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Preprints Archive: Abstract of IC2010075 (2010)

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A note on the g-cyclic operators over a bounded semigroup

by Nuha H. Hamada and Zeana Z. Jamil

Document info: Pages 5, Figures 0.

Let h be an infinite-dimensional separable complex Hilbert space, and BH be the Banach algebra of all linear bounded operators on h. Let s be a multiplication semigroup of C with 1, an operator $T\in BH$ is called g-cyclic operator over s if there is a vector x in h such that $\{\alpha T^n x| \alpha \in s, n \geq 0 \}$ is dense in h. In this case x is called a g-cyclic vector for T over s. If T is g-cyclic operator and $s=\{1\}$ then T is a hypercyclic operator. In this paper, we study the spectral properties of a g-cyclic operators over a bounded s under the condition that zero is not in the closure of s. We show that the class of all g-cyclic operators is contained in the norm-closure of the class of all hypercyclic operators.

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