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MATHEMATICS
V. B. LIDSKII
THEOREMS ON THE COMPLETENESS OF THE SYSTEM OF EIGEN- AND ASSOCIATED ELEMENTS OF OPERATORS WITH DISCRETE SPECTRUM
(Presented by Academician M. V. Keldysh on 30 XII 1957)
Let (T) be a completely continuous operator acting in a separable Hilbert space (\mathfrak H). The operator (T) is called an operator of Hilbert–Schmidt type (of H.–S. type) if
[
\sum_s (T\varphi_s, T\varphi_s)<\infty,
\tag{1}
]
where (\varphi_s) ((s=1,2,\ldots)) is an orthonormal basis in (\mathfrak H). In the case of the integral operator
[
Kf=\int_0^1 k(x,t)f(t)\,dt,
]
which acts in (\mathscr L_2(0,1)) (in the Hilbert space of functions whose square is Lebesgue integrable on the interval ((0,1))), condition (1) is equivalent to the condition
[
\int_0^1\int_0^1 |k(x,t)|^2\,dx\,dt<\infty.
\tag{2}
]
Theorem 1. Let an operator (T) of H.–S. type, acting in a separable Hilbert space (\mathfrak H), be written in the form
[
T=A+iB,
\tag{3}
]
and let the self-adjoint operators (A) and (B) be sign-definite (the quadratic forms ((Af,f)) and ((Bf,f)) preserve their sign). Then the eigen- and associated elements of the operator (T), corresponding to the nonzero points of the spectrum, form a system complete in the range of values of the operator (T). If to the indicated system one adds a basis in the subspace of solutions of the equation (Tf=0), then one obtains a system complete in (\mathfrak H).
For the proof, denote by (Q_1) the closed subspace spanned by all eigen- and associated elements of the operator (T) corresponding to nonzero points of the spectrum. Let (Q_2) be the orthogonal complement to (Q_1). It is easy to show that the operator (T^) adjoint to (T) is invariant on (Q_2). Denote by (T^{q_2}) the operator induced by the operator (T^) on (Q_2). One can show that zero is the only point of the spectrum of the operator (T^) (cf. ((^3))).
After these remarks, let us map (Q_2), preserving the scalar product, onto (\mathscr L_2(0,1)) (or onto some subspace of it, if (Q_2) is finite-dimensional) and consider in (\mathscr L_2(0,1)) an integral operator (K), unitarily equivalent to the operator (T^*_{q_2}). Such an operator is easy to construct. Let
(D^*(\lambda)) is the Fredholm determinant of the integral operator (K) ((1), p. 196):
[
D^*(\lambda)=1+\sum_{n=2}^{\infty}\frac{(-\lambda)^n}{n!}\delta_n,
\tag{4}
]
where
[
\delta_n=\int_0^1 \cdots \int_0^1
\begin{vmatrix}
0 & k(x_1,x_2) & \cdots & k(x_1,x_n)\
k(x_2,x_1) & 0 & \cdots & k(x_2,x_n)\
\cdots & \cdots & \cdots & \cdots\
k(x_n,x_1) & \cdots & \cdots & 0
\end{vmatrix}
\,dx_1\cdots dx_n .
\tag{5}
]
As Carleman showed ((1), p. 217), (D^*(\lambda)) is an entire function for which the representation
[
D^*(\lambda)=1\cdot\prod_{s=1}^{\infty}(1-\lambda\lambda_s)e^{\lambda\lambda_s}
\tag{6}
]
is valid.
Here (\lambda_s) denote the eigenvalues of the operator. Since in our case the operator (K) has no eigenvalues different from zero, on the basis of (6) we conclude that (D^*(\lambda)\equiv 1).
From formulas (4) and (5), then, in particular, it follows that
[
\delta_2=-\int_0^1\int_0^1 k(x_1,x_2)k(x_2,x_1)\,dx_1dx_2=0.
