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Boundary spectrum of nonnegative operators. (English. Russian original) Zbl 0646.47002
Sib. Math. J. 26, 798-802 (1985); translation from Sib. Mat. Zh. 26, No. 6(154), 24-28 (1985).
The authors consider real n-dimensional space \(R^ n\) (n\(\geq 3)\) equipped with a closed solid cone K. Let A be a linear operator which is non-negative, i.e., AK\(\subset K\). If A has spectral radius r(A) equal to one, then the part of the spectrum of A lying on the unit circle is called the boundary spectrum. The space is said to possess property F if for each operator \(A\geq 0\) \((r(A)=1)\), all points of the boundary spectrum are roots of unity. If there is an additional requirement of boundedness of the semigroup of powers of A, then the space is said to possess property \(F_ 1\). In a previous article, Yu. I. Lyubich [Sib. Mat. Zh. 11, 358-369 (1970; Zbl 0209.448)] gave analogous definitions for contractions in Minkowski space (in which case properties F and \(F_ 1\) are equivalent), and he stated necessary and sufficient geometric conditions that a Minkowski space possesses property F. In the present article the analogous problem is solved for spaces with property \(F_ 1\). The principal result is that an n-dimensional space with cone K possesses property \(F_ 1\) if and only if there does not exist in it a K-complemented three-dimensional subspace with a Euclidean cone. Here, a cone K is called Euclidean if it is possible in a certain coordinate system \(\xi_ 1,\xi_ 2,...,\xi_ n\) to describe the cone by the inequality \(\xi_ 1\geq (\sum^{n}_{k=2}\xi^ 2_ k)^{1/2}\). In any space with cone K a subspace is called K-complemented if it is the image of a certain projector \(P\geq 0\). The above result is used to prove that a space with an \(\ell_ p\)-cone \(\xi_ 1\geq (\sum^{n}_{k=2}| \xi_ k|^ p)^{1/p}\) \((p>1)\) possesses property \(F_ 1\) if \(p\neq 2.\)
It is noted that the basic result of the present article carries over to completely continuous operators in a Banach space with a closed solid cone.
Reviewer: R.C.Gilbert
MSC:
47A10 Spectrum, resolvent
46A40 Ordered topological linear spaces, vector lattices
47B60 Linear operators on ordered spaces
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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[1] I. M. Glazman and Yu. N. Lyubich (Ju. I. Ljubic), Finite-Dimensional Linear Analysis, MIT Press, Cambridge, Mass. (1974). · Zbl 0332.15001
[2] Yu. I. Lyubich, ?On the boundary spectrum of contractions in Minkowski spaces,? Sib. Mat. Zh.,11, No. 2, 358-369 (1970).
[3] M. A. Krasnosel’skii, ?On a spectral property of completely continuous linear operators in the space of continuous functions,? in: Problems of Mathematical Analysis of Composite Systems [in Russian], No. 2, Voronezh (1968), pp. 68-71.
[4] M. Yu. Lyubich and Yu. I. Lyubich, ?The spectral theory of almost periodic representations of semigroups,? Ukr. Mat. Zh.,36, No. 5, 632-636 (1984).
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