MATHEMATICS

M. G. KREIN, G. K. LANGER

ON THE THEORY OF QUADRATIC PENCILS OF SELF-ADJOINT OPERATORS

(Presented by Academician L. S. Pontryagin, 21 X 1963)

Various problems of mathematical physics lead to the study of the spectral properties of the quadratic pencil \(L(\lambda)=\lambda^{2}I+\lambda B+C\), where \(B,C\) are certain closed operators (\(I\) is the identity operator) acting in a Hilbert space \(\mathfrak H\).

The methods set forth below for investigating the pencil \(L(\lambda)\), as well as a number of propositions obtained with their aid, apparently are new even for a quadratic matrix pencil.

1. Denote by \(\mathfrak R\) the ring of all bounded operators in \(\mathfrak H\), and by \(\mathfrak C_\infty\) the ideal of all completely continuous operators from \(\mathfrak R\).

In what follows we assume that \(B=B^*\) (i.e. \(B\) is any self-adjoint operator) and that \(\mathfrak D(B)\subset \mathfrak D(C)\).

\(1^\circ\). For \(\lambda\ne0\) the operator \(L(\lambda)\) is defined on \(\mathfrak D(B)\), and the set \(\rho(L)\) of all regular points \(\lambda\) (\(\ne0\)) of the pencil (i.e. points \(\lambda\) (\(\ne0\)) for which there exists a continuous inverse operator \(L^{-1}(\lambda)\in\mathfrak R\)) is an open set.

The complement \(\sigma(L)\) in the complex plane to the set \(\rho(L)\) is called the spectrum of the pencil \(L(\lambda)\).

\(2^\circ\). If \(C=C^*\), then \(\sigma(L)=\overline{\sigma(L)}\) (i.e. the spectrum \(\sigma(L)\) is symmetric with respect to the real axis). For a nonreal isolated eigenvalue \(\lambda\) of the pencil \(L(\lambda)\), the root subspaces* of the pencil \(\mathfrak L_\lambda(L)\) and \(\mathfrak L_{\bar\lambda}(L)\) have the same dimension (\(\leqslant\infty\)) (and, moreover, in a certain sense have the same structure).

\(3^\circ\). If \(B\) and \(C\) are nonnegative self-adjoint operators (\(B,C\geqslant0\)), then the spectrum \(\sigma(L)\) lies in the left half-plane.

\(4^\circ\). If the operator \(C\) is \(B\)-completely continuous \((^2)\), then every point \(\lambda_0\in\sigma(L)\), \(\lambda_0\ne0\), not belonging to the condensation spectrum** of the operator \(B\), is an isolated point of the spectrum \(\sigma(L)\), and moreover an eigenvalue of finite algebraic multiplicity (i.e. the root subspace \(\mathfrak L_{\lambda_0}(L)\) has finite dimension and \(L(\lambda_0)\) is a \(\Phi\)-operator \((^2)\)).

At the basis of our further investigation lies the study of the quadratic operator equation:

\[ Z^2+BZ+C=0. \tag{1} \]

If \(C\in\mathfrak R\) and \(Z_0\bigl(Z_0\in\mathfrak R,\ \mathfrak R(Z_0)\subset\mathfrak D(B)\bigr)\) is some root of equation (1), then

\[ \lambda^2I+\lambda B+C=(\lambda I-\hat Z_0)(\lambda I-Z_0), \]

where \(\hat Z_0=-B-Z_0\). If \(B=B^*\) and \(C=C^*\in\mathfrak R\), then together with \(Z_0\) the operator \(\hat Z_0^*\) will also be a root of equation (1) (in the case of unbounded \(B\), in the sense that \(Z^2x+BZx+Cx=0\) for every \(x\in\mathfrak D(B^2)\)). It may happen that \(\hat Z_0^*=Z_0\) (see Theorem 2).

* If \(\lambda_0\) is an eigenvalue of the pencil \(L(\lambda)\), then the root subspace \(\mathfrak L_{\lambda_0}(L)\) of the pencil is the linear span of all eigenvectors and associated vectors of the pencil \(L(\lambda)\) corresponding to the number \(\lambda_0\) (see \((^1)\)).

** The condensation spectrum of the operator \(B\) comprises all limit points of its spectrum \(\sigma(B)\) and its eigenvalues of infinite multiplicity.

