Abstract
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MATHEMATICS
A. R. ESAYAN, V. Ya. STETSKENKO
ESTIMATES OF THE SPECTRUM OF INTEGRAL OPERATORS AND INFINITE MATRICES
(Presented by Academician A. Yu. Ishlinskii on 20 II 1964)
Let \(E\) be a real Banach space, and let \(A\) be a linear bounded operator acting in \(E\). If the spectrum of the operator \(A\) is contained in the disk \(|\lambda| \leqslant r\), then, as is known \({}^{1}\), for every \(f \in E\) \((f \ne 0)\) the equation \(\lambda x = Ax + f\), for \(|\lambda| > r\), has a unique solution \(x^*\), to which the method of successive approximations converges,
\[ x_{n+1}=\frac{1}{\lambda}(Ax_n+f)\qquad (n=0,1,\ldots), \]
starting from an arbitrary element \(x_0 \in E\). In connection with this, it is of interest to estimate the spectrum of a linear bounded operator \(A\). Such an estimate is given, for example, by the norm \(\|A\|\) of the operator \(A\). The exact value of the spectral radius \(r(A)\) of the operator \(A\) can be computed by the formula \({}^{1}\)
\[ r(A)=\lim_{n\to\infty}\sqrt[n]{\|A^n\|}. \tag{1} \]
Estimating the spectrum of an operator by means of its norm is, as a rule, crude, while determining the spectral radius by formula (1) is very difficult.
In the present article a method is proposed for estimating the spectral radius of linear integral operators acting in the spaces \(C(\Omega)\), \(L_p(\Omega)\) \((p>1)\), and also the spectral radius of infinite matrices generating linear bounded operators acting in one or another space of sequences. This method is based on the following theorem \({}^{2}\).
Theorem 1. Let the cone \(K\) be normal and reproducing \({}^{(3,4)}\), the linear operator \(A\) be positive, \(u_0\)-bounded above, and
\[ A^m u_0 \leqslant \lambda_0 u_0\qquad (u_0\in K). \]
Let, further, the linear operator \(A_0\) satisfy the condition
\[ -A \leqslant A_0 \leqslant A. \]
Then the spectrum of the operator \(A_0\) lies in the disk
\[ |\lambda|\leqslant \sqrt[m]{\lambda_0}. \]
1. Let \(E=C(\Omega)\), where \(\Omega\) is a closed bounded set of a finite-dimensional space. Let the function \(K(t,s)\) \((t,s\in\Omega)\) be such that the operators
\[ A_0x(t)=\int_{\Omega}K(t,s)x(s)\,ds,\qquad Ax(t)=\int_{\Omega}|K(t,s)|x(s)\,ds \tag{2} \]
act in \(C(\Omega)\) and are bounded. We distinguish in \(C(\Omega)\) the cone \(K\) of nonnegative functions. This cone is normal and solid. Put
\[ u_0(t)=\int_{\Omega}|K(t,s)|\,ds. \]
The operator \(A\) is \(u_0(t)\)-bounded above, since for every \(x(t)\in K\)
\[ Ax(t)=\int_{\Omega}|K(t,s)|\,x(s)\,ds\leq \max_{s\in\Omega}|x(s)|\,u_0(t)=\|x\|u_0. \tag{3} \]
From Theorem 1 it then follows that
Theorem 2. Let \(\lambda_0\) be a real number such that
\[ \int_{\Omega}|K(t,s)|\left[\lambda_0-\int_{\Omega}|K(s,\tau)|\,d\tau\right]ds\geq 0. \tag{4} \]
Then the spectral radius of the operator \(A_0\) does not exceed \(\lambda_0\):
\[ r(A_0)\leq \lambda_0. \]
Remark. Relation (4) is always satisfied if for \(\lambda_0\) one takes the quantity
\[ \lambda_0=\|A_0\|=\max_s\int_{\Omega}|K(s,\tau)|\,d\tau. \]
However, as a rule, \(\lambda_0\) can be chosen considerably smaller.
