Abstract
We show that a spherical contraction T with a spherical dilation and isometric H∞-functional calculus on the unit ball belongs to the class \({\mathbb{A}_{{1,{\aleph _{0}}}}}\) if and only if it possesses an analytic invariant subspace, or if and only if, there is a compression of T to a semi-invariant subspace which is of type C.0 and possesses dominating Harte spectrum in the unit ball. If any of these conditions holds, then T is proved to be reflexive. As a byproduct we obtain that the dual algebra generated by a spherical isometry has property (\({\mathbb{A}_{{1,{\aleph _{0}}}}}\)). Applied to the particular case of subnormal tuples our results lead to a new reflexivity proof for subnormal spherical contractions with rich Taylor spectrum in the unit ball.
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Eschmeier, J. (2001). On the structure of spherical contractions. In: Kérchy, L., Gohberg, I., Foias, C.I., Langer, H. (eds) Recent Advances in Operator Theory and Related Topics. Operator Theory: Advances and Applications, vol 127. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8374-0_12
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DOI: https://doi.org/10.1007/978-3-0348-8374-0_12
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