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Invariant subspaces of focusing plus-operators. (English. Russian original) Zbl 0565.47022
Math. Notes 30, 845-848 (1982); translation from Mat. Zametki 30, 695-702 (1981).
Let L be a real or a complex Hilbert J-space with an indefinite metric $$[x,y]=((P_+-P_-)x,y),$$ x,y$$\in L$$, and let $$P_{\pm}$$ be mutually complementary orthoprojections: $$L_{\pm}=P_{\pm}L$$, $$P_++P_-=I$$. We set $$M_+$$ as the set of all maximal nonnegative subspaces.
The present paper is the continuation of Funkts. Anal. Prilozh. 12, No.1, 88-89 (1978; Zbl 0395.47026), in which the author has investigated the problems of the existence and uniqueness of an invariant maximal nonnegative subspace for a focusing plus-operator A, acting in a real Hilbert J-space and satisfying the following condition:
A maps maximal nonnegative subspaces onto maximal nonnegative subspaces.
The mentioned condition can be written in the equivalent form: $(1)\quad (P_+AP_++P_+AP_-K_+)L_+=L_+,$ where $$K_+\in {\mathcal K}_+$$ (the set of all angular operators, corresponding to the subspaces in $$M_+)$$. The author in this paper considers the case of a complex J-space and establishes the possibility of the extension of each nonnegative subspace S with the property $$\overline{AS}=S$$ to an invariant maximal nonnegative subspace, where the condition (1) is replaced by the more general condition $${\mathcal R}(P_+AP_-)\subset {\mathcal R}(_+AP_+).$$
Reviewer: Th.Rassias
##### MSC:
 47B50 Linear operators on spaces with an indefinite metric 47A15 Invariant subspaces of linear operators
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##### References:
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