Abstract
Full Text
MATHEMATICS
J. E. Allahverdiev
ON THE COMPLETENESS OF THE SYSTEM OF EIGEN AND ASSOCIATED ELEMENTS OF NON-SELF-ADJOINT OPERATORS CLOSE TO NORMAL ONES
(Presented by Academician M. V. Keldysh on 20 February 1957)
In the paper ((^{1})), M. V. Keldysh studied the question of the completeness of the system of eigen and associated elements of non-self-adjoint equations
[
y=\left(A_{0}+\lambda H^{1/n}A_{1}+\ldots+\lambda^{\,n-1}H^{(n-1)/n}A_{n-1}+\lambda^{n}H\right)y,
\tag{1}
]
where (A_i) ((i=0,1,\ldots,n-1)) are completely continuous operators; (H) is a complete completely continuous self-adjoint operator of finite order. In the same paper the resolvent of equations of the form
[
y=(K_{1}+\lambda K_{2}+\ldots+\lambda^{n}K_{n})y+f,
]
where (K_i) ((i=1,2,\ldots,n)) are completely continuous operators, was investigated.
Following M. V. Keldysh, in the present note we consider the equation
[
y=\left(A+\lambda H^{1/n}A_{1}+\ldots+\lambda^{\,n-1}H^{(n-1)/n}A_{n-1}+\lambda^{n}H\right)y\equiv L(\lambda)y,
\tag{2}
]
where (A) is a bounded operator; (A_i) ((i=1,2,\ldots,n-1)) are completely continuous operators; (H) is a complete completely continuous normal operator, and it is proved that, under certain conditions imposed on (A) and (H), the system of eigen and associated elements of equation (2) (of the operator (L(\lambda))) is (n)-fold complete in the Hilbert space (\mathcal H).
Everywhere we shall adhere to the terminology of the article ((^{1})).
Let (H) be a completely continuous normal operator. The order of the operator (H) is called the lower bound (\rho) of the numbers (\alpha) for which (H^\alpha) has finite absolute norm; for (\rho<\infty), (H) is called an operator of finite order.
Let (G) be the plane of the complex variable, in which the eigenvalues of the operator (\lambda H) are located.
All rays discussed below issue from the origin. We shall say that a ray belongs to the class (\mathfrak K_\beta) if it is the bisector of some angle with aperture (2\beta), whose vertex is at the origin and inside which there may lie only a finite number of eigenvalues of the operator (\lambda H). We shall say that a ray with argument (\varphi) belongs to the class (\mathfrak K_\beta) if, upon rotation through the angle ((n-1)\varphi), it coincides with some ray of the class (\mathfrak K_\beta) (the argument of a ray is the angle between this ray and the positive direction of the real axis). We shall say that the class of rays (\mathfrak K) is (\varepsilon)-dense in (G) if inside every angle with aperture less than (\varepsilon) there is at least one ray from (\mathfrak K).
Let (A) be a bounded operator; then the operator (A) can be represented in the form of a sum (A=A' + B'), where (A') is completely continuous and (B') is a bounded operator. Such a representation is not unique. We shall call (B_0) the purely bounded part of the operator (A), if (A=A_0+B_0), where (A_0) is a completely continuous operator, and, moreover, from the representation (A=A_1+B_1), where (A_1) is a completely continuous operator, it follows that (|B_1|\geqslant |B_0|). Concerning the completeness of the eigenfunctions and associated functions of equation (2), the following theorem is valid.
Theorem 1. Let (H) be a completely continuous complete normal operator of finite order (\rho); let (A) be a bounded operator with purely bounded part (B_0); and let (A_i) ((i=1,2,\ldots,n-1)) be completely continuous operators.
