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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Domingo A. Herrero and Alan Lambert
Journal: Trans. Amer. Math. Soc. 185 (1973), 229-235
MSC: Primary 46L20; Secondary 47A15
MathSciNet review: 0328619
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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\Vert {A{x_j}} \right\Vert \leq \left\Vert {A{x_0}} \right\Vert$, 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}$.

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Keywords: Banach algebras, algebras of operators, $ \mathcal{P}$-algebras, strictly cyclic algebras, reflexive algebra, weighted shifts, invariant subspaces
Article copyright: © Copyright 1973 American Mathematical Society