Square roots of complex symmetric operators

Abstract

In this paper we study square roots of complex symmetric operators. In particular, we prove that if $T\in \mathcal{L}(\mathcal{H})$ is a square root of a complex symmetric operator, then $T^{\ast }$ has the single-valued extension property if and only if so does T. Moreover, in this case, T has the Bishop's property $(\beta )$ if and only if T is decomposable. Finally, we show that if T has a nontrivial hyperinvariant subspace, then $T^{\ast }$ has a nontrivial invariant subspace.

Keywords:

• Square roots of complex symmetric operators
• the single-valued extension property
• the Bishop's property $(\beta )$
• the Dunford's property (C)
• the dunford's boundedness condition (B)
• nontrivial invariant subspace

2020 Mathematics Subject Classifications:

• Primary 47A05
• Secondary 47B20

Acknowledgments

The authors wish to thank the referees for their invaluable comments on the original draft.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [grant number 2019R1F1A1058633] and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education [grant number 2019R1A6A1A11051177].