Abstract Generated abstract
This paper studies spectra associated with uniformly bounded families of finite-dimensional operators acting in normed spaces, introducing several notions of spectral kernels defined through asymptotic approximate eigenvalues. It establishes basic topological and inclusion properties of these kernels, shows that all such kernels together recover the spectrum, and relates their radii to upper and lower estimates for powers of the operators. The paper also examines dependence on the choice of norms, proving invariance of the absolute kernel and of exponential kernels under suitable polynomial equivalence of norms, while showing that the full spectrum may depend on the norms. Special cases for unitary and Hermitian operator families are treated, and a difference-scheme example illustrates the distinction between the spectrum, arithmetic kernel, exponential kernels, and absolute kernel.
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
\[ \sup_{\|u\|^{(1)}=1}\|u\|_N^{(2)} \leq N^s \inf_{\|u\|^{(1)}=1}\|u\|_N^{(2)}, \]
\[ \sup_{\|u\|^{(2)}=1}\|u\|_N^{(1)} \leq N^s \inf_{\|u\|^{(2)}=1}\|u\|_N^{(1)}. \tag{2} \]
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.