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On the reflexivity, hyperreflexivity and transitivity of algebrasand subspaces of operators

The recent reflexivity, transitivity and hyperreflexivity results for subspaces and algebras of operators will be presented. We start with the  situation when underlying Hilbert space is finite dimensional and giving some examples show that even in this case the notion of reflexivity is interesting. It will be presented that reflexivity and hypereflexivity are equivalent for finite dimensional subspaces of operators even the underlying Hilbert space is not finite dimensional (positive answer for Larson--Kraus problem). We will study the dichotomic behavior (reflexivity versus transitivity) of  subspaces of Toeplitz operators on  the Hardy space. The Toeplitz operators on the Bergman space will be also considered. We discuss also algebras generated by isometries, power partial isometries and quasinormal operators. The multivariable case will be also presented.
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