Absolute continuity for commuting row contractions (Q290595)

The paper under review studies the so-called absolutely continuous row contractions. The essential motivation comes on the one hand from Sz.-Nagy-Foiaş theory, and on the other hand from the theory of the Drury-Arveson \(d\)-shift. NEWLINENEWLINENEWLINE NEWLINERecall that a \(d\)-tuple \(T\) of commuting bounded linear Hilbert space operators \(T_1, \dots, T_d\) is a \textit{row contraction} if \(T_1T^*_1 + \dots + T_dT^*_d \leq I\), while a row contraction \(T\) is said to be \textit{absolutely continuous} if the functional calculus \(\phi \mapsto M_\phi\) extends to a weak*-continuous map on the multiplier algebra \(\mathcal M_d\). One of the main results (Theorem 3.2) describes all absolutely continuous row contractions:NEWLINENEWLINENEWLINE{Theorem.} If \(T\) is a commuting row contraction on a separable Hilbert space with the spherical unitary part \(U\) of its minimal co-extension, then the following are equivalent: {\parindent=0.7cm\begin{itemize}\item[(1)] \(T\) is absolutely continuous. \item[(2)] \(U\) is absolutely continuous. \item[(3)] The spectral measure of \(U\) is Henkin. NEWLINENEWLINE\end{itemize}} NEWLINEA substantial part of this paper is devoted to interesting decomposition results. For instance, it is proved that any row contraction can be decomposed into an absolutely continuous row contraction and a totally singular spherical unitary (Theorem 3.7). This result generalizes Theorem 1.5 of [\textit{J. Eschmeier}, Proc. Lond. Math. Soc. (3) 75, No. 1, 157--176 (1997; Zbl 0878.47001)] (cf.\ Theorem 4 of [\textit{A. Athavale}, J. Funct. Anal. 154, No. 1, 117--129, Art. No. FU973196 (1998; Zbl 0919.47006)]). In view of the dilation theorem of \textit{V. Müller} and \textit{F. H. Vasilescu} [Proc. Am. Math. Soc. 117, No. 4, 979--989 (1993; Zbl 0777.47009)], it is likely that many of these considerations have counterparts for the class of row \(\nu\)-hypercontractions.