On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces (Q1600103)

Let \(C\) be the reproducing kernel of the Hilbert space \(\mathcal{H}\) of functions defined on an abstract set \(\Lambda\), such that the kernel \(1-1/C\) is also positive definite on \(\Lambda\times\Lambda\). The multiplier algebra \(\mathcal{M}\) is the algebra of all bounded operators on \(\mathcal{H}\) of the form \((Tf)(\lambda)=\varphi(\lambda)f(\lambda)\); the function \(\varphi\) is called the symbol of \(T\). A closed subspace \(\mathcal{X}\) of \(\mathcal{H}\) is called invariant if \(T^*\mathcal{X}\subset\mathcal{X}\) for all \(T\in\mathcal{M}\). The main result of this paper is an intertwining lifting theorem stating that any operator \(T\) in the commutant of \(\mathcal{M}_\mathcal{X}=P_\mathcal{X}\mathcal{M}|\mathcal{X}\) can be lifted to a multiplier with the same norm as \(T\). This theorem was previously obtained, in a slightly more general setting, by \textit{J. A. Ball, T. T. Trent} and \textit{V. Vinnikov} in [Oper. Theory, Adv. Appl. 122, 89-138 (2001; Zbl 0983.47011)], but the proof in the paper under review is shorter, more elegant, and avoids many unnecessary technicalities.