On complex symmetric weighted shifts

TL;DR

This paper investigates unbounded complex symmetric weighted shifts, showing they can be decomposed into sums of selfadjoint shifts, and extends known results to bilateral cases with new examples and open problems.

Contribution

It generalizes previous results by demonstrating the decomposition of complex symmetric weighted shifts into selfadjoint components, including bilateral cases.

Findings

Unbounded complex symmetric weighted shifts can be decomposed into orthogonal sums of selfadjoint shifts.

The paper extends decomposition results to bilateral weighted shifts.

Provides new examples and discusses open problems in the area.

Abstract

Unbounded complex symmetric weighted shifts are studied. Complex symmetric unilateral weighted shifts whose $C^\infty$ 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.