If, for some (\varepsilon \leqslant \pi/\rho n), the class of rays (\mathfrak K_\beta^n) for (\sin\beta>|B_0|) is (\varepsilon)-dense in (G), then the systems of eigen and associated elements of each of equations (2) and (2) (of the operators (L(\lambda)) and (L^(\lambda)))
[
y=\bigl[A^+\lambda A_1^(H^{1/n})^+\ldots+\lambda^{\,n-1}A_{n-1}^(H^{(n-1)/n})^+\lambda^n H^\bigr]y\equiv L^(\lambda)y
\tag{2}
]
are (n)-fold complete in the Hilbert space (\mathfrak H).
We shall precede the proof of Theorem 1 by the following lemma.
Lemma. If the conditions of Theorem 1 are fulfilled, then the operator
[
C(\lambda)=\bigl[E+T(\lambda^n)\bigr]\bigl[A+\lambda H^{1/n}A_1+\ldots+\lambda^{\,n-1}H^{(n-1)/n}A_{n-1}\bigr]
\tag{3}
]
has a meromorphic resolvent (B(\lambda)), which is bounded on all rays of (\mathfrak K_\beta^n) (generally speaking, by a number depending on the ray), and the resolvents (R(\lambda)) and (R^(\lambda)) of the operators (L(\lambda)) and (L^(\lambda)) are representable in the form
[
R(\lambda)=T(\lambda^n)+B(\lambda)\bigl[E+T(\lambda^n)\bigr],
\tag{4}
]
[
R^(\lambda)=T^(\lambda^n)+\bigl[E+T^(\lambda^n)\bigr]\cdot B^(\lambda),
\tag{4'}
]
where (T^(\lambda)) is the resolvent of the operator (\lambda H).*
We outline the proof. Using the conditions of the lemma, we prove that on each ray of (\mathfrak K_\beta^n), (|C(\lambda)|<1-\varepsilon) for some (\varepsilon>0) ((\varepsilon<1)) and for sufficiently large (|\lambda|). Consequently, for sufficiently large (|\lambda|), (B(\lambda)) exists and is bounded on each ray of (\mathfrak K_\beta^n). From (4) we obtain that (R(\lambda)) exists (and is bounded for sufficiently large (|\lambda|)) on each ray of (\mathfrak K_\beta^n). Hence it follows that (R(\lambda)) exists on the whole plane and is a meromorphic function of (\lambda). From (4) we have
[
B(\lambda)=\bigl[R(\lambda)-T(\lambda^n)\bigr][E-\lambda H].
]
Consequently, (B(\lambda)) exists on the whole plane and is a meromorphic function of (\lambda). Thus the lemma is completely proved.
For the proof of Theorem 1 we shall use the following theorem, due to M. V. Keldysh.
Theorem. Let (H) be a completely continuous self-adjoint operator such that (\sum \frac{1}{|h_i|^\rho}<\infty) ((h_i) are the eigenvalues of the operator (\lambda H)); let (K_1,K_2,\ldots,K_n) be bounded operators. Denote by (\lambda_i) the eigenvalues of the equation
[
y=(\lambda K_1H+\ldots+\lambda^n K_nH^n)y,
]
and by (R(\lambda)) its resolvent. We have:
-
(\displaystyle \sum \frac{1}{|\lambda_j|^\rho}<\infty.)
-
(\displaystyle R(\lambda)=\frac{D(\lambda)}{\Delta(\lambda)},) where (D(\lambda)) is an operator function of order not exceeding (\rho);
[
\Delta(\lambda)=\prod_j\left(1-\frac{\lambda}{\lambda_j}\right)
\exp\left[\sum_{i=1}^{m}\frac{1}{i}\left(\frac{\lambda}{\lambda_j}\right)^i\right],
]
(m) being the largest integer satisfying the inequality (m<\rho).
Let us prove Theorem 1. Suppose that the theorem is false. Then there exist (n) elements (f_0,\ldots,f_{n-1}) such that
[
\sum_{\nu=0}^{n-1}(f_\nu,Y_h^{k,\nu})=0,
\tag{5}
]
where (Y_h^{k,\nu}) are the derived chains of eigen and associated elements of the operator (L(\lambda)).