\tag{7}
]
On the other hand, (-\operatorname{Im}\delta_2) is the trace of the imaginary part* of the operator (K^2). Thus, (\operatorname{Sp}(\operatorname{Im}K^2)=0). Hence, by unitary equivalence, we conclude that also
[
\operatorname{Sp}(\operatorname{Im}T_{q_2}^{*2})=0.
\tag{8}
]
Represent the operator (T_{q_2}^) in the form (T_{q_2}^=A_{q_2}-iB_{q_2}), where (A_{q_2}) and (B_{q_2}) are self-adjoint operators defined in (Q_2), and compute the left-hand side of (8) in an orthonormal basis of eigenvectors of the operator (B_{q_2}). After simple transformations we obtain
[
\operatorname{Sp}(\operatorname{Im}T_{q_2}^{*2})
=
-\sum_s 2\mu_s\,(A_{q_2}\psi_s,\psi_s)=0,
\tag{9}
]
where (\mu_s) ((s=1,2,\ldots)) are the eigenvalues of (B_{q_2}).
The operators (A_{q_2}) and (B_{q_2}) are sign-definite together with (A) and (B) in (3). Therefore from equality (9) we conclude that
[
\mu_s(A_{q_2}\psi_s,\psi_s)=0
\tag{10}
]
for all (s=1,2,\ldots). We shall show that all eigenvalues of the operator (B_{q_2}) are equal to zero.
Suppose the contrary. Let (\mu_s^0\ne 0). Then from (10) it follows that ((A_{q_2}\psi_s^0,\psi_s^0)=0), where (\psi_s^0) is the eigen-element of (B_{q_2}) corresponding to (\mu_s^0). In view of the sign-definiteness of (A_{q_2}), this means that (A_{q_2}\psi_s^0=0). Applying further the operator (T_{q_2}^*) to (\psi_s^0), we shall have
[
T_{q_2}^*\psi_s^0
=
A_{q_2}\psi_s^0-iB_{q_2}\psi_s^0
=
-i\mu_s^0\psi_s^0 .
\tag{11}
]
* By the imaginary part of an operator (C) is meant the operator (\dfrac{1}{2i}(C-C^*)), denoted by (\operatorname{Im}C).
Equality (11) contains a contradiction, since, by hypothesis, (T_{q_2}^{}) has only the zero point of the spectrum, while (\mu_s^0 \ne 0). Thus all (\mu_s=0) ((s=1,2,\ldots)). Consequently, (B_{q_2}=0). Similarly one can show that (A_{q_2}=0). As a result, the operator (T_{q_2}^{}=0), and this already means completeness in the range of values of the operator (T). Indeed, if (g\in Q_2), then ((Th,g)=(h,T^{}g)=(h,T_{q_2}^{}g)=0); consequently, for any (h\in\mathfrak H) the element (Th) is orthogonal to (Q_2), and hence (Th\in Q_1).
The validity of the remark at the end of the theorem follows from the fact that the operator (T), just as (T^{}), maps (Q_2)—the orthogonal complement to (Q_1)—into zero. The theorem is proved.
It should be noted that examples can be given which demonstrate the essential nature of the definiteness conditions put forward in Theorem 1.
From Theorem 1 the following theorem can be obtained:
Theorem 2. Let the operators (L_1) and (L_2) be symmetric on some dense manifold (\mathfrak D) in (\mathfrak H), and suppose that for some (\lambda) the manifold ((L_1+iL_2-\lambda E)\mathfrak D) is dense in (\mathfrak H).
Suppose, further, that one of the following two conditions is satisfied:
a) The operators (L_1) and (L_2) are semibounded on (\mathfrak D) (both, for definiteness, from below) and, in addition, one of the three self-adjoint operators (\widetilde L_1), (\widetilde L_2), or (\widetilde{L_1+L_2}) has a resolvent of type (\Gamma).—III.**
b) For all (f\in\mathfrak D)
[
(L_1 f,f)-|(L_2 f,f)|\ge -\gamma^2(f,f)
]
and, in addition, the operator (\widetilde L_1) has a resolvent of type (\Gamma).—III.
