Let B(H) denote the Banach algebra of all bounded linear operators on an infinite-dimensional separable complex Hubert space H, and let I2 be the Hilbert space of all square-summable complex sequences x = {*", xx, x2,... }. For an injective operator A in B(H) and a nonzero vector / in H, put wm = \\Amf\\/\\Am-xf\\, m = 1, 2, .... The operator TAJ on I2, defined by TA/x) = [wxxx, w2x2,. . . }, is called a weighted (backward) shift with the weight sequence {wm}m-i-This paper is concerned with the

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... investigation of the existence of cyclic vectors of TAJ. Also it is shown that if A satisfies certain nice conditions, then every transitive subalgebra of B(H) containing TAJ coincides with B(H).