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UDC 513.88
MATHEMATICS
V. M. ENI
ON THE STABILITY OF THE ROOT NUMBER OF AN ANALYTIC OPERATOR-FUNCTION AND ON PERTURBATIONS OF ITS CHARACTERISTIC NUMBERS AND EIGENVECTORS
(Presented by Academician I. N. Vekua on 9 VI 1966)
In the present note, the theorem of I. Ts. Gohberg and M. G. Krein ((¹), Theorem 4.3) on the stability of the root number of a linear operator and the theorem on analytic perturbations of characteristic numbers and eigenvectors of a linear operator (see, for example, (²)) are extended to the case of an analytic operator-function.
- Let \(G\) be some domain in the complex plane, and let \(A(\lambda)\) be an operator-function analytic in the domain \(G\), whose values are bounded linear operators acting in a Banach space \(\mathfrak{B}\). A number \(\lambda_0 \in G\) is called a characteristic number of the operator-function \(A(\lambda)\) if the equation \(A(\lambda_0)\varphi = 0\) has a nontrivial solution \(\varphi_0\), and the vector \(\varphi_0\) is called an eigenvector of the operator-function \(A(\lambda)\) corresponding to \(\lambda_0\). The number \(a(A(\lambda_0)) = \dim \mathfrak{Z}(A(\lambda_0))\), where \(\mathfrak{Z}(A(\lambda_0))\) is the subspace of all eigenvectors corresponding to \(\lambda_0\), will be called the geometric multiplicity of the characteristic number \(\lambda_0\). A set of vectors \(\varphi_1, \varphi_2, \ldots, \varphi_k\) is called a chain of vectors associated with the eigenvector \(\varphi_0\) of the operator-function \(A(\lambda)\), corresponding to the number \(\lambda_0\), if
\[ \sum_{q=0}^{j} \frac{1}{q!} A^{(q)}(\lambda_0)\varphi_{j-q} = 0 \qquad (j = 1, 2, \ldots, k). \]
The number \(k+1\) is called the length of this chain. By the multiplicity of the eigenvector \(\varphi_0\) we shall mean the number \(m(\varphi_0)\), equal to the maximal length of a chain of vectors associated with \(\varphi_0\). A canonical system of eigenvectors and associated vectors of the operator-function \(A(\lambda)\), corresponding to the number \(\lambda_0\), is a system \(\varphi_{j0}, \varphi_{j1}, \ldots, \varphi_{j,m_j-1}\) \((j = 1, 2, \ldots, \alpha = \alpha(A(\lambda_0)))\), where \(\varphi_{j0}\) is an eigenvector of multiplicity \(m_j\), and \(\varphi_{j1}, \varphi_{j2}, \ldots, \varphi_{j,m_j-1}\) is a chain of vectors associated with it, with \(m_1\) being the maximal of the multiplicities of all eigenvectors corresponding to \(\lambda_0\), and \(m_j\) \((j = 2, 3, \ldots, \alpha)\) being the maximal of the multiplicities of all eigenvectors from some direct complement in \(\mathfrak{Z}(A(\lambda_0))\) to the subspace spanned by \(\varphi_{10}, \ldots, \varphi_{j-1,0}\). The number
\[ m(\lambda_0, A) = \sum_{j=1}^{\alpha} m_j \]
is called the algebraic multiplicity of the characteristic number \(\lambda_0\) of the operator-function \(A(\lambda)\). A point \(\lambda\) will be called a regular point of the operator-function \(A(\lambda)\) if the operator \(A(\lambda)\) is continuously invertible, and a \(\Phi\)-point of the operator-function \(A(\lambda)\) if the operator \(A(\lambda)\) is a \(\Phi\)-operator (see (¹), p. 52).