Then the non-self-adjoint operator (L=L_1+iL_2) admits a closure (\widetilde L); the operator (\widetilde L) has a resolvent of type (\Gamma).—III., and the system of its eigen and associated elements is complete in (\mathfrak H).
Theorems 1 and 2 can be used in the study of various integral and differential operators. We give two examples.
Example 1 (cf. ((2^{-4}))). Consider a strongly elliptic operator ((^5)):
[
Lu=\sum_{[i,j]}\frac{\partial^m}{\partial x_{i_1}\cdots\partial x_{i_m}}
C^{[i,j]}(x)\frac{\partial^m u}{\partial x_{j_1}\cdots\partial x_{j_m}}+
]
[
+\sum_{[i,j]}\frac{\partial^m}{\partial x_{i_1}\cdots\partial x_{i_m}}
K^{[i,j]}(x)\frac{\partial^m u}{\partial x_{j_1}\cdots\partial x_{j_m}}+Pu,
]
which acts on the manifold (\mathfrak D) of vector-functions (u(x)) satisfying, on the boundary (\Gamma) of a certain bounded domain (G) of (n)-dimensional Euclidean space, the boundary conditions
[
u(x)\big|{\Gamma}=\frac{\partial u}{\partial \nu}\bigg|=\cdots=
\frac{\partial^{m-1}u}{\partial \nu^{m-1}}\bigg|_{\Gamma}=0,
]
where (\nu) is the normal to (\Gamma). For brevity we write the operator (L) in the form:
[
Lu=Cu+Ku+Pu,
]
where by (C) and (K) we have denoted, respectively, the semibounded symmetric and skew-symmetric operators of order (2m), and by (P) an arbitrary operator of order (<2m).
* Indeed, if (g\in Q_2), then (T^{*}g=Ag-iBg=0), and hence, by the definiteness of (A) and (B): (Ag=0) and (Bg=0), whence (Tg=0).
** A wavy line denotes a self-adjoint extension of the corresponding operator.
Let (4m-n>0), and suppose that for all (u\in\mathfrak D)
[
(Cu,u)-|(Ku,u)|\ge -\gamma^2(u,u).
\tag{12}
]
Then the system of eigen- and associated elements of the operator (L) is complete in (\mathscr L_2(G)). If the operator (iK) is semibounded, then condition (12) may be omitted.
Example 2 (cf. ((6^{--}8))). Consider the operator
[
Ly=-y''+(q(x)+ir(x))y,
\tag{13}
]
defined on some set dense in (\mathscr L_2(-\infty,+\infty)). Let the function (q(x)) be bounded below, and let (r(x)) be semibounded (i.e., bounded either above or below). Then, for completeness of the system of eigen- and associated elements of the operator (L) in (\mathscr L_2(-\infty,+\infty)), it is sufficient that, for some (\alpha>2/3),
[
\lim_{|x|\to\infty}\frac{q(x)+|r(x)|}{|x|^\alpha}\ge C>0.
]
We emphasize that this thereby establishes completeness, in the case of operators of the form (13), also when the “imaginary potential” (r(x)) tends to infinity.*
I express my gratitude to Corresponding Member of the Academy of Sciences of the USSR I. M. Gel'fand and to Prof. M. A. Naimark for their attention to the present work.
Moscow Institute of Physics and Technology
Received
28 XII 1957
REFERENCES
- T. Carleman, Math. Zs., 9, 196 (1921).
- M. V. Keldysh, DAN, 77, 11 (1951).
- F. E. Browder, Proc. Nat. Acad. Sci. USA, 39, 433 (1953).
- M. Naimark, DAN, 112, 198 (1957).
- M. L. Vishik, Mat. sborn., 29 (71), 3, 615 (1951).
- M. A. Naimark, DAN, 98, 727 (1954).
- J. Schwartz, Pacific J. Math., 4, 2, 415 (1954).
- V. B. Lidskii, DAN, 110, 172 (1956).
* The corresponding problem was posed by I. M. Gel'fand.