0, 1], we have λ 1 η = λ η. Since λ 1 0 and λ 0 are the roots of 4.4, one of them, say λ 1 0, lies in the open left half plane and the other in the open right half plane. According to Lemmas 4.1, 4. and 4.3, we have inf L 1 λy, Y > 0, λ Γ Y DA DA, Y =1 where Γ {λ C : Re λ > 0} is the rectangle Γ := {λ + : λ Γ}. Now we choose ρ 1 > ρ such that ρ 1 > λ 0. Then λ η remains inside Γ and λ 1 η outside Γ for all η [0, 1]. The roots of LλY, Y are given by λ 1 1 and λ 1. Hence the first one lies outside Γ and the second one inside Γ. In order to prove results about the eigenvectors and associated vectors corresponding to the branches of eigenvalues possibly accumulating at, we introduce à 1 := 1 A A 1

13 On a class of non-self-adjoint quadratic matrix operator pencils 337 and consider the pencil L λ := λ 1 Ã 1/ L λ I β β I = λ 1 Ã 1/ + λ + 1 I + 1 A 1/ K 1 A 1/ 0 A A 1 0 I + 1 A 1/ K A 1/ for λ C. By Γ we denote the closed simple curve obtained from the rectangle Γ after the transformation λ 1 λ. Let G + G denote the interior exterior of Γ. Theorem 4.5. i The set of eigenvectors and associated vectors corresponding to the eigenvalues of L in G + is minimal in H H. ii If A 1 S p for some p <, then the set of eigenvectors and associated vectors corresponding to the eigenvalues of L in G + is complete in H H. If, in addition, nτ, 1 A 1 c 1 τ c as τ with some 0 < c 1, c <, then nτ, G +, L c 1 τ c. iii If nτ, A 1 = Oτ γ for some γ 0, 1 ], then the set of eigenvectors and associated vectors corresponding to the eigenvalues of L in G + is a Riesz basis with parentheses in H H. If, in addition, nτ, 1 A 1 = c 1 τ c + Oτ β for some 0 < c 1, c <, 0 β < β +γ, then also nτ, G +, L = c 1 τ c +Oτ β. Proof. Again we invoke the results contained in [10], Theorem.13 and Corollary 6.0, and apply them with H = 1 A A 1, T = 0 for ii and H = 1 D 0 = 0, A A 1 D 1 = 1, A 1/ K 1 A 1/+γ 0 0 A 1/ K A 1/+γ where γ 0, 1 ] for iii. Theorem 4.6. The set of eigenvectors and associated vectors of L corresponding to the eigenvalues in the half plane {λ C : Re λ < } is minimal in the space H A 1 H A 1 where H A 1 = H, A 1, A 1. It is complete in H A 1 H A 1 if A 1 S p for some p <, and a Riesz basis with parentheses in H A 1 H A 1 if nτ, A 1 = Oτ γ for some γ 0, 1 ]. In particular, the eigenvalues in {λ C : Re λ < } accumulate at if A 1 S p for some p <. Proof. The statements of the theorem follow from Theorem 4.5 and Theorem.1.