Theorem 1. Let \(B=B^*\) (\(\|B\|\leqslant \infty\)), and let \(C\in\mathfrak{R}\) be a nonnegative operator having a \(B\)-completely continuous nonnegative root \(C^{1/2}\). Then, for any partition of the nonreal spectrum \(\sigma_0(L)\) of the pencil \(L(\lambda)\) into two disjoint parts \(\Lambda\) and \(\overline{\Lambda}=\sigma_0(L)\setminus\Lambda\), symmetrically situated with respect to the real axis, equation (1) has a root \(Z_\Lambda\) \((Z_\Lambda\in\mathfrak{R},\ \mathfrak{R}(Z_\Lambda)\subset\mathfrak{D}(B))\) with the properties: 1) \(Z_\Lambda^*Z_\Lambda\leqslant C\), 2) the nonreal part of the spectrum \(\sigma(Z_\Lambda)\) coincides with \(\Lambda\).

Let us note that, under the conditions of the theorem, properties 1) and 2) of the root \(Z_\Lambda\) imply the property: 3) \(\sigma(Z_\Lambda)\subset \sigma(L)\cup\{0\}\), and for every \(\lambda_0\in\Lambda\) the root subspace \(\mathfrak{L}_{\lambda_0}(Z_\Lambda)\) coincides with the root subspace \(\mathfrak{L}_{\lambda_0}(L)\) of the pencil \(L(\lambda)\).

We give some explanations for the proof of the theorem. Take two copies \(\mathfrak{H}_1\) and \(\mathfrak{H}_2\) of the space \(\mathfrak{H}\) and form their direct orthogonal sum \(\widetilde{\mathfrak{H}}=\mathfrak{H}_1\oplus\mathfrak{H}_2\). Put \(J=P_1-P_2\), where \(P_j\) \((j=1,2)\) is the projector orthogonally projecting \(\widetilde{\mathfrak{H}}\) onto \(\mathfrak{H}_j\). A subspace \(M\subset\widetilde{\mathfrak{H}}\) is called maximal \(J\)-nonnegative if it is a maximal subspace with the property \((J\tilde{x},\tilde{x})\geqslant 0\) for \(\tilde{x}\in M\). Every such subspace \(M\) is given by its angular operator \(K\) according to the rule: \(M=\{x_1+Kx_1:\ x_1\in\mathfrak{H}_1\}\), where \(K\) is a certain nonexpanding operator \((\|K\|\leqslant 1)\) mapping \(\mathfrak{H}_1\) into \(\mathfrak{H}_2\) \(({}^3,{}^4)\). Consider the operator \(H=\|H_{jk}\|_1^2\) \((H_{jk}=P_jHP_k;\ j,k=1,2)\), for which
\[ H_{11}=0,\qquad H_{12}=C^{1/2},\qquad H_{21}=-C^{1/2},\qquad H_{22}=-B. \]
This operator is \(J\)-self-adjoint (i.e. \((JH)^*=JH\)). The conditions of the theorem mean that Theorem 4 of \(({}^5)\) is applicable to \(H\), by virtue of which \(H\) has at least one maximal \(J\)-nonnegative subspace \(M\) \((\subset\mathfrak{D}(H))\), invariant with respect to \(H\). It is easy to verify that the angular operator \(K\) of every such subspace \(M\) satisfies the relation
\[ KC^{1/2}K+BK+C^{1/2}=0, \]
and therefore \(Z_0=KC^{1/2}\) satisfies equation (1) and has property 1).

A more complete use of Theorem 4 of \(({}^5)\) makes it possible to establish the existence of a root \(Z_\Lambda\) which also has property 2).

