Journal of Operator Theory

Volume 42, Issue 1, Summer 1999 pp. 37-76.

Algebras of subnormal operators on the unit ball

Authors: Jorg Eschmeier
Author institution: Fachbereich Mathematik, Universitaet des Saarlandes, Postfach 15 11 50, D--66041 Saarbrucken, Germany

Summary: In this paper we show that each subnormal $n$-tuple $T\in L(H)^n$ with the property that the Taylor spectrum of $T$ is contained in the closed Euclidean unit ball and is dominating in the open ball, is reflexive. The proof is based on the observation that the dual algebra generated by $T$ possesses the factorization property $({\bbb A}_{1,\aleph_0})$. The same results are shown to hold for subnormal tuples that possess an isometric ${\rm w}^*$-continuous $H^\infty$-functional calculus over the unit ball. Thus we extend a result of Olin and Thomson on the reflexivity of arbitrary single subnormal operators to the case of subnormal systems with rich spectrum in the Euclidean unit ball.

Keywords: Subnormal systems, reflexive operators, dual operator algebras, Henkin measures, $H^\infty$-functional calculus

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