On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum

Details On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum

2010-11-30

arxiv.org/abs/1011.6584v2

SIGMA 7 (2011), 016, 9 pages

Mathematics

Functional Analysis

Scientific paper

10.3842/SIGMA.2011.016

In this paper 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 a set of all such operators is a proper subset of a 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 $H$.

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