Criteria for algebraic operators to be unitary

Criteria for an algebraic operator $T$ on a complex Hilbert space $\mathcal{H}$ to be unitary are established. The main one is written in terms of the convergence of sequences of the form $\{\mathrm{\parallel }{T}^{n}h\mathrm{\parallel }{\}}_{n=0}^{\mathrm{\infty }}$ with $h\in \mathcal{H}$. Related questions are also discussed.