On complex symmetric weighted shifts

Unbounded complex symmetric weighted shifts are studied. Complex symmetric unilateral weighted shifts whose $C^{\mathrm{\infty }}$ vectors contain the image of the canonical orthonormal basis under the conjugation are shown to be decomposable into an orthogonal sum of infinitely many complex selfadjoint truncated weighted shifts, which generalizes a result of S. Zhu and C. G. Li. The bilateral case is discussed as well. Additional results, examples, and open problems are supplied.