Remarks on Complex Symmetric Operators
An operator {T\in{\mathcal{L}\mathcal{H}$TL(H) is said to be complex symmetric if there exists a conjugation C on \mathcal H}$H such that {T= CT^\ast}C}$T=CT*C. In this paper, we study the spectral radius algebras for complex symmetric operators. In particular, we prove that if A is a complex symmetric operator, then the spectral radius algebra \mathcal B}_{A}$BA associated with A has a nontrivial invariant subspace under some conditions. Finally, we give some relations between {P_{\tilde{A}$PA~ and {P_{\widetilde{A^\ast}$PA* (defined below) when A is complex symmetric.