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MATHEMATICS
V. Ya. STETSENKO
ON ESTIMATING THE SPECTRUM OF CERTAIN CLASSES OF LINEAR OPERATORS
(Presented by Academician A. N. Kolmogorov on 28 III 1964)
The paper considers the question of estimating the spectrum of linear operators acting in a Banach space \(E\), semi-ordered by a cone \(K\) in the sense of M. G. Krein. Estimates of this kind for linear positive operators were obtained earlier by M. G. Krein \((^1)\) and M. A. Krasnosel’skii \((^2)\). For example, in \((^1)\) it was shown that if \(K\) is a solid normal cone in \(E\) and \(A(E \to E)\) is a linear positive \((AK \subset K)\) operator, and for some \(u\), which is an interior element of \(K\), \(Au = \rho u\), then the spectrum of the operator \(A\) is contained in the disk \(|\lambda| \leqslant \rho\). It turns out that M. G. Krein’s idea of estimating the spectrum of a linear operator from the behavior of the operator on just one interior element of a solid cone \(K\) admits an essential development.
Everywhere below it is assumed that \(K\) is a normal \((^1)\) cone in \(E\).
1. Lemma 1. Let \(A\) be a linear positive operator. Suppose that for a fixed element \(f \in K\), for some \(\lambda_0\),
\[ A^m f \leqslant \lambda_0 f . \tag{1} \]
Then, for \(|\lambda| > \sqrt[m]{\lambda_0}\), the equation
\[ \lambda x = Ax + f \tag{2} \]
has at least one solution \(x^*\):
\[ x^* = \sum_{n=0}^{\infty} \frac{A^n f}{\lambda^{n+1}} . \]
2. Let \(g\) be a fixed element of the cone \(K\). Consider the set \(E_g\) of those elements \(x \in E\) for which, for some \(a \geqslant 0\), the inequalities
\[ -ag \leqslant x \leqslant ag \tag{3} \]
hold.
Obviously, \(E_g\) is a linear manifold in \(E\). For the elements of this linear manifold one may introduce a new norm \(\|\cdot\|_g\), setting \(\|x\|_g = \inf\{a\}\), where the infimum is taken over the set of those values of \(a\) for which the inequalities (3) hold (see \((^2)\)). With the aid of Lemma 1 the following is proved.
Theorem 1. Let a positive linear operator \(A\) satisfy the inequality
\[ A^m g \leqslant \lambda_0 g, \tag{4} \]
where \(g\) is some fixed element of \(K\).
Then, for \(|\lambda|>\sqrt[m]{\lambda_0}\), equation (2) has, for every \(f\in E_g\), a solution \(x^*\), unique in \(E_g\), and
\[ x^*=R_\lambda f=\sum_{n=0}^{\infty}\frac{A^n f}{\lambda^{n+1}}. \]
Corollary. If \(g\) is an interior element of the cone \(K\) and \(A^m g\leqslant \lambda_0 g\), then the spectrum of the linear positive operator \(A\) is contained in the disk
\[ |\lambda|\leqslant \sqrt[m]{\lambda_0}. \]
- In some cases the corollary to Theorem 1 makes it possible to obtain a more precise estimate for the radius \(\rho_0\) of the disk in which the spectrum of the operator \(A\) lies, in comparison with the estimate given by the norm of the operator. We give the corresponding example.
Let
\[ A(\varphi(t))=\int_0^1 K(t,s)a(s)\,ds, \tag{5} \]
where the continuous kernel \(K(t,s)\) has the form
\[ K(t,s)=ts^2K_1(t,s), \]
and the function \(K_1(t,s)\), continuous in the square, is such that
\[ 0\leqslant K_1(t,s)\leqslant M \qquad (0\leqslant t,s\leqslant 1). \]
It is not difficult to see that then, in the space \(C(0,1)\), \(\|A\|=M/3\). Under the assumptions made, the operator \(A\) is positive with respect to the cone of nonnegative functions of this space. The functions \(\varphi_n(t)=(1+nt)\) \((0\leqslant t\leqslant 1)\) are interior elements of this cone. Obviously,
\[ A\varphi_n(t)\leqslant t\int_0^1 Ms^2(1+ns)\,ds = M\left(\frac13+\frac n4\right)t, \]
whence it follows that
\[ A\varphi_n\leqslant M\left[\frac{1}{3(n+1)}+\frac{n}{4(n+1)}\right]\varphi_n, \tag{6} \]
and therefore the spectrum of the operator \(A\) lies in the disk \(|\lambda|\leqslant M/4\). Note that, for kernels \(K(t,s)\) of the form indicated in this example, the estimate \(|\lambda|\leqslant M/4\) for the spectrum of the operator cannot be improved: if \(K_1(t,s)=M\), then the operator (5) has the eigenvalue \(\lambda_0=M/4\).
