I wonder if it is possible to characterize the bounded linear operators on some Hilbert space which are similar to the unilateral shift. I would like to have necessary and sufficient conditions on the operator which can be directly checked. Here are some necessary conditions on $T\in L(\mathcal H)$:
(a) $T^{*n}f\to 0$ as $n\to\infty$ for each $f\in\mathcal H$.
(b) For each $z\in\mathbb D$ (open disk) $T-z$ is Fredholm, injective, and $\dim\ker(T^*-\overline z) = 1$.
(c) $\|(T-z)^{-1}\|$ grows linearly to $\mathbb T = \{z : |z|=1\}$ from outside $\overline{\mathbb D}$.
(d) $\mathbb T = \sigma_c(T)$ (continuous spectrum).
However, I don't know whether these are sufficient. Does anyone know more?