UDC 513.88:513.83+517.948.35

MATHEMATICS

V. Ya. STETSENKO

ON THE SPECTRAL PROPERTIES OF INDECOMPOSABLE OPERATORS

(Presented by Academician A. Yu. Ishlinskii, 28 III 1967)

Let \(E\) be a semi-ordered Banach space with cone \(K\) \((^{1,2})\); let \(A(E \to E)\) be a linear positive operator; \(r(A)\) its spectral radius. Throughout the paper it is assumed that \(K\) is a normal reproducing cone. By \(\nu(A)\), \(\sigma(A)\), \(\sigma_c(A)\) we denote respectively the resolvent set, the spectrum, and the continuous spectrum of the operator \(A\); \(r(A)\) is its spectral radius. The paper studies the spectral properties of an indecomposable operator.

1. In this section a variational characterization of the spectral radius of an indecomposable operator will be given. By \(P, Q\) we denote respectively the collection of all nonzero vectors \(u, v \in K\) for which, for some \(\mu, \lambda \ge 0\), the inequalities \(Au \ge \mu u\), \(Av \le \lambda v\) hold. We first present one assertion valid for the case of a solid minihedral cone \(K\).

Theorem 1. Let \(K\) be solid and minihedral, and let the operator \(A\) be indecomposable. Suppose there exists a sequence of real numbers \(\lambda_n\) such that \(\lambda_n \le r(A)\), \(\lambda_n \in \nu(A) \cup \sigma_c(A)\), and \(\lim\limits_{n\to\infty}\lambda_n = r(A)\). Then

\[ r(A)=\sup_{u\in P}\sup\{\mu:\ Au\ge \mu u\} =\inf_{v\in Q}\inf\{\lambda:\ Av\le \lambda v\}. \tag{1} \]

The analogues of Theorem 1 for nonsolid cones are the following two assertions.

Theorem 2. Let \(K\) be a minihedral cone, and let \(A\) be a linear \(u_0\)-positive operator \((^2)\). Suppose there exists a sequence of real numbers \(\lambda_n<r(A)\) such that \(\lambda_n \in \sigma(A)\) and \(\lim\limits_{n\to\infty}\lambda_n=r(A)\). Then relation (1) holds.

Theorem 3. Let \(K\) be a minihedral cone, and let the operator \(A\) be indecomposable, quasifully continuous, and \(r(A)>0\). Then relation (1) holds.

We note that the assumption of indecomposability of the operator \(A\) in Theorems 1 and 3, and also the assumption of \(u_0\)-positivity of the operator \(A\) in Theorem 2, are essential—without these assumptions the assertions of Theorems 1–3 lose their force.

We give one corollary of Theorem 1. Let a nonlinear, completely continuous monotone operator \(F\) act in \(E\) and leave invariant the solid cone \(K\). Suppose there exist linear positive operators \(A\) and \(B\), \(\rho>0\), and elements \(g,u_0\in K\) such that \(Fx\ge Ax\) for all \(x\in K\), \(\|x\|\le \rho\), and \(Fx\le Bx+g\) for all \(x\ge u_0\). Finally, suppose that the operator \(A\) satisfies the hypotheses of Theorem 1 and \(r(A)>1\), while \(r(B)<1\). Then the operator \(F\) has in \(K\) at least one fixed point \(x^*\ne\theta\). A fixed point \(x^*\) of the operator \(F\) can be obtained by the method of successive approximations \(u_n=Fu_{n-1}\), \(v_n=Fv_{n-1}\), with a suitable choice of initial approximations \(u_0,v_0\in K\). Moreover,

\[ u_0\le u_1\le \cdots \le u_n\le \cdots \le x^*\le \cdots \le v_n\le \cdots \le v_1\le v_0 . \]

1. In this section we consider the question of when the inequality \(r(A)<r(B)\) follows from the inequality \(\theta\le A\le B\), where \(A,B\) are linear operators. Below we put \(C=B-A\).

Theorem 4. Let \(K\) be solid, and let the operator \(C\) be indecomposable. Then every positive eigenvalue of the operator \(A\) (if it exists) is less than \(r(B)\).