- We shall call the operator \(A\) \(g\)-bounded above \((g\in K,\ g\ne \theta)\) if for every \(x\in K\) there exist a natural number \(m\) and a number \(a>0\) such that
\[ A^m x\leqslant ag \; ({}^2). \]
Theorem 2. Let \(K\) be a reproducing cone in \(E\), and let \(A\) be a linear positive operator \(g\)-bounded above, where
\[ A^s g\leqslant \lambda_0 g. \tag{7} \]
Then the spectrum of the operator \(A\) is contained in the disk
\[ |\lambda|\leqslant \sqrt[s]{\lambda_0}. \tag{8} \]
Proof. Suppose first that \(f\in K\). By virtue of the \(g\)-boundedness of the operator \(A\) above, there is a natural number \(m\) such that
\[ \theta\leqslant A^m f\leqslant ag \qquad (a>0). \]
Applying to this inequality the positive operator \(A^{ps}\) and taking (7) into account, we obtain
\[ \theta \ll A^{m+ps} f \ll a\lambda_0^{p}g \qquad (p=1,2,\ldots). \tag{9} \]
Let \(n>m+s\); then \(n\) can be represented in the form
\[ n=(m+p_1s)+p_2, \tag{10} \]
where \(p_1\) and \(p_2\) are integers, and \(0\leq p_2<s\). Denote
\[ \max\{\|Ag\|,\ \|A^2g\|,\ldots,\|A^s g\|\}=M. \tag{11} \]
If \(n\) satisfies (10), then
\[ \|A^n f\|=\|A^{(m+p_1s)+p_2}f\|=\|A^{p_2}A^{m+p_1s}f\|. \]
But
\[ \theta \ll A^n f \ll A^{p_2}A^{m+p_1s}f \ll a\lambda_0^{p_1}A^{p_2}g. \]
Therefore, by virtue of the normality of the cone \(K\),
\[ \|A^n f\|\leq Ca\lambda_0^{p_1}\|A^{p_2}g\|\leq C_1\lambda_0^{p_1}, \]
where \(C\) is the constant entering into the definition of the semimonotonicity of the norm \((^2)\), and
\[ C_1=CaM. \]
Next consider the numerical positive-term series
\[ \sum_{n=0}^{\infty}\frac{\|A^n f\|}{|\lambda|^{n+1}}. \tag{12} \]
For \(|\lambda|>\sqrt[s]{\lambda_0}\), from the following estimates it is not difficult to see that it converges:
\[ \sum_{n=0}^{\infty}\frac{\|A^n f\|}{|\lambda|^{n+1}} = \sum_{n=0}^{m+s-1}\frac{\|A^n f\|}{|\lambda|^{n+1}} + \sum_{p_1=1}^{\infty}\sum_{p_2=0}^{s-1} \frac{\|A^{m+p_1s+p_2}f\|}{|\lambda|^{m+p_1s+p_2+1}} \leq \]
\[ \leq \sum_{n=0}^{m+s-1}\frac{\|A^n f\|}{|\lambda|^{n+1}} + \sum_{p_2=0}^{s-1}\frac{C_1}{|\lambda|^{m+p_2+1}} \sum_{p_1=1}^{\infty}\left(\frac{\lambda_0}{|\lambda|^s}\right)^{p_1}. \]
From the convergence of the series (12) follows the strong convergence of the series
\[ R_\lambda f \equiv \sum_{n=0}^{\infty}\frac{A^n f}{\lambda^{n+1}}. \tag{13} \]
Further, it is obvious that
\[ \lambda R_\lambda f=AR_\lambda f+f, \]
i.e. \(R_\lambda f\) is a solution of equation (2). We now show that the series (12) converges for every \(f\in E\).
Let \(f\in E\); then
\[ f=f_1-f_2\qquad (f_1,f_2\in K), \]
and, consequently, by virtue of the convergence, for \(|\lambda|>\sqrt[s]{\lambda_0}\), of the series
\[ \sum_{n=0}^{\infty}\frac{\|A^n f_1\|}{|\lambda|^{n+1}}, \qquad \sum_{n=0}^{\infty}\frac{\|A^n f_2\|}{|\lambda|^{n+1}}, \]
there follows the convergence of the series (12), and along with it the strong convergence of the series
(13). Thus, it has been shown that for \(|\lambda|>\sqrt[s]{\lambda_0}\) and any \(f\in E\) the series (13) is a solution of equation (2). The uniqueness of this solution is obvious. The theorem is proved.
Theorem 3. Let \(A\) be a linear bounded positive operator such that for each \(x\in E\) there exist natural numbers \(m=m(x)\) and \(r=r(x)\) such that, for some \(a=a(x)>0\) and \(b=b(x)>0\), the relations
\[ A^m x \leq ag,\quad -bg \leq A^r x,\quad A^s g \leq \lambda_0 g \]
hold.
Then the spectrum of the operator \(A\) is contained in the disk
\[ |\lambda|\leq \sqrt[s]{\lambda_0}. \]
- We now dispense with the requirement that the operator \(A\) be positive. Instead, we shall assume that there exist positive operators \(A_1\) and \(A_2\) such that, for every \(x\in K\), the relation
\[ -A_1x \leq A^p x \leq A_2x \]
holds.
Theorem 4. Let the positive operators \(A_1\) and \(A_2\) satisfy, at some element \(g\ne \theta\) of the normal cone \(K\), the inequalities
\[ A_1g \leq \lambda_1 g,\quad A_2g \leq \lambda_2 g. \]
Then equation (2), for all \(f\in E_g\), has a unique solution in \(E_g\) whenever \(|\lambda|>\sqrt[p]{\lambda_1+\lambda_2}\).
If, moreover, the operators \(A_1, A_2\), and \(A_1+A_2\) are \(g\)-bounded from above, and the cone \(K\) is reproducing, then the spectrum of the operator \(A\) lies in the disk
\[ |\lambda|\leq \sqrt[p]{\lambda_1+\lambda_2}. \]
The author, jointly with A. R. Esayan, has established that in the case \(A_1=A_2\) the last estimate can be improved: the spectrum of the operator \(A\) lies in the disk
\[ |\lambda|\leq \sqrt[p]{\max(\lambda_1,\lambda_2)}. \]
Tajik State University
named after V. I. Lenin
Received
7 I 1964
REFERENCES
- M. G. Krein, M. A. Rutman, UMN, 3, issue 1 (23) (1948); 2. M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Moscow, 1962.