Consider the equation
[
y=\bigl[A^+\lambda A_1^(H^{1/n})^+\cdots+\lambda^{n-1}A^(H^{(n-1)/n})^
+\lambda^n H^\bigr]y+f(\lambda),
\tag{6}
]
where (f(\lambda)=f_0+\lambda f_1+\cdots+\lambda^{n-1}f_{n-1}).
From (6) we obtain (y=[E+R^(\lambda)]f(\lambda)), where (R^(\lambda)) is the resolvent of (L^(\lambda)). Using the theorem of M. V. Keldysh, it is easy to prove that (y(\lambda)=[E+R^(\lambda)]f(\lambda)) grows on each ray of (\mathcal K_\beta^n) no faster than a polynomial of degree (n-1). Calculating the principal parts of (R^(\lambda)f(\lambda)) for the function (f(\lambda)=f_0+\lambda f_1+\cdots+\lambda^{n-1}f_{n-1}) and using the equalities (5), we find that ([E+R^(\lambda)]f(\lambda)) is an entire function of (\lambda).
By Lindelöf’s theorem, taking into account the behavior of (y(\lambda)) on the rays from (\mathcal K_\beta^n), we find that (y(\lambda)) grows in the whole plane no faster than (\lambda^{n-1}). Hence,
[
y(\lambda)=y_0+\lambda y_1+\cdots+\lambda^{n-1}y_{n-1}.
]
Substituting (y(\lambda)) into equation (6) and comparing the coefficients of equal powers on the left- and right-hand sides of the equation, we conclude that all (y_i) ((i=0,1,\ldots,n-1)) are equal to zero. But this contradicts the fact that at least one of (f_i) ((i=0,1,\ldots,n-1)) is nonzero and that (y(\lambda)) is a solution of equation (6).
Thus the theorem is proved for equation (2). The proof for ((2^*)) is analogous.
Concerning the distribution of the eigenvalues of equation (2) (and also of ((2^*))), the following theorem is valid.
Theorem 2. If, under the conditions of Theorem 1, the eigenvalues of the operator (\lambda H) are situated on a finite number of rays (we denote the arguments of the rays by (\varphi_1,\ldots,\varphi_k)), then, for any (\varepsilon) for which (\sin\varepsilon>|B_0|), outside the angles
[
\psi_{ij}-\varepsilon\leq \arg\lambda\leq \psi_{ij}+\varepsilon,
\tag{7}
]
where
[
\psi_{ij}=\frac{2\pi j+\varphi_i}{n},\quad
i=1,2,\ldots,k;\quad j=0,1,\ldots,n-1,
]
there can be only a finite number of eigenvalues of equation (2).
Let us outline the proof. From the definition of (\mathcal K_\beta^n) it follows that outside the angles (7) all rays belong to (\mathcal K_\beta^n) when (\sin\beta>|B_0|). Consequently, for (\sin\varepsilon>|B_0|), outside the angles (7) there can be only a finite number of eigenvalues of equation (2). If (B_0=0), then we obtain that for any (\varepsilon>0)
outside the angles (7) there can be only a finite number of eigenvalues of equation (2). Consequently, the eigenvalues of equation (2) approach asymptotically the rays
[
\arg \lambda=\psi_{ij},\qquad i=1,2,\ldots,k;\ j=0,1,\ldots,n-1.
]
Let us note that Theorem 2 is also true in the case where the eigenvalues (\lambda H) approach asymptotically the rays (\arg \lambda=\varphi_i,\ i=1,2,\ldots,k).
In conclusion, I sincerely thank Academician M. V. Keldysh, under whose supervision the present work was carried out.
Moscow State University
named after M. V. Lomonosov
Received
19 II 1957
REFERENCES
- M. V. Keldysh, DAN, 77, No. 1 (1951).
- J. E. Allakhverdiev, Dokl. Acad. Sci. Azerbaijan SSR, 13, No. 5, 469 (1957).