Corollary. Let the hypotheses of Theorem 4 be satisfied, and suppose that \(A\) is a quasibounded continuous operator. Then \(r(A)<r(B)\).

We note that in the case of a minihedral cone, under the hypotheses of Theorem 4 it can be proved that the absolute values of all eigenvalues of the operator \(A\) are less than \(r(B)\).

Theorem 5. Let the hypotheses of Theorem 4 be satisfied, and let \(r(B)\) be a positive eigenvalue of the operator \(B\). Then \(r(A)<r(B)\).

Let us make one more remark concerning Theorems 4 and 5. If the operators \(A, B\) are quasiboundedly continuous, then the assertions of these theorems remain valid also for nonsolid cones. The assumption that \(K\) is solid can be omitted in the hypotheses of these theorems also in the case when one of the operators is \(u_0\)-bounded from above.

1. In this section we give one assertion supplementing assertions of the “incompatible inequalities” type proved in \((^2)\), and indicate one of its applications. We shall write \(x\ll y\) if \(y-x\in K\).

Theorem 6. Let a linear positive operator \(A\) be indecomposable. Suppose that one of the following three conditions is satisfied: a) the cone \(K\) is solid; b) the operator \(A\) is \(u_0\)-bounded from above; c) the operator \(A\) is quasiboundedly continuous.

Then: 1) for no nonzero \(u,v\in K\) and real \(\lambda,\varepsilon>0\) are the relations \(Av\ll \lambda v,\ Au\gg(\lambda+\varepsilon)u\) compatible; 2) from the inequalities \(Av_0\ll \lambda_0v_0,\ Au_0\gg \lambda_0u_0\) it follows that \(\lambda_0=r(A)\), \(v_0=cv_0\) for some real \(c\), and \(Au_0=r(A)u_0\); 3) for \(\lambda<r(A)\) and arbitrary \(v\in K,\ v\ne0\), the relation \(Au\ll\lambda v\) holds.

From Theorem 6 and Theorem 4.12 of the monograph \((^2)\) the following assertion follows:

Theorem 7. Let a nonlinear positive operator \(F\) be completely continuous and satisfy the following conditions: there exist \(0<r<R\) and such linear positive operators \(A_1\) and \(A_2\) that \(A_1x\ll Fx\) \((x\in K,\ 0<\|x\|<r)\) and \(Fx\ll A_2x\) \((x\in K,\ \|x\|\ge R)\). Suppose that one of conditions a)–c) of Theorem 6 is satisfied, \(A_1\) is indecomposable and \(r(A_1)>1,\ r(A_2)\le1\). Then the operator \(F\) has in \(K\) at least one fixed point \(x^*\), and moreover \(r\le\|x^*\|\le R\).

Under the hypotheses of Theorem 7 the operator \(F\) is a compression operator of the cone \((^2)\). Starting from Theorem 6 and Theorem 4.14 \((^2)\), it is easy to formulate an assertion analogous to Theorem 7 concerning an operator \(F\) that is an expansion of the cone.

1. Let the cone \(K\) be solid, and let \(A\) be an indecomposable completely continuous operator. Then \(r(A)>0\), and the number \(r(A)\) is a positive and simple eigenvalue of the operator \(A\) \((^5)\). Denote by \(x^*(\|x^*\|=1)\) the eigenvector corresponding to \(r(A)\), lying in \(K\). In this section a method is indicated for approximately finding \(r(A)\) and \(x^*\).

