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Completion, extension, factorization, and lifting of operators

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Abstract

The well-known results of M. G. Kreĭn concerning the description of selfadjoint contractive extensions of a hermitian contraction \(T_1\) and the characterization of all nonnegative selfadjoint extensions \({{\widetilde{A}} }\) of a nonnegative operator A via the inequalities \(A_K\le {{\widetilde{A}} } \le A_F\), where \(A_K\) and \(A_F\) are the Kreĭn–von Neumann extension and the Friedrichs extension of A, are generalized to the situation, where \({{\widetilde{A}} }\) is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the negative index of the operators \(I-T_1^*T_1\) and A, respectively; these conditions are automatically satisfied if \(T_1\) is contractive or A is nonnegative, respectively. The approach developed in this paper starts by establishing first a generalization of an old result due to Yu. L. Shmul’yan on completions of nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M. G. Kreĭn and, in addition, to solve some related lifting problems for J-contractive operators in Hilbert, Pontryagin and Kreĭn spaces in a simple manner. Also some new factorization results are derived, for instance, a generalization of the well-known Douglas factorization of Hilbert space operators. In the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non-contractive selfadjoint operators under Cayley transforms properly.

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Notes

  1. After the Math ArXiv version of the present paper we inquired contents of that work from H. Langer who then kindly provided us their initial work in [53].

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Acknowledgments

We wish to thank Heinz Langer for making the paper [53] available to us. Main part of this work was carried out during the second author’s sabbatical year 2013–2014; he is grateful about the Professor Pool’s grant received from the Emil Aaltonen Foundation.

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  1. Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101, Vaasa, Finland

    Dmytro Baidiuk & Seppo Hassi

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Baidiuk, D., Hassi, S. Completion, extension, factorization, and lifting of operators. Math. Ann. 364, 1415–1450 (2016). https://doi.org/10.1007/s00208-015-1261-5

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