Dilation theory and systems of simultaneous equations in the predual of an operator algebra. II
Let T be a contraction on the Hilbert space \({\mathcal H}\) such that the weak*-closed algebra generated by T is isomorphic with the Hardy space \(H^{\infty}\) via the Sz.-Nagy-Foias functional calculus. We say that T has property \({\mathbb{A}}_{\aleph_ o}\) if for every family \(\{K_{ij}: 0\leq i,j<\infty \}\) of trace-class operators on \({\mathcal H}\) there exist vectors \(\{x_ i,y_ i: 0\leq i<\infty \}\) in \({\mathcal H}\) such that \(tr(T^ nK_{ij})=(T^ nx_ i,y_ j)\) for \(0\leq i,j,n<\infty\). We show that every contraction with property \({\mathbb{A}}_{\aleph_ 0}\) is reflexive, thus extending a previous result pertaining to a more restricted class of operators. [For part I see Mich. Math. J. 30, 335-354 (1983; Zbl 0541.47007).]
scientific article; zbMATH DE number 3901461