Let \(\Gamma\) be a closed rectifiable contour situated in the domain \(G\) and having, with respect to the operator-function \(A(\lambda)\), the following properties:
a) inside the contour \(\Gamma\) there is a finite number of characteristic numbers \(\lambda_1,\lambda_2,\ldots,\lambda_s\) of the operator function \(A(\lambda)\), which are its \(\Phi\)-points;
b) all the remaining points \(\lambda\) \((\lambda\ne\lambda_j,\ j=1,2,\ldots,s)\) situated inside the contour and on the contour \(\Gamma\) are regular points of the operator function \(A(\lambda)\).
From Theorem 1 of A. S. Markus (³) it follows that \(m(\lambda_j,A)<\infty\) \((j=1,2,\ldots,s)\). The root number of the operator function \(A(\lambda)\), corresponding to the contour \(\Gamma\), will mean the sum of the algebraic multiplicities of all characteristic numbers \(\lambda_j\) \((j=1,2,\ldots,s)\) of the operator function \(A(\lambda)\) located inside the contour \(\Gamma\), i.e. the number
\[ m(A,\Gamma)=m(\lambda_1,\Gamma)+m(\lambda_2,\Gamma)+\cdots+m(\lambda_s,A). \]
Theorem 1. Let \(\Gamma\) be a closed rectifiable contour bounding a domain \(G\), and let \(A(\lambda)\) be an operator function, analytic in \(G\) and continuous in the closed domain \(G\cup\Gamma\), with the contour \(\Gamma\) possessing, relative to the operator function \(A(\lambda)\), properties a), b). Then there exists a number \(\delta>0\) such that, whatever the operator function \(B(\lambda)\), analytic in \(G\), continuous in \(G\cup\Gamma\), and satisfying the condition
\[ \max_{\lambda\in\Gamma}\|B(\lambda)-A(\lambda)\|<\delta, \tag{1} \]
the contour \(\Gamma\) possesses properties a), b) also relative to \(B(\lambda)\), and moreover
\[ m(B,\Gamma)=m(A,\Gamma). \]
For the proof of Theorem 1 the following two lemmas are needed:
Lemma 1. Let \(\{A_j\}_0^\infty\) be a sequence of linear bounded operators acting in a Banach space \(\mathfrak{B}\) and satisfying the condition \(\sup\|A_j\|<\infty\), and
\[ A(\lambda)=\sum_{j=0}^{\infty}\lambda^j A_j \qquad (|\lambda|<1). \]
Let, further, \(\widetilde{\mathfrak{B}}\) be the Banach space of all sequences \(\widetilde{x}=\{x_j\}_1^\infty\) of elements from \(\mathfrak{B}\) satisfying the condition
\[ \|\widetilde{x}\|=\sum_{j=1}^{\infty}\|x_j\|<\infty, \]
and let \(\widetilde{A}_0,\widetilde{C}\) be the operators defined in \(\widetilde{\mathfrak{B}}\) by the matrices
\[ \widetilde{A}_0= \begin{pmatrix} A_0 & A_1 & \cdots & A_n & \cdots\\ 0 & I & \cdots & 0 & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & \cdots & I & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots \end{pmatrix}, \qquad \widetilde{C}= \begin{pmatrix} 0 & 0 & \cdots & 0 & \cdots\\ -I & 0 & \cdots & 0 & \cdots\\ 0 & -I & \cdots & 0 & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots \end{pmatrix}. \]
A set of vectors \(\widetilde{\varphi}_j=\{\varphi_j^{(k)}\}_{k=0}^{\infty}\) \((j=0,1,\ldots,p)\) is then and only then a chain of eigenvectors and associated vectors of the linear pencil \(\widetilde{A}(\lambda)=\widetilde{A}_0+\lambda\widetilde{C}\), corresponding to the characteristic number \(\lambda\) \((|\lambda|<1)\), when \(\varphi_j^{(0)}\) \((j=0,1,\ldots,p)\) is a chain of eigenvectors and associated vectors of the operator function \(A(\lambda)\), corresponding to the characteristic number \(\lambda\), and
