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By the well-known result of Yood, every strictly transitive algebra of operators on a Banach space is WOT- dense. This motivated us to investigate the relationships between SOT and WOT largeness of sets of operators and the transitivity behavior of them. We show that, to obtain Yood's result, strict transitivity may not be replaced by the weaker condition of hypertransitivity. We prove that, for a wide class of topological vector spaces, every SOT-dense set of operators is hypertransitive. The general form of SOT-dense sets that are not strictly transitive is presented. We also describe the form of WOT-dense sets that are not hypertransitive. It is shown that a set is hypertransitive if and only its SOT-closure is hypertransitive. We introduce strong topological transitivity and we show that every separable infinite-dimensional Hilbert space supports an invertible topologically transitive operator that is not strongly topologically transitive. [ABSTRACT FROM AUTHOR]

We characterize the commutants of some multiplication operators on a Banach space of analytic functions defined on a bounded domain in the plane. Under certain conditions on the symbol of a multiplication operator, we show that its commutant is a set of multiplication operators. This partially answers a question of Axler, Cuckovic and Rao. Next, the hyper-reflexivity of these multiplication operators are proved. The paper is concluded by proving the hyper-reflexivity of the multiplication operators with symbols φ(z)= z, k =1, 2,... . [ABSTRACT FROM AUTHOR]