Abstract
Full Text
UDC 517.949.2+517.948.35
MATHEMATICS
V. S. RYABEN'KII
ON THE KERNELS OF SPECTRA OF FAMILIES OF OPERATORS
(Presented by Academician A. N. Tikhonov on 9 VII 1968)
Consider a family of operators ({R_N}) mapping the normed spaces (U_N) of dimension (N), (N=1,2,\ldots), into themselves. We shall assume that the operators (R_N) are uniformly bounded:
[
|R_N| < C .
\tag{1}
]
Definition 1. In view of (1), we shall say that a complex number (\lambda) belongs to the resolvent set of the family of operators ({R_N}) if there exist (\varepsilon>0) and (N_0) such that for all (N), (N>N_0), and all (u), (u \in U_N), the inequality
[
|R_Nu-\lambda u| \geq \varepsilon |u|
]
is satisfied.
By the spectrum of the family of operators ({R_N}) we shall mean the complement of the resolvent set in the whole complex plane.
Theorem 1. The spectrum of the family of operators ({R_N}) is a closed set.
Definition 2. Let ({\varepsilon_N}) be some nonincreasing sequence of numbers tending to zero. We shall call the ({\varepsilon_N})-kernel of the spectrum of the family of operators ({R_N}) the set (\Lambda_{{\varepsilon_N}}), defined as follows:
[
\Lambda_{{\varepsilon_N}}=\bigcap_{k=1}^{\infty}\ \overline{\bigcup_{N>k}\Lambda_N},
]
where (\Lambda_N) is the set of points (\lambda) for which the inequality (|R_Nu-\lambda u|\leq \varepsilon_N|u|) has a nontrivial solution (u), and (\overline{\bigcup \Lambda_N}) is the closure of the set (\bigcup \Lambda_N).
Theorem 2. The ({\varepsilon_N})-kernel of the spectrum of the family of operators ({R_N}) lies in the disk (|\lambda|\leq C), where (C) is introduced in (1), is closed, nonempty, and belongs to the spectrum of the family of operators ({R_N}).
Theorem 3. The totality of ({\varepsilon_N})-kernels of the spectrum, constructed for all possible nonincreasing sequences ({\varepsilon_N}) converging to zero, coincides with the whole spectrum.
Definition 3. Let (a), (a\geq 1), be a certain constant. We shall call the kernel with exponent (a) of the spectrum of the family of operators ({R_N}) the intersection of the ({\varepsilon_N})-kernels, (\varepsilon_N=N^{-k}a^{-N}), constructed for all natural (k). The kernel with exponent (a=1) will sometimes be called the arithmetic kernel of the spectrum.
Definition 4. We shall call the absolute kernel of the spectrum of the family of operators ({R_N}) the intersection of all ({\varepsilon_N})-kernels whatsoever.
Theorem 4. The kernel with exponent (a), (a\geq 1), and the absolute kernel are closed. If (a_1\geq a_2), then the kernel with exponent (a_1) is contained in the kernel with exponent (a_2). The absolute kernel coincides with the ({\varepsilon_N})-kernel for (\varepsilon_N\equiv 0).
Definition 5. We shall call the radius of the spectrum (the radius of the kernel) of the family of operators ({R_N}) the radius of the smallest closed disk with center at the point (\lambda=0) that contains the spectrum (the kernel of the spectrum).
Theorem 5. For any (\varepsilon>0) there exists a number (A=A(\varepsilon)), independent of (N), such that
[
\left|R_N^m\right|\leq A(\varepsilon)(\rho+\varepsilon)^m,\qquad m=1,2,\ldots,
]
where (\rho) is the radius of the spectrum of the family of operators ({R_N}).
Theorem 6. Let (\rho) be the radius of the ({\varepsilon_N})-kernel of the spectrum. Then for every (\varepsilon>0) there exists an index (N_0), (N_0=N_0(\varepsilon)), such that for (N>N_0) the inequality
[
\left|R_N^m\right|\leq(\rho+\varepsilon)^{m+1}/\varepsilon_N,\qquad m=1,2,\ldots
]
holds.
Theorem 7. Let (\rho) be the radius of the ({\varepsilon_N})-kernel of the spectrum. Then, for any (\varepsilon>0) and any (N_0), there is an (N), (N>N_0), such that
[
\left|R_N^m\right|\geq
(\rho-\varepsilon)^m
\left(
1-\frac{\varepsilon_N(C+\varepsilon)^m}{(\rho-\varepsilon)^m}
\left(
\min m,\frac{1}{1-(\rho+\varepsilon)/C}
\right)
\right),
]
where (C) is the number introduced in (1).
