
SIGMA 7 (2011), 016, 9 pages arXiv:1011.6584
https://doi.org/10.3842/SIGMA.2011.016
On the Complex Symmetric and SkewSymmetric 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 threediagonal complex symmetric matrix with
nonzero elements on the first subdiagonal
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 threediagonal complex skewsymmetric matrix with
nonzero elements on the first subdiagonal
in an orthonormal basis in H.
Key words:
complex symmetric operator; complex skewsymmetric operator; cyclic operator; simple spectrum.
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