Abstract
We give, in terms of the Berezin number, necessary and sufficient conditions providing unitarity of an invertible operator. Also we obtain in terms of the Berezin number a new inequality for the norm of the Hankel operator H-phi which is better than the classical inequality parallel to H-phi parallel to <= parallel to phi parallel to(infinity). The Berezin number is also used to generalize the Douglas lemma on zero Toeplitz products.