Let
φ
\varphi
be a nonconstant analytic self-map of the open unit disk in
C
\mathbb {C}
, with
‖
φ
‖
∞
>
1
\|\varphi \|_{\infty }>1
. Consider the operator
D
φ
D_{\varphi }
, acting on the Hardy space
H
2
H^{2}
, given by differentiation followed by composition with
φ
\varphi
. We obtain results relating to the adjoint, norm, and spectrum of such an operator.