I. Marek (I. MAREK)

THE SPECTRAL RADIUS OF INDECOMPOSABLE POSITIVE OPERATORS

(Presented by Academician S. L. Sobolev on 22 XII 1966)

In this note we shall use the notation and definitions introduced in \((^1)\).

It is known \((^2)\) that if \(T \in [Y]\) is a semi-non-supporting operator and if \(T\) has property \((S)\), then the point \(\lambda = \rho(T)\) belongs to the spectrum \(\sigma(T)\), and there correspond to it eigen-elements \(x_0 \in K,\ x_0' \in K'\) such that \(\langle x, x_0' \rangle > 0\) for all nonzero vectors \(x \in K\), and \(\langle x_0, x' \rangle > 0\) for all nonzero functionals \(x' \in K'\). In other words, the semi-non-supporting operator is indecomposable in the sense of the definition of Stetsenko \((^3)\). It is also known that the converse assertion holds \((^3)\).

Everywhere in what follows it is assumed that \(K\) is a reproducing and normal cone \((^4)\).

Theorem 1. Suppose that:

1. \(T \in [Y]\) is a positive operator such that its spectral radius \(\rho(T)\) is an eigenvalue. Let this eigenvalue have corresponding to it an eigenvector \(x_0 \in K\).

2. \(x \in K,\ x' \in K'\) are elements such that there exist \(\alpha > 0\) and a positive integer \(q\) such that \(T^q x > \alpha x_0\) and \(\langle x_0, x' \rangle > 0\).

Then

\[ \rho(T)=\lim_{p\to\infty}\left[\langle T^p x, x'\rangle\right]^{1/p}. \]

The validity of this formula is obvious in the case \(\rho(T)=0\), and in the case \(\rho(T)>0\) follows from the relations

\[ 0<\rho(T)\alpha^{1/p}[\rho(T)]^{-q/p}\left[\langle x_0,x'\rangle\right]^{1/p} \leq \left[\langle T^p x,x'\rangle\right]^{1/p} \leq \]

\[ \leq \rho(T)\left\|\frac{1}{[\rho(T)]^p}T^p\right\|^{1/p}\|x\|^{1/p}\|x'\|^{1/p}. \]

Remark. Condition 1 of Theorem 1 is fulfilled if:

(a) The operator \(T \in [Y]\) has property \((S)\) and \(TK \subset K\).

Condition 2 is fulfilled if (a) holds and

(b) \(T\) is semi-non-supporting and \(u_0\) is an operator bounded below \((^5)\), where \(u_0 \in K,\ \|u_0\|=1\), and \((u_0 - x x_0) \in K,\ x_0 = [\rho(T)]^{-1}T x_0,\ x_0 \in K,\ x_0 \ne 0\).

Then the only restrictions on \(x \in K,\ x' \in K'\) are the conditions \(x \ne 0,\ x' \ne 0\).

Theorem 2. Suppose that:

\((\alpha)\) \(T \in [Y]\) is a positive operator.

\((\beta)\) \(\hat{x}\) is a non-supporting element of the cone \(K\).

\((\gamma)\) \(H'\) is a \(K\)-total set such that \(\langle \hat{x}, x' \rangle = 1\) for all \(x' \in H'\).

If we put

\[ r(T)=\inf_{x'\in H'}\langle T\hat{x},x'\rangle,\qquad R(T)=\sup_{x'\in H'}\langle T\hat{x},x'\rangle, \]

then the inequalities

\[ r(T) \leq [r(T^2)]^{1/2} \leq \cdots \leq [r(T^{2^p})]^{2^{-p}} \leq \cdots \leq \rho(T) \leq \cdots \]
\[ \cdots [R(T^{2^p})]^{2^{-p}} \leq \cdots \leq [R(T^2)]^{1/2} \leq R(T). \]

Suppose, in addition:

\[ (\delta)\quad T \text{ has property } (S). \]
\[ (\varepsilon)\quad T \text{ is a semi-nonsupporting operator}. \]
\[ (\eta)\quad \text{There exist positive numbers } \alpha,\ \beta \text{ and a positive integer } q \]
such that \(\alpha x_0 < T^q x < \beta x_0\), where \(x_0 = [\rho(T)]^{-1}Tx_0,\ x_0 \in K,\ x_0 \ne 0\). Then the equalities hold

\[ \rho(T)=\lim_{p\to\infty}[r(T^{2^p})]^{2^{-p}} =\lim_{p\to\infty}[R(T^{2^p})]^{2^{-p}}. \]

