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Zbl 1099.47058
Strictly cyclic operator algebras on Banach spaces.
(English)
[J] Integral Equations Oper. Theory 48, No. 4, 557-560 (2004). ISSN 0378-620X; ISSN 1420-8989/e

Let $X$ be a complex Banach space and let $B(X)$ be the set of all bounded linear operators on~$X$. Recall that a subalgebra $A\subset B(X)$ is said to be {\it strictly cyclic\/} if the orbit of some vector in~$X$ coincides with~$X$. It is proved that every norm closed commutative semisimple strictly cyclic algebra in a Banach space is reflexive, which extends a result of {\it A.~Lambert} [Pac.\ J.\ Math.\ 39, 717--726 (1971; Zbl 0213.40701)] to the case of Banach spaces. The reflexivity of algebras with strictly cyclic and strictly separating vectors and of some generalizations of these algebras was studied in many papers in the 1990s, but this literature is not reflected in the present paper.
[Alexander Isaakovich Shtern (Moskva)]
MSC 2000:
*47L10 Algebras of operators on Banach spaces, etc.
47A16 (Hyper-)cyclic vectors
47L30 Abstract operator algebras on Hilbert spaces
46L80 K-theory and operator algebras

Keywords: operator algebra; strictly cyclic algebra; reflexive algebra

Citations: Zbl 0213.40701

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