14 338 Vadim Adamjan, Vjacheslav Pivovarchik and Christiane Tretter 5. APPLICATION TO THE PROBLEM OF VIBRATIONS OF A ROTATING BEAM In this section we are going to apply the results of the previous sections to the system 1., 1.3 of partial differential equations with boundary conditions 1.4. After the separation of variables 1.5 and assuming m 1 for simplicity, it takes the form with boundary conditions EI y ωκei y 4 + λ κei y ε 1 y 1 + λ y 1 = 0, EI y 4 ωκei y λ κei y 4 + ε y + λ y = y 1 0 = y 1 l = y 1 0 = y 1 l = 0, y 0 = y l = y 0 = y l = 0. Here the operators A and K 1, K are determined by 1.6 and 1.7, and the constants, β by = κ the coefficient of inner damping and β = ωκ where ω is the angular frequency of the rotation of the beam. Obviously, the operator A has compact resolvent, and it is not difficult to see that the eigenvalues λ k, k = 1,,..., of A are all simple and given by 4 kπ λ k = EI, k = 1,,.... l Hence the lower bound δ of A is its least eigenvalue, π 4 δ = EI. l It is also easy to see that the number nτ, A 1 of eigenvalues of A 1 greater than τ 1, i.e., the number of eigenvalues of A less than τ satisfies 5.4 nτ, A 1 l 1/4 1 τ 1/4. π EI An immediate consequence of Theorem.1 is the following statement. Theorem 5.1. The essential spectrum of the problem consists of the two points 1 κ iω, 1 κ + iω. The other points of the spectrum of the problem are normal eigenvalues which accumulate at most at the points 1 κ iω, 1 κ + iω, and at. From Theorem.3 we immediately get the following stability result. Theorem 5.. Set µ := min min {ε ix}. Then the spectrum of the problem lies in the open left half plane if i=1, x [0,l] π 4 µ κei and µ > ω κ l 4, or if π 4 π 4 π 4 µ < κei and κei + µ > ω κ EI. l l l

15 On a class of non-self-adjoint quadratic matrix operator pencils 339 Concerning results about the minimality, completeness and basis properties of the eigenvectors of the problem corresponding to certain branches of eigenvalues, we have to distinguish the case when the outer medium is homogeneous, i.e., ε 1 ε : ε and hence K 1 = K, and the case when the outer medium is inhomogeneous, i.e., ε 1 ε and hence K 1 K. In the case of a homogeneous outer medium, according to Section 3, the eigenvalues of the given problem split into 4 branches {λ 1 k } {λ k } {λ 1 k } {λ k } where λ1 k and λ k, k = 1,,..., are the eigenvalues of the problem 1 + iωκeiy 4 + λκeiy 4 + εy + λ y = 0, 5.5 y0 = yl = y 0 = y l = 0, located in the upper and lower half plane, respectively counted according to their algebraic multiplicities. The respective eigenfunctions and associated functions of the problem of can be obtained from the eigenfunctions and associated functions yk 1 and y k of the problem 5.5 and from the eigenfunctions and associated functions yk 1 and y k of the problem 1 iωκeiy 4 + λκeiy 4 + εy + λ y = 0, 5.6 y0 = yl = y 0 = y l = 0. For instance, the eigenfunctions of are given by the formulas y 1 k, iy 1 k t, y k, iy k t, iy 1 k, y1 k t, iy k, y k t. In the following we denote by W 4 0, l the Sobolev space of order 4 associated with L 0, l. Theorem 5.3. i The set of eigenfunctions and associated functions of problem corresponding to the eigenvalues λ 1 k and λ1 k, k = 1,,..., forms a Riesz basis with parentheses in the space W 4 0, l W 4 0, l. ii The eigenvalues λ 1 k and hence λ1 k accumulate at ; if enumerated such that λ 1 k λ1 k+1, they satisfy the asymptotics 4 kπ λ 1 k = κei + iω + ξk 1 + iη 1 l k, k, where ξk 1, η1 k are real and ξ1 k = ok4, ηk 1 = o1. Proof. The assertion in i and the first assertion in ii follow from Theorem 3.10 which we can apply due to 5.4 with γ = 1 4. From Theorem 3.10 we also obtain that 4 kπ 5.7 λ 1 k κei, k. l Now let yk 1, y1 k = 1, be an eigenfunction of 5.5 i.e., of L + at λ 1 k. Then 5.8 λ 1 k + λ 1 k Ay 1 k, yk 1 + Kyk, 1 yk iβayk, 1 yk 1 = 0. From this it follows that Ayk 1, y1 k, k, because otherwise λ 1 k λ 1 k + Ayk, 1 yk 1 + Kyk, 1 yk 1 would also be bounded, a contradiction to 5.7. Then the assertion follows from the formula for the solutions of the quadratic equation 5.8 see 3. and from 5.7.