For the proof of Theorem 2 we introduce functionals, defined on \(K\setminus\{0\}\),

\[ r_x(T)=\inf_{\substack{x'\in H'\\ \langle x,x'\rangle\ne 0}} \frac{\langle Tx,x'\rangle}{\langle x,x'\rangle}, \qquad r^x(T)=\sup_{x'\in H'} \frac{\langle Tx,x'\rangle}{\langle x,x'\rangle}. \]

An obvious consequence of the \(K\)-totality of the set \(H'\) is the inequalities

\[ r_x(T)\leq r_{Tx}(T)\leq \cdots \leq r_{T^q x}(T)\leq \cdots \leq r^{T^q x}(T)\leq \cdots \leq r^{Tx}(T)\leq r^x(T). \]

Using these inequalities and the hypotheses of Theorem 2, we obtain the relations

\[ 0\leq r_{T^{2^p}\hat{x}}(T^{2^p})-r_{\hat{x}}(T^{2^p}) =\inf_{x'\in H'}\frac{\langle T^{2^p+1}\hat{x},x'\rangle} {\langle T^{2^p}\hat{x},x'\rangle} -\inf_{x'\in H'}\langle T^{2^p}\hat{x},x'\rangle, \]

\[ 0\leq r^x(T^{2^p})-r^{T^{2^p}\hat{x}}(T^{2^p}) =\sup_{x'\in H'}\langle T^{2^p}\hat{x},x'\rangle -\sup_{x'\in H'}\frac{\langle T^{2^p+1}\hat{x},x'\rangle} {\langle T^{2^p}\hat{x},x'\rangle} \]

and, after simple calculations, the desired result.

The class of operators that we consider is quite broad. Let us give two typical examples. Some classes of indecomposable operators are considered in \((^3)\).

Example 1. Let \(Y\) be an \(m\)-dimensional Banach space. Then, in a certain basis of the space \(Y\), the semi-nonsupporting operator \(T\) is represented by an indecomposable matrix \((t_{jk})\), where \(t_{jk}\geq 0,\ 1\leq j,k\leq m\). If \(\hat{x}(t,\ldots,1)^*\), where the asterisk denotes that \(x\) is a column vector,

\[ x'_j=(0,\ldots,\underbrace{1}_{j},0,\ldots,0), \]

\[ H'=\{x'_j\mid j=1,\ldots,m\},\qquad r_j(T)=\sum_{k=1}^{m} t_{jk}, \]

\[ r(T)=\min_{j=1,\ldots,m} r_j(T),\qquad R(T)=\max_{j=1,\ldots,m} r_j(T), \]

then we obtain the theorems to which Yamamoto refers in \((^6)\).

Example 2. Let \(Y=C(\langle 0,1\rangle)\), and let \(T\) be an integral operator with kernel \(\tau=\tau(s,t)\geq 0\), continuous on \(\langle 0,1\rangle\times\langle 0,1\rangle\). Let \(\hat{x}=\hat{x}(s)\equiv 1\),

\[ H'=\{\delta_s\mid 0\leq s\leq 1,\ \delta_s=\delta(s)\text{ is the Dirac delta function}\}. \]

Then

\[ r(T)=\min_{s\in\langle 0,1\rangle}\int_0^1 \tau(s,t)\,dt \leq \rho(T)\leq \max_{s\in\langle 0,1\rangle}\int_0^1 \tau(s,t)\,dt =R(T). \]

If, moreover, the kernel $\tau$ is such that for an arbitrary nonnegative function $y \in C(\langle 0,1\rangle)$, $y(s) \ne 0$, there is a positive integer $q=q(y)$ such that

\[ \int_0^1 \int_0^1 \cdots \int_0^1 \tau(s,t_q)\cdots \tau(t_1,t)y(t)\,dt\,dt_1\cdots dt_q>0, \]

then

\[ \lim_{p\to\infty}\{[R(T^{2^p})]^{2^{-p}}-[r(T^{2^p})]^{2^{-p}}\}=0. \]

Mathematical Institute of Charles University
Prague, Czechoslovak Socialist Republic

Received
13 XII 1966

References

1. I. Marek, DAN, 176, No. 4 (1967).
2. I. Sawashima, Nat. Sci. Rep. Ochanomizu Univ., 15, No. 1 (1964).
3. V. Ya. Stetsenko, UMN, 21, No. 5 (1966).
4. M. G. Krein, M. A. Rutman, UMN, 3, No. 1 (1948).
5. M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Moscow, 1962.
6. T. Yamamoto, Numer. Math., 8, No. 4 (1966).