Hypercyclic algebras for convolution and composition operators

We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator $D$ of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\mbox{cos}(D)$, $De^D$, or $e^D-aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras. Non-trivial translations on the space of complex-valued, smooth functions on the real line do support hypercyclic algebras.