A lifting theorem for symmetric commutants (Q2706584)

Let \(H\) be a Hilbert space, and let \(T_1,\dots ,T_n\in {\mathcal B}(H)\) be bounded operators on \(H\). We say that they form a row contraction if \(T_1T_1^*+T_2T_2^*+\dots +T_nT_n^*\leq I_H\), where \(I_H\) denotes the identity operator on \(H\). The main result of the paper is the following NEWLINENEWLINENEWLINETheorem: Let \({\mathcal T}=[T_1,\dots ,T_n]\) be a row contraction on \(H\) and let \({\mathcal V}=[V_1,\dots ,V_n]\) be its minimal isometric dilation on \({\mathcal K}\) (where \(H\subset {\mathcal K}\)). We set \(H_0=H\), and NEWLINE\[NEWLINEH_k=H\oplus\bigvee_{|\alpha|=1}V_{\alpha}H_{k-1}NEWLINE\]NEWLINE if \(k\geq 1\). Let \(j\) be a symmetry on \(H\) and let \(J\in {\mathcal B}({\mathcal K})\) be a contraction such that each \(H_k\) reduces \(J\) and \(J_{|H}=j\). If \(A\in {\mathcal B}(H)\) is in the j-commutant of \(\{T_1,\dots ,T_n\}\) then there exists \(A_J\in{\mathcal B}({\mathcal K})\) in the \(J\)-commutant of \(\{V_1,\dots ,V_n\}\) such that \(A_{J^*H}=A*\) and \(\|A_J\|=\|A\|\).