\[ \varphi_0^{(k)}=\lambda\varphi_0^{(k-1)} \qquad (k=1,2,\ldots,\infty); \quad \varphi_j^{(k)}=\lambda\varphi_j^{(k-1)}+\varphi_{j-1}^{(k-1)} \]
\[ (j=1,2,\ldots,p;\ k=1,2,\ldots,\infty). \]
Lemma 2. Let \(A(\lambda)\) and \(B(\lambda)\) be operator-functions, analytic inside the circle \(|\lambda|<r\) and continuous in the closed circle \(|\lambda|\le r\), and let \(\Gamma\) be a closed rectifiable contour lying inside the circle \(|\lambda|<r\). If the contour \(\Gamma\) has, with respect to the operator-function \(A(\lambda)\), properties a), b), then there exists a number \(\delta>0\) such that, when the condition
\[ \max_{|\lambda|=r}\|B(\lambda)-A(\lambda)\|<\delta \]
is satisfied, the contour \(\Gamma\) has properties a), b) also with respect to the operator-function \(B(\lambda)\), and moreover
\[ m(B,\Gamma)=m(A,\Gamma). \]
Lemma 1 is a generalization of a well-known property of polynomial operator pencils (see (4), p. 326). Lemma 2 is proved on the basis of Lemma 1 and the theorem from (5).
Proof of Theorem 1. Let \(\lambda_1,\lambda_2,\ldots,\lambda_s\) be all the characteristic numbers of the operator-function \(A(\lambda)\) in the domain \(G\) (without regard to multiplicity). Choose numbers \(\varepsilon_j>0\) \((j=1,2,\ldots,s)\) such that the disks \(G_j=\{\lambda:\ |\lambda-\lambda_j|<\varepsilon_j\}\) lie in \(G\) and are pairwise disjoint. Put \(\gamma_j=\{\lambda:\ |\lambda-\lambda_j|=\varepsilon_j/2\}\). By Lemma 2, for each \(j=1,2,\ldots,s\) there exists a number \(\delta_j>0\) such that from the inequality
\[ \max_{|\lambda-\lambda_j|=\varepsilon_j}\|B(\lambda)-A(\lambda)\|<\delta_j \tag{2} \]
it follows that the contours \(\gamma_j\) have, with respect to \(B(\lambda)\), properties a), b), and moreover
\[ m(B,\gamma_j)=m(A,\gamma_j). \tag{3} \]
Denote by \(F\) the closed set obtained by removing from \(G\cup\Gamma\) the open disks \(|\lambda-\lambda_j|<\varepsilon_j/2\) \((j=1,2,\ldots,s)\). For \(\lambda\in F\) the operator \(A(\lambda)\) is invertible. Put
\[ M=\max_{\lambda\in F}\|A^{-1}(\lambda)\|. \]
We now show that the number
\[ \delta=\min(\delta_1,\delta_2,\ldots,\delta_s,1/M) \]
has the required property. Indeed, if inequality (1) is satisfied, then for any point \(\lambda\in F\)
\[ \|B(\lambda)-A(\lambda)\|<\delta\le 1/M\le 1/\|A^{-1}(\lambda)\|, \]
and therefore the operator \(B(\lambda)\) is invertible.
On the other hand, from (1) it follows that, for all \(j=1,2,\ldots,s\), (2) holds, and therefore all contours \(\gamma_j\) have, with respect to \(B(\lambda)\), properties a), b), and the equalities (3) hold.
From what has been said above it follows directly that the contour \(\Gamma\) has properties a), b) with respect to the operator-function \(B(\lambda)\) and
\[ m(B,\Gamma)=\sum_{j=1}^{s}m(B,\gamma_j)=\sum_{j=1}^{s}m(A,\gamma_j)=m(A,\Gamma). \]
The theorem is proved.
