Abstract
We consider Berezin symbols and Hankel operators on the Hardy space H-2(D) over the unit disc D = {z is an element of C : |z| < 1} and give their some applications. Namely, we estimate in terms of Hankel operators and Berezin symbols the distances from a given operator to the algebra of all analytic Toeplitz operators and to the set of all Toeplitz operators on H-2(D). We use Hankel operator also to prove some lower estimate for the so-called Berezin number of bounded linear operators on H-2. Some other related questions are also discussed.