Theorem 2. If, under the conditions of Theorem 1, the linear manifolds \(\mathfrak{L}_\lambda(L)\) \((\lambda\in\Lambda)\) form a complete system in \(\mathfrak{H}\), then the root \(Z_\Lambda\) is uniquely determined by the given \(\Lambda\). In this case \(Z_\Lambda\in\mathfrak{S}_\infty\), and the factorization
\[ \lambda^2 I+\lambda B+C=(\lambda I-Z_\Lambda^*)(\lambda I-Z_\Lambda), \]
holds, and consequently
\[ B=-(Z_\Lambda+Z_\Lambda^*)\in\mathfrak{S}_\infty \quad\text{and}\quad C=Z_\Lambda^*Z_\Lambda\in\mathfrak{S}_\infty . \]

1. If \(B\in\mathfrak{R}\), then the condition that \(C^{1/2}\) be \(B\)-completely continuous is equivalent to the condition \(C\in\mathfrak{S}_\infty\). In what follows, we consider mainly the case when \(C\in\mathfrak{S}_\infty,\ C\geqslant 0\). In this case, for any \(B=B^*\), all the conditions of Theorem 1 are satisfied,* and the root \(Z_\Lambda\) given by this theorem, by virtue of its property 1), will be completely continuous. Let
   \[ \lambda_1(C)\geqslant \lambda_2(C)\geqslant\cdots \]
   be the successive eigenvalues of the operator \(C\), and let \(\{\lambda_j(Z_\Lambda)\}_1^N\) \((N\leqslant\infty)\) be the complete sequence of all nonzero eigenvalues (counting their algebraic multiplicities) of the operator \(Z_\Lambda\), arranged in some order of nondecreasing moduli. By a result of T. Weyl \(({}^8)\), from property 1) of the root \(Z_\Lambda\) it follows that for any function \(f(r)\) \((0\leq r<\infty;\ f(+0)=0)\), to which there corresponds a function \(f(e^t)\) convex downward \((-\infty<t<\infty)\), the inequalities
   \[ \sum_{j=1}^{n} f\bigl(|\lambda_j(Z_\Lambda)|\bigr) \leqslant \sum_{j=1}^{n} f\bigl(\sqrt{\lambda_j(C)}\bigr) \qquad (n=1,2,\ldots,N). \tag{2} \]

* Let us note that in this important case Theorem 1 is proved in the same way on the basis of the results of \(({}^6,{}^7)\), earlier than \(({}^5)\).

It turns out that if \(C\;(\in \mathfrak S_\infty)\) is a positive operator \(((Cx,x)>0\) for \(x\ne0)\), \(N=\infty\), and for some strictly convex function \(f(e^t)\) the right-hand side in (2) is finite for \(n=\infty\), then for this function the sign \(=\) occurs in (2) for \(n=\infty\) if and only if the operator \(B\) is permutable with the operator \(C\) and the condition \((Bx,x)^2 \leq 4(x,x)(Cx,x)\) is satisfied.

1. We shall be interested in the condition of weak damping of the pencil \(L(\lambda)\):

\[ (Bx,x)^2 < 4(x,x)(Cx,x)\qquad (x\ne0). \tag{3} \]

It is easy to see that if this condition is satisfied (for \(x\in\mathfrak D(B)\), \(x\ne0\)), then, together with the operator \(C\in\mathfrak S_\infty\), also \(B\in\mathfrak S_\infty\) \((\mathfrak D(B)=\mathfrak H)\). Condition (3) is equivalent to the condition of positivity of the operator \(L(\lambda)\) for every real \(\lambda\). If it is known in advance that \(B=B^*\in\mathfrak S_\infty\) and \(C=C^*\in\mathfrak S_\infty\), then the fulfillment of (3) is a necessary and sufficient condition that the spectrum \(\sigma(L)\) contain no real points \(\lambda\ne0\). In this case \(\sigma(L)\) consists of \(\Lambda\cup\overline{\Lambda}\) and the point \(\lambda=0\), and inequalities (2) give a certain characterization of the distribution of the moduli of the eigenvalues of the whole spectrum \(\sigma(L)\).

Theorem 3. Let \(B\) be a nonnegative nuclear operator \((\operatorname{Sp} B<\infty)\), and let condition (3) be satisfied. Then the relation

\[ -\sum_j \operatorname{Re}\lambda_j \leq \operatorname{Sp} B, \tag{4} \]

holds, where the summation extends over all eigenvalues (counting their multiplicities) of the pencil \(L(\lambda)\). Equality in relation (4) holds if and only if the system of root vectors of the root \(\lambda\) is complete in \(\mathfrak H\). This case occurs whenever \(\liminf n^2\lambda_n(C)=0\), and, in particular, when \(\operatorname{Sp}(C^{1/2})<\infty\).