Let \(u_0\) be an arbitrary interior element of \(K\). It is easy to see that for every \(n\), for some \(\alpha,\beta>0\), the inequality \(\beta u_0\ll A^n u_0\ll \alpha u_0\) holds. Denote by \(\beta_n^{\,n}\) \((\alpha_n^{\,n})\) the exact upper (lower) bound of all such numbers \(\beta\) \((\alpha)\) for which the last inequality holds. Obviously,
\[ \beta_n^{\,n}u_0\ll A^n u_0\ll \alpha_n^{\,n}u_0. \]
It turns out \((^6)\) that \(\beta_n\le r(A)\le \alpha_n\), and moreover
\[ \lim \beta_n=\lim \alpha_n=r(A). \]
Put
\[ u_n=\sum_{k=1}^{n}\beta_n^{\,n-k}A^{k-1}u_0,\qquad v_n=\sum_{k=1}^{n}\alpha_n^{\,n-k}A^{k-1}u_0. \]
Then, as is not difficult to see,
\[ \beta_nu_n\ll Au_n \quad\text{and}\quad Av_n\ll \alpha_nv_n, \]
whence
\[ \beta_n A\frac{u_n}{\|Au_n\|} \ll A\left(A\frac{u_n}{\|Au_n\|}\right), \qquad A\left(A\frac{v_n}{\|Av_n\|}\right) \ll \alpha_n A\frac{v_n}{\|Av_n\|}. \]

Theorem 8. Let the cone \(K\) be solid, and let \(A\) be an indecomposable completely continuous operator. Then the sequences \(Au_n/\|Au_n\|\), \(Av_n/\|Av_n\|\) converge to the unique normalized positive eigenvector \(x^*\) of the operator \(A\): \(Ax^*=r(A)x^*\).

From the relations defining \(u_n, v_n\), it follows that Theorem 8 is close to the ergodic theorems \((^3)\). We note that the assertion of Theorem 8 will also hold in the case where, instead of the assumption of complete continuity of the operator \(A\), one makes the following assumption: the space \(E\) is weakly complete, the unit ball in \(E\) is weakly compact, and the cone \(K\) admits plastering \((^2)\). Similarly, the assumption that the cone is solid may be replaced by the assumption that the operator \(A\) is \(u_0\)-positive.

1. We present assertions supplementing the results of M. G. Krein \((^1)\) and I. A. Bakhtin \((^7)\) on commutative families of linear positive operators.

Theorem 9. Let \(K\) be a solid cone. Then, whatever commutative family \(\Gamma=\{A\}\) of linear positive operators \(A\) is given, there is always a positive functional \(\varphi_0\in E^*\), \(\varphi_0\ne\theta\), such that \(A^*\varphi_0=r(A)\varphi_0\) for all \(A\in\Gamma\).

An analogous assertion will hold if the assumption that the cone \(K\) is solid is replaced by the following condition: there exists an element \(u_0\in K\) such that for every \(A\in\Gamma\), for some \(a=a(x,A)\), the inequality \(Ax\le au_0\) is satisfied.

We give one more assertion about a commutative family \(\Gamma=\{A\}\) of linear positive operators.

Theorem 10. Let \(K\) be a solid cone and let at least one of the operators \(A\in\Gamma\) be quasi-completely continuous and indecomposable. Then in \(E\) one can introduce a new equivalent norm \(\|\ \|_1\) such that for every \(A\in\Gamma\) the equality \(\|A\|_1=r(A)\) holds.

It turns out that, under some additional conditions (for example, in the case of a reflexive space \(E\)), the new equivalent norm can be introduced so that it is strictly monotone: from \(0\le x<y\) it follows that \(\|x\|_1<\|y\|_1\).

The author expresses his gratitude to M. A. Krasnosel’skii for his constant attention and advice.

Voronezh Civil Engineering Institute

Received
17 III 1967

CITED LITERATURE

1. M. G. Krein, M. A. Rutman, UMN, 3, no. 1, 3 (1948).
2. M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Moscow, 1962.
3. N. Dunford, J. Schwartz, Linear Operators, 1, IL, 1962.
4. V. Ya. Stetsenko, UMN, 21, no. 5 (31), 265 (1966).
5. S. N. Mukhtarov, V. Ya. Stetsenko, Dokl. AN TadzhSSR, 8, no. 2, 7 (1965).
6. V. Ya. Stetsenko, Matem. sborn., 67 (109), 2, 210 (1965).
7. I. A. Bakhtin, Matem. sborn., 67 (109), 2, 267 (1965).