Corollary. From Theorems 3 and 7 there follows the known theorem ((^1)), asserting that, for the norms of the operators (R_N^m) to be bounded uniformly in (m) and (N), it is necessary that the spectrum of the family of operators ({R_N}) lie in the unit disk (|\lambda|\leq 1).
Let us note that in Theorem 7 one cannot put (\varepsilon=0).
We shall discuss the question of to what extent the spectrum of a family of operators ({R_N}) is independent of the choice of norms in the spaces (U_N).
Theorem 8. The absolute kernel of the spectrum of a family of operators does not depend on the choice of norms in the spaces (U_N).
Theorem 9. For a given family of operators ({R_N}), whose eigenvalues are bounded in the aggregate, the sequence of norms in the spaces (U_N) can be chosen so that the operators (R_N) are bounded in the aggregate and so that the spectrum of the family of operators ({R_N}) coincides with its absolute kernel.
Theorem 10. Let (R_N) be unitary operators in unitary spaces (U_N). Then the spectrum of the family of operators ({R_N}) lies on the unit circle and coincides with its absolute kernel.
Theorem 11. Let (R_N) be Hermitian operators in unitary spaces (U_N), bounded in the aggregate. Then the spectrum of the family of operators ({R_N}) is real and coincides with its absolute kernel.
Generally speaking, the spectrum of a family of operators ({R_N}) bounded in the aggregate is not exhausted by its absolute kernel, as we shall show below, relying on the example from ((^1)), Ch. VI, § 1. Comparing this with Theorem 9, we see that the spectrum of a family of operators depends, in general, on the choice of norms in the spaces (U_N).
Theorem 12. Suppose that for each (N) two norms (|\cdot|_N^{(1)}) and (|\cdot|_N^{(2)}) are given in the space (U_N), and suppose that there exists a number (s), independent of (N), such that for all sufficiently large values of (N) the inequalities
[
\sup_{|u|^{(1)}=1}|u|N^{(2)}
\leq
N^s
\inf,}=1}|u|_N^{(2)
]
[
\sup_{|u|^{(2)}=1}|u|N^{(1)}
\leq
N^s
\inf.}=1}|u|_N^{(1)
\tag{2}
]
hold.
Suppose, further, that the operators (R_N) are bounded in the aggregate in each of these norms. Then the kernels, with any exponent (a), (a\geq 1), of the spectrum of the family of operators ({R_N}) in both sequences of norms coincide.
Condition (2) is satisfied for any pair of norms ordinarily used in the theory of difference schemes:
[
|u|=\max |u_k|;\qquad
|u|=\left(\sum_k |u_k|^p\right)^{1/p}
]
and so on.
Example. Consider the spectrum of the family of operators ({R_N}) ((1), Ch. VI, § 1). Define the operator (R_N) as follows: this operator assigns to an ((N+1))-dimensional vector (u=(u_0,u_1,\ldots,u_N)), with norm (|u|=\max |u_k|), a certain vector (v=(v_0,v_1,\ldots,v_N)) according to the formulas
[
v_n=(1-\xi)u_n+\xi u_{n+1},\quad n=0,1,\ldots,N-1;\qquad v_N=0,
]
where (\xi) is a certain positive constant. This family of operators arises when replacing the boundary-value problem for the equation (u_t-u_x=0) by a difference analogue. In (1) it is shown that the spectrum of the family of operators ({R_N}) consists of the closed disk of radius (\xi) with center at the point (\lambda=1-\xi) and of the point (\lambda=0). It can be shown that the kernel with exponent (a) consists of the closed disk with center at the same point (\lambda=1-\xi) and radius (\xi/a), and also of the point (\lambda=0). Taking (a=1), we see that the arithmetic kernel of the spectrum coincides with the entire spectrum. The absolute kernel of the spectrum of this family of operators consists of the two points (\lambda=0) and (\lambda=1-\xi).
One can construct an example of such a family of operators for which the arithmetic kernel of the spectrum does not coincide with the entire spectrum.
Received
4 VI 1968
REFERENCES
- S. K. Godunov, V. S. Ryaben’kii, Introduction to the Theory of Difference Schemes, Moscow, 1962.