We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences β every symbol φ : D → D with φ ∈ H 2 (β) induces a bounded composition operator C φ on the weighted Hardy space H 2 (β). We give partial answers and investigate when H 2 (β) is an algebra. We answer negatively to another question in showing that there are a sequence β and φ ∈ H 2 (β) such that ∥φ∥ ∞ < 1 and the composition operator C φ is not bounded on H 2 (β). In a second part, we show that for p ̸ = 2, no automorphism of D, except those that fix 0, induces a bounded composition operator on the Beurling-Sobolev space ℓ p A , and even on the weighted versions of this space.