A strictly cyclic operator
algebra.𝒜 on a complex Banach space X(dimX ≧ 2) is a uniformly closed
subalgebra of ℒ(X) such that 𝒜x = X for some x in X. If Ax = 0,A ∈𝒜
implies that A = 0, then 𝒜 is separated. In this paper it is shown that i) if
𝒜 is strictly cyclic and separated, then 𝒜 has a maximal, proper, closed
invariant subspace, ii) if 𝒜 is strictly cyclic, then Ψ^{r} (the commutant of Ψ) is
intransitive, iii) if A ∈ℒ(X),A≠zI and {A}′ is strictly cyclic, then A has a
hyperinvariant subspace, and iv) that a transitive subalgebra of ℒ(X), containing
a strictly cyclic algebra which contains I, is strongly dense in ℒ(X). An
example demonstrates the existence of abelian and nonabelian strictly cyclic,
separated operator algebras on each Banach space of dim ≧ 2. A second
example classifies the strictly cyclic weighted shifts on l_{1} and shows that the
commutant of each such operator consists of uniform limits of polynomials of the
operator.
If 𝒜 is a strictly cyclic operator algebra on X and 𝒜x = X, then x is called a
strictly cyclic vector for 𝒜. If it is also true that Ax = 0 implies A = 0 for A in 𝒜 we
say that x is a separating vector for 𝒜 For many results in this paper it is not
required that 𝒜 be separated or that the identity operator I on X be an element of
𝒜.