A generalization of Theorem 2 is
Theorem 3. Let
\[ R_m(t,s)=\int_{\Omega}\cdots\int_{\Omega}|K(t,s_1)|\cdot |K(s_1,s_2)|\cdots |K(s_{m-1},s)|\,ds_1\cdots ds_{m-1}, \]
where \(m\) is an arbitrary positive integer, and let \(\lambda_1\) be such that
\[ \int_{\Omega}|K(t,s)|\cdot\left[\lambda_1^m-\int_{\Omega}R_m(s,\tau)\,d\tau\right]ds\geq 0. \tag{5} \]
Then the estimate holds
\[ r(A_0)\leq \lambda_1. \]
Remark. Inequality (5) is satisfied for
\[ \lambda_1=\left[\max_t\int_{\Omega}R_m(t,s)\,ds\right]^{1/m}. \]
If one takes a positive function from \(C(\Omega)\) such that
\[ \min_{t\in\Omega}\psi(t)=\delta>0, \]
then the operator \(A\) will be \(\vartheta_0(t)\)-bounded above, where
\[ \vartheta_0(t)=\psi(t)\int_{\Omega}|K(t,s)|\,ds=\psi(t)u_0(t). \]
Indeed, for any \(x(t)\in K\),
\[ \int_{\Omega}|K(t,s)|\,x(s)\,ds\leq \|x\|_{C(\Omega)}u_0(t)\leq \frac{\|x\|}{\delta}u_0(t)\psi(t)=a(x)\vartheta_0(t) \]
\[ \left(a(x)=\frac{\|x\|}{\delta}\right). \]
Therefore \(r(A_0)\leq \lambda_\psi\), where \(\lambda_\psi\) is such that
\[ \int_{\Omega}|K(t,s)|\left[\lambda_\psi^m\psi(t)-\psi(s)\int_{\Omega}R_m(s,\tau)\,d\tau\right]ds\geq 0. \]
Theorem 4. Let \(\lambda_2\) satisfy the inequality
\[ \int_{\Omega}R_m(t,s)\left[\lambda_2-\int_{\Omega}|K(s,\tau)|\,d\tau\right]ds\geq 0. \]
Then
\[ r(A_0)\leqslant \lambda_2. \]
For the proof it is enough to note that the operator \(A\) is \(g_m(t)\)-bounded from above, where
\[ g_m(t)=\int_\Omega R_m(t,s)\,ds. \]
Indeed, for any \(x(t)\in K\),
\[ \int_\Omega R_m(t,s)x(s)\,ds \leqslant \|x\|_C g_m(t). \]
Let us note that the same arguments allow one to assert that \(r(A_0)\leqslant \lambda_\psi\), where \(\lambda_\psi(n,m)\) satisfies the condition
\[ \int_\Omega R_m(t,s)\left[\lambda_\psi^n\psi(t)-\psi(s)\int_\Omega R_n(s,\tau)\,d\tau\right]ds \geqslant 0 \]
(\(n\) is an arbitrary natural number). We note that \(\lambda_\psi(n,m)\) is monotone in \(n,m\) and \(\lim_{n\to\infty}\lambda_\psi(n,m)=r(A_0)\).
- Let now the space \(E=L_p(\Omega)\) be semiordered by the cone \(K\) of nonnegative functions. This cone is normal and reproducing. Let the operators \(A_0\) and \(A\) act in \(L_p(\Omega)\). It is clear that \(AK\subset K\).
Theorem 5. Let
\[ u_0(t)\equiv \left(\int_\Omega |K(t,s)|^q\,ds\right)^{1/q}\in L_p(\Omega) \qquad \left(\frac1p+\frac1q=1\right). \]
Then \(r(A_0)\leqslant \mu_0\), where \(\mu_0\) is such that
\[ \int_\Omega |K(t,s)|\left(\int_\Omega |K(s,\tau)|^q\,d\tau\right)^{1/q}ds \leqslant \mu_0\left(\int_\Omega |K(t,s)|\,ds\right)^{1/q}. \]
As \(\mu_0\) in this relation one may take the quantity
\[ \mu_0= \left[\int_\Omega\left(\int_\Omega |K(t,s)|^q\,ds\right)^{p/q}dt\right]^{1/p} \equiv C, \]
however, in concrete cases \(\mu_0\) can be chosen in a better way, i.e. \(\mu_0<C\).
