  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
$H$.
