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On the complex symmetric and skew-symmetric operators with a simple spectrum. (English) Zbl 1218.44002

Summary: We obtain necessary and sufficient conditions for a linear bounded operator in a Hilbert space \(H\) to have a three-diagonal complex symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in \(H\). It is shown that the set of all such operators is a proper subset of the set of all complex symmetric operators with a simple spectrum. Similar necessary and sufficient conditions are obtained for a linear bounded operator in \(H\) to have a three-diagonal complex skew-symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in \(H\).

MSC:

44A60 Moment problems
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