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

Abstract:

An operator algebra $\mathfrak {A} \subset \mathcal {L}(\mathcal {X})$ (the algebra of all operators in a Banach space $\mathcal {X}$ over the complex field C) is called a “strictly cyclic algebra” (s.c.a.) if there exists a vector ${x_0} \in \mathcal {X}$ such that $\mathfrak {A}({x_0}) = \{ A{x_0}:A \in \mathfrak {A}\} = \mathcal {X};{x_0}$ is called a “strictly cyclic vector” for $\mathfrak {A}$. If, moreover, ${x_0}$ separates elements of $\mathfrak {A}$ (i.e., if $A \in \mathfrak {A}$ and $A{x_0} = 0$, then $A = 0$), then $\mathfrak {A}$ is called a “separated s.c.a." $\mathfrak {A}$ is a $\mathcal {P}$-algebra if, given ${x_1}, \ldots ,{x_n} \in \mathcal {X}$, there exists ${x_0} \in \mathcal {X}$ such that $\left \| {A{x_j}} \right \| \leq \left \| {A{x_0}} \right \|$, for all $A \in \mathfrak {A}$ and for $j = 1, \ldots ,n$. Among other results, it is shown that if the commutant $\mathfrak {A}’$ of the algebra $\mathfrak {A}$ is an s.c.a., then $\mathfrak {A}$ is a $\mathcal {P}$-algebra and the strong and the uniform operator topology coincide on $\mathfrak {A}$; these results are specialized for the case when $\mathfrak {A}$ and $\mathfrak {A}’$ are separated s.c.a.’s. (Here, and in what follows, algebra means strongly closed subalgebra on $\mathcal {L}(\mathcal {X})$ containing the identity I on $\mathcal {X}$.) 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 Hilbert space are reflexive. The result is used to show that “reflexivity” is neither a “restriction property” nor a “quotient property." Recall that an algebra $\mathfrak {A}$ is called reflexive if, whenever $T \in \mathcal {L}(\mathcal {X})$ and the lattice of invariant subspaces of T contains the corresponding lattice of $\mathfrak {A}$, then $T \in \mathfrak {A}$.