Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 016, 9 pages      arXiv:1011.6584      http://dx.doi.org/10.3842/SIGMA.2011.016

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

Sergey M. Zagorodnyuk
School of Mathematics and Mechanics, Karazin Kharkiv National University, 4 Svobody Square, Kharkiv 61077, Ukraine

Received December 14, 2010, in final form February 11, 2011; Published online February 16, 2011

Abstract
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.

Key words: complex symmetric operator; complex skew-symmetric operator; cyclic operator; simple spectrum.

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