In this paper, we find complex symmetric composition operators on the classical Hardy space H 2 whose symbols are linear-fractional but not automorphic. In doing so, we answer a recent question of Noor, and partially answer the original problem posed by Garcia and Hammond.

Let φ \varphi be an analytic self-map of the open unit disk D \mathbb {D} . We study the complex symmetry of composition operators C φ C_\varphi on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of D \mathbb {D} that is not an elliptic automorphism, we establish that if C φ C_{\varphi } is complex symmetric, then either φ ( 0 ) = 0 \varphi (0)=0 or φ \varphi is linear. In the case of weighted Bergman spaces A α 2 A^{2}_{\alpha } , we find the non-automorphic linear fractional symbols φ \varphi such that C φ C_{\varphi } is complex symmetric.

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We consider the Spectral radius algebra associated with a weighted shift of finite multiplicity. When the weighted shift is injective, we describe the structure of this algebra. This leads to a necessary and sufficient condition for there to exist a nontrivial invariant subspace for the Spectral radius algebra. This result is then generalized to noninjective weighted shifts of finite multiplicity.

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