16 340 Vadim Adamjan, Vjacheslav Pivovarchik and Christiane Tretter Theorem 5.4. The set of eigenfunctions and associated functions of problem corresponding to the eigenvalues λ k and λ k, k = 1,,..., is minimal and complete in the space W 4 0, l W 4 0, l. In particular, λ k 1 κ iω, λ k 1 κ + iω for k. Proof. The statements are immediate from Theorem In the case of an inhomogeneous outer medium we obtain: Theorem 5.5. Assume that EI π 4 l > 4 κ. Then: i The set of eigenfunctions and associated functions of problem corresponding to the eigenvalues located in the half plane {λ C : Re λ < κ } forms a Riesz basis with parentheses in the space W 4 0, l W 4 0, l. ii The eigenvalues λ k of problem in {λ C : Re λ < κ } accumulate at ; if enumerated such that λ k λ k+1, they satisfy the asymptotics 4 kπ λ k = κei + ok 4, k. l Proof. The first assertion follows immediately from Theorem 4.6. The proof of the second statement is similar to the proof of Theorem 5.3 ii. Acknowledgements. This work was supported by the Deutsche Forschungsgemeinschaft, DFG, within a German Ukrainian cooperation. V. Adamjan and V. Pivovarchik also wish to thank the University of Regensburg for hospitality while this work was done. REFERENCES 1. V.M. Adamjan, H. Langer, Spectral properties of a class of rational operator valued functions, J. Operator Theory , F.V. Atkinson, H. Langer, R. Mennicken, A.A. Shkalikov, The essential spectrum of some matrix operators, Math. Nachr , V.V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, [Russian], GIFML, Moscow 1961; Engl. transl. Pergamon Press, New York I.C. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators. I, Oper. Theory Adv. Appl., vol. 49, Birkhäuser Verlag, Basel I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., vol. 18, Amer. Math. Soc., Providence, RI, T. Kato, Perturbation Theory for Linear Operators, Springer Verlag, Berlin Heidelberg New York M.G. Krein, H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. I, II, Integral Equations Operator Theory 11978, , H. Langer, C. Tretter, Spectral decomposition of some nonselfadjoint block operator matrices, J. Operator Theory , H. Langer, C. Tretter, Diagonalization of certein block operator matrices and applications to Dirac operators, Oper. Theory Adv. Appl. 1001,

17 On a class of non-self-adjoint quadratic matrix operator pencils A.S. Markus, Introduction to the Theory of Polynomial Operator Pencils, Transl. Math. Monogr., vol. 71, Amer. Math. Soc., Providecnce, RI, A.S. Markus, V.I. Matsaev, On the spectral properties of holomorphic operatorvalued functions in Hilbert space, [Russian], Mat. Issled , A.S. Markus, V.I. Matsaev, On the spectral theory of holomorphic operatorvalued functions in Hilbert space, Funktsional. Anal. i Prilozhen , 76 77; Engl. transl. Funct. Anal. Appl , A.S. Markus, V.I. Matsaev, G.I. Russu, Some generalizations of the theory of strongly damped pencils to the case of pencils of arbitrary order, Acta Sci. Math. Szeged , VADIM ADAMJAN VJACHESLAV PIVOVARCHIK Department of Theoretical Physics Department of Higher Mathematics University of Odessa Odessa State Academy of Civil Engineering Ul. Dvorjanskaja and Architecture Odessa Ul. Didrikhsona 4 UKRAINE Odessa UKRAINE vadamjan@m-vox.odessa.ua v.pivovarchik@paco.net CHRISTIANE TRETTER Department of Mathematics University of Bremen Bibliothekstr. 1 D 8359 Bremen GERMANY ctretter@math.uni-bremen.de Received September 5, 1999; revised December 1, 1999.