# Journal of Operator Theory

Volume 87, Issue 1, Winter 2022  pp. 137-156.

Universal composition operators

Authors:  Joao R. Carmo (1), S. Waleed Noor (2)
Summary: A Hilbert space operator $U$ is called \textit{universal} (in the sense of Rota) if every Hilbert space operator is similar to a multiple of $U$ restricted to one of its invariant subspaces. It follows that the \textit{invariant subspace problem} for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for $U$ are one dimensional. In this article we characterize all linear fractional composition operators $C_{\phi} f=f\circ\phi$ that have universal translates on both the classical Hardy spaces $H^2(\mathbb{C}_+)$ and $H^2(\mathbb{D})$ of the half-plane and the unit disk, respectively. The new example here is the composition operator on $H^2(\mathbb{D})$ with affine symbol $\phi_a(z)=az+(1-a)$ for \$0