Vol. 45, No. 2, 1973

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Strictly cyclic operator algebras on a Banach space

Mary Rodriguez Embry

Vol. 45 (1973), No. 2, 443–452

A strictly cyclic operator algebra.𝒜 on a complex Banach space X(dimX 2) is a uniformly closed subalgebra of (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 Ψr (the commutant of Ψ) is intransitive, iii) if A ∈ℒ(X),AzI and {A}′ is strictly cyclic, then A has a hyperinvariant subspace, and iv) that a transitive subalgebra of (X), containing a strictly cyclic algebra which contains I, is strongly dense in (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 l1 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.

Mathematical Subject Classification 2000
Primary: 47A15
Received: 18 November 1971
Published: 1 April 1973
Mary Rodriguez Embry