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We study some problems of operator theory by using Berezin symbols approach. Namely, we investigate in terms of Berezin symbols invariant subspaces of isometric composition operators on H Ω . We discuss operator corona problem, in particular, the Toeplitz corona problem. Further, we characterize unitary operators in terms of Berezin symbols. We show that the well known inequality w A ≥ 1 2 A for numerical radius is not true for the Berezin number of operators, which is defined by ber A : = sup λ ∈ Ω A ~ λ , where A ~ λ : = A k ^ λ , k ^ λ is the Berezin symbol of operator A : H Ω → H Ω . Finally, we provide a lower bound for ber A .

We consider the space C n Ω , the Banach space of continuous functions with n derivatives and the n th derivative continuous in Ω ¯ , where Ω ⊂ C is a starlike region with respect to α ∈ Ω . We use the so-called α -Duhamel product f ⊛ α g ( z ) : = d dz ∫ α z f ( z + α - t ) g ( t ) d t = d dz f ∗ α g z to describe usual ∗ α -generators of the Banach algebra C n Ω , ∗ α , to estimate I - V α m and to estimate below the norm δ A m , where V α is the Volterra integration operator defined by V α f z = ∫ α z f t d t and δ A is the inner derivation operator defined by δ A X : = X , A . We give a new proof of Aleman-Korenblum theorem in one particular case. Namely, we describe V -invariant subspaces in the Hardy space H p by using Duhamel product.