1. Let us introduce some general definitions for the pencil \(L(\lambda)\) with \(C\gg0\). Let \(x_0\) be an eigenvector of the pencil \(L(\lambda)\) corresponding to the eigenvalue \(\lambda_0\). Then three cases are possible: the quantity \(|\lambda_0|^2\) may be equal to, less than, or greater than the ratio \((Cx_0,x_0)/(x_0,x_0)\). Corresponding to these cases, the eigenvector \(x_0\) is called neutral, of the first kind, or of the second kind.

If \(\lambda_0\) is nonreal, then the eigenvector \(x_0\) will necessarily be neutral.

If the eigenvalue \(\lambda_0\) has corresponding eigenvectors of one and the same kind (first or second), then their totality (the eigensubspace \(\mathfrak E_{\lambda_0}(L)\)) will coincide with the root subspace \(\mathfrak L_{\lambda_0}(L)\).

We now consider a pencil \(L(\lambda)\) for which the condition of strong damping is satisfied:

\[ (Bx,x) > 2\sqrt{(Cx,x)(x,x)}\qquad (x\in\mathfrak D(B),\ x\ne0). \tag{5} \]

This condition entails the uniform positivity of the operator \(B\), i.e. \(\inf [(Bx,x)/(x,x)]>0\). When it is fulfilled, the spectrum of the pencil \(L(\lambda)\) is situated on the negative semiaxis and the pencil has no neutral eigenvectors.

Theorem 4. Let the positive operator \(C\in\mathfrak S_\infty\), \(B\in\mathfrak R\), and let condition (5) be satisfied.

Then

\(1^\circ\). The quadratic equation (1) has one and only one root \(Z_1\) \((Z_2)\) with the property \(Z_1^*Z_1\leq C\) \((Z_2^*Z_2\geq C)\), and in addition

\[ Z_2=-B-Z_1^*,\qquad Z_2^*Z_1=C. \]

\(2^\circ\). The operator

\[ S=B+Z_1+Z_1^*=-(B+Z_2+Z_2^*)=Z_1-Z_2 \]

is a uniformly positive operator.

\(3^\circ\). The roots \(Z_k\) \((k=1,2)\) are symmetrized by the operator \(S\) \((SZ_k=Z_k^*S,\ k=1,2)\), and, moreover, they are similar to negative operators; furthermore,

\[ (SZ_2y,y)<(SZ_1x,x)<0 \]

for any \(x,y\in\mathfrak H\) with \(\|x\|=\|y\|=1\).

4°. The spectrum
\[ \sigma(L)=\sigma(Z_1)\cup\sigma(Z_2), \]
and
\[ \max\{\lambda:\lambda\in\sigma(Z_2)\}\leq \min\{\lambda:\lambda\in\sigma(Z_1)\}. \]

5°. The eigenvectors of the root \(Z_1\) \((Z_2)\) exhaust all eigenvectors of the first (second) kind of the pencil \(L(\lambda)\).

Theorem 4 can easily be reformulated for the case of nonnegative \(C\in \mathfrak S_\infty\), and also for the case of an unbounded \(B\;(=B^*)\).

We also omit possible generalizations of Theorem 4 to the case of bounded operators \(C\;( \geq 0)\) with a \(B\)-completely continuous square root \(C^{1/2}\).

Let us note that the task of establishing Theorem 4 was somewhat facilitated for the authors by the original work \((^9)\), in which, by other methods, oscillations of strongly damped systems with a finite number of degrees of freedom were investigated.

In conclusion we point out that our methods make it possible to establish a number of propositions for the pencil \(L(\lambda)\) also in the case of an indefinite operator \(C=C^*\). In this connection we note that the case of a nonpositive \(C\) admits treatment within the framework of the usual spectral theory of self-adjoint operators \((^{10,11})\); however, even in this case the study of the quadratic equation (1) leads to more general and precise results.

A more detailed exposition of the present results will be given in the “Proceedings of the International Symposium on Applications of the Theory of Functions of a Complex Variable in Continuum Mechanics,” where the authors will also dwell on questions of the twofold completeness of eigenvectors and associated vectors of the pencil
\[ \lambda^2L(\lambda^{-1})=I+\lambda B+\lambda^2C, \]
studied under other assumptions by M. V. Keldysh \((^1)\).

Odessa Civil Engineering Institute

Dresden Technical University
Dresden, GDR

Received
17 X 1963

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