For any given symmetric measure μ on the closed unit disk D, we apply the Berezin transform to characterizing semi-commuting and commuting Toeplitz operators with bounded harmonic symbols on A2（D, dμ）.
： We consider an approximation problem related to strongly irreducible operators, that is, does the direct sum of a strongly irreducible operator in B∞（Ω） and certain operator have a small compact perturbation which is a strongly irreducible operator in B∞（Ω）？ In this paper, we prove that the direct sum of any strongly irreducible operator in B∞（Ω） and certain biquasitriangular operator have small compact perturbations which are strongly irreducible operators in B∞（Ω）.
Two operators A, B ∈ B（H） are said to be strongly approximatively similar, denoted by A -sas B, if （i） given ε 〉 0, there exist Ki ∈ B（H） compact with ｜｜Ki｜｜ 〈ε（i = 1,2） such that A＋K1 and B ＋ K2 are similar; （ii） σ0（A） = σ0（B） and dim H（λ; A） = dim H（λ; B） for each λ ∈ σ0（A）. In this paper, we prove the following result. Let S,T ∈ B（H） be quasitriangular satisfying： （i） σ（T） = σ（S） = σw（S） is connected and σe（S） = σlre（S）; （ii） ρs-F（S） ∩ σ（S） consists of at most finite components and each component Ω satisfies that Ω = int Ω, where int Ω is the interior of Ω. Then, S -sas T if and only if S and T are essentially similar.
