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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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