We note that \(\mu_0\) can also be chosen from the relation
\[ \int_\Omega R_n(t,s)\left[\int_\Omega R_m^q(s,\tau)\,d\tau\right]^{1/q}ds \leqslant \lambda_0^n\left(\int_\Omega R_m^q(t,s)\,ds\right)^{1/q}. \]
Here it is assumed that
\[ u_m(t)\equiv \left(\int_\Omega R_m^q(t,s)\,ds\right)^{1/q}\in L_p(\Omega). \]
- We turn to estimating the spectral radius of an infinite matrix \(A_0=(a_{ij})\) \((i,j=1,2,\ldots)\). Let \(m\) be the space of bounded numerical sequences. Put \(A=(|a_{ij}|)\).
The infinite matrices \(A_0,A\) generate linear bounded operators acting in \(m\) if and only if
\[ \sum_{k=1}^{\infty}|a_{ik}|\leqslant M<+\infty. \]
We construct the element
\[ \vartheta_0=\left(\sum_{k=1}^{\infty}|a_{1k}|,\ \sum_{k=1}^{\infty}|a_{2k}|,\ldots,\sum_{k=1}^{\infty}|a_{ik}|,\ldots\right). \]
Obviously, \(\vartheta_0 \in m\), and the operator \(A\) is \(\vartheta_0\)-bounded above; moreover,
\[ A\vartheta_0 \leqslant \sup_{i=1,2,\ldots} \frac{\displaystyle \sum_{j=1}^{\infty}\sum_{k=1}^{\infty} |a_{ij}|\cdot |a_{jk}|} {\displaystyle \sum_{j=1}^{\infty} |a_{ij}|}\,\vartheta_0 . \]
In view of what has been said, the following holds.
Theorem 6. Let \(\lambda_0\) satisfy the condition
\[ \sum_{j=1}^{\infty} |a_{ij}| \left[ \lambda_0-\sum_{k=1}^{\infty}|a_{jk}| \right]\geqslant 0 \qquad (i=1,2,\ldots). \]
Then
\[ r(A_0)\leqslant \lambda_0 . \]
Theorem 6 admits a generalization in the sense of Theorems 3 and 4.
4. Suppose that the matrix \(A=(|a_{ij}|)\) satisfies the Riesz condition
\[ \sum_{i=1}^{\infty} \left( \sum_{k=1}^{\infty}|a_{ik}|^{p/(p-1)} \right)^{p-1}<\infty . \]
Then the matrix \(A_0\) generates a linear completely continuous operator acting in the space \(l_p\).
Theorem 7. The estimate
\[ r(A_0)\leqslant \sup_{i=1,2,\ldots} \frac{\displaystyle \sum_{j=1}^{\infty}|a_{ij}| \left( \sum_{k=1}^{\infty}|a_{jk}|^{q} \right)^{1/q}} {\displaystyle \left( \sum_{j=1}^{\infty}|a_{ij}|^{q} \right)^{1/q}} . \]
is valid.
One may also consider the operator \(A_0\) acting in other Banach spaces semi-ordered by normal and reproducing cones, and, with the aid of Theorem 1, obtain various kinds of estimates for the spectrum of this operator. Moreover, one may consider an operator \(A_0\) such that \(A_0E_1\to E_2\), where \(E_1\) and \(E_2\) are Banach spaces semi-ordered respectively by cones \(K_1\) and \(K_2\), the first of which is reproducing and the second normal.
We note that the results presented remain valid also in the case when a complex Banach space \(E\) is considered.
In conclusion, the authors express their gratitude to M. A. Krasnosel’skii for his attention.
Tajik State University
Received
5 II 1964
CITED LITERATURE
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- M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Moscow, 1962.
- Fr. Riesz, Leçons sur les systèmes d’équations linéaires à une infinité d’inconnues, Paris, 1913.