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An operator algebra S C AO() (the algebra of all operators in a Banach space 9( over the complex field C) is called a "strictly cyclic algebra" (s.c.a.) if there exists a vector x0 e D( such that Sl(.Xo) — [Ax0:A e 31} = D(; x» is called a "strictly cyclic vector" for 21. If, moreover, x0 separates elements of SI (i.e., if A e 21 and Ax<, « 0, then A » 0), then SI is called a "separated s.c.a." SI is a £P-algebra if, given x,.x. 6 % there exists x0 e £x such that \\Axj\\ < \\A x0 II, for all A E 21 and for y = 1.n. Among other results, it is shown that if the commutant SI' of the algebra SI is an s.c.a., then St is a iP-algebra and the strong and the uniform operator topology coincide on SI; these results are specialized for the case when 21 and SI' are separated s.c.a.'s. (Here, and in what follows, algebra means strongly closed subalgebra on £(<X) containing the identity I on %) In the second part of the paper, it is shown that a large class of bilateral weighted shifts (which includes all the invertible ones) in a Hubert space are reflexive. The result is used to show that "reflexivity" is neither a "restriction property" nor a "quotient property." Recall that an algebra 21 is called reflexive if, whenever T 6 X(0() and the lattice of invariant subspaces of T contains the corresponding lattice of SI, then T B SI. (2) 1. A strictly cyclic algebra and its commutant Let 91 be a strictly cyclic algebra on % with strictly cyclic vector x0. We have a commutative diagram:

On strictly cyclic algebras, $\mathcal{P}$-algebras and reflexive operators