2. Theorem 2. Let \(\{A_j(\varepsilon)\}_0^\infty\) be a sequence of operator-functions, analytic in some disk \(|\varepsilon|<\delta\) and satisfying the condition
\[ \sup_j\|A_j(\varepsilon)\|<\infty\qquad (|\varepsilon|<\delta); \]
\[ A_\varepsilon(\lambda)=\sum_{j=0}^{\infty} A_j(\varepsilon)\lambda^j \qquad (|\lambda|<1,\ |\varepsilon|<\delta). \]
Let \(\Gamma\) be a closed rectifiable contour lying in the circle \(|\lambda|<1\), inside which there is a single characteristic number \(\lambda_0\) of the operator-function \(A_0(\lambda)\), of finite algebraic multiplicity \(m\). Then:
1) There exists a number \(\rho\) \((0<\rho\leq \delta)\) and a natural number \(r\) \((\leq m)\) such that, for \(0<|\varepsilon|<\rho\), the operator-function \(A_\varepsilon(\lambda)\) has exactly \(r\) distinct characteristic numbers inside the contour \(\Gamma\). These characteristic numbers can be arranged into groups \(\lambda_{ij}(\varepsilon)\) \((i=1,2,\ldots,l;\ j=1,2,\ldots,p_i;\ \sum_{i=1}^{l} p_i=r)\) in such a way that the functions of one group (i.e.
\[ \lambda_{i1},\lambda_{i2},\ldots,\lambda_{ip_i}) \]
constitute the complete set of branches of a \(p_i\)-valued function; moreover, in the circle \(|\varepsilon|<\rho\) they are represented in the form of series
\[ \lambda_{ij}(\varepsilon) = \lambda_0+ \sum_{k=1}^{\infty} a_{ik}\left[(\varepsilon^{1/p_i})_j\right]^k \quad (j=1,2,\ldots,p_i;\ i=1,2,\ldots,l), \]
where \((\varepsilon^{1/p_i})_j\) \((j=1,2,\ldots,p_i)\) is the complete set of branches of the function \(\varepsilon^{1/p_i}\).
2) There are numbers \(m_i\) \((i=1,2,\ldots,l)\) such that \(m(\lambda_{ij}(\varepsilon),A_\varepsilon)=m_i\) \((j=1,2,\ldots,p_i;\ 0<|\varepsilon|<\rho)\), and
\[ \sum_{i=1}^{l}\sum_{j=1}^{p_i} m(\lambda_{ij}(\varepsilon),A_\varepsilon) = \sum_{i=1}^{\infty} p_i m_i = m . \]
3) There are numbers \(\alpha_i\leq \alpha(A_0(\lambda_0))\) \((i=1,2,\ldots,l)\) such that
\[ \alpha(A_\varepsilon(\lambda_{ij}))=\alpha_i \quad (j=1,2,\ldots,p_i;\ 0<|\varepsilon|<\rho). \]
4) A basis of the eigenspace \(\mathfrak{Z}(A_\varepsilon(\lambda_{ij}(\varepsilon)))\) can be represented, for \(0<|\varepsilon|<\rho\), in the form of series
\[ \varphi_{ij}^{(q)}(\varepsilon) = \sum_{k=0}^{\infty} \varphi_{ik}^{(q)} \left[(\varepsilon^{1/p_i})_j\right]^k \quad (q=1,2,\ldots,\alpha_i), \]
where \(\varphi_{ik}^{(q)}\) are certain vectors, and \(\varphi_{i0}^{(q)}\in \mathfrak{Z}(A_0(\lambda_0))\).
The proof of this theorem is carried out analogously to the proof of Theorem 2 of [6], with the aid of Lemma 1.
The author expresses his gratitude to I. Ts. Gokhberg and A. S. Markus for their constant attention to the work.
Institute of Mathematics with Computing Center
Academy of Sciences of the Moldavian SSR
Received
8 VI 1966
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