Proof. By \(q\)-commutativity of \(\underline{T}\), we have that \(T_1T_2=q_{12}T_2T_1\) for some \(q_{12} \in \mathbb{T}\). Suppose \(\underline{T}\) has a minimal \(*\)-regular \(q\)-isometric dilation. By Theorem 3, \(\underline{T}\) admits a minimal \(q\)-isometric dilation \(\underline{V}=(V_1, V_2)\) on a Hilbert space \(\mathcal{K}_+\), which is doubly \(q\)-commuting. By the minimality condition, \(\mathcal{K}_+=\overline{\text{span}}\left\{V_1^m V_2^n x: x \in \mathcal{H}, m, n \in \mathbb{Z}_+\right\}.\) It follows from Lemma 2.11 in [15] that \((T_1^*, T_2^*)=(V_1^*|_{\mathcal{H}}, V_2^*|_{\mathcal{H}})\). Let \(W_1\) acting on a Hilbert space \(\mathcal{K}_1\) be the minimal isometric dilation of \(T_1\). Again by minimality, \(\mathcal{K}_1=\overline{\text{span}}\{V_1^mx: x \in \mathcal{H}, m \in \mathbb{Z}_+\}\) and \(W_1=V_1|_{\mathcal{K}_1}\). Here, we have used the fact that the minimal isometric dilation of a contraction is unique (up to a unitary equivalence). Define \(W_2=P_{\mathcal{K}_1}V_2|_{\mathcal{K}_1}\), which is a contraction on \(\mathcal{K}_1\). Since \((T_1^*, T_2^*)=(V_1^*|_{\mathcal{H}}, V_2^*|_{\mathcal{H}})\), we have \[\begin{align} \label{eqn95WT} W_i^*x=P_{\mathcal{K}_1}V_i^*x=P_{\mathcal{K}_1}T_i^*x=T_i^*x \quad \text{for all x \in \mathcal{H} and so,} \quad W_i^*|_{\mathcal{H}}=V_i^*|_{\mathcal{H}}=T_i^* \end{align}\tag{3}\] for \(i=1, 2\). We now prove that \(W_2^*W_1=\overline{q}_{21}W_1W_2^*\). For \(m \in \mathbb{Z}_+\) and \(x \in \mathcal{H}\), we have that \[\begin{align} W_2^*W_1V_1^mx=W_2^*V_1^{m+1}x=P_{\mathcal{K}_1}V_2^*V_1^{m+1}x =\overline{q}_{21}^{m+1}P_{\mathcal{K}_1}V_1^{m+1}V_2^*x =\overline{q}_{21}^{m+1}V_1^{m+1}T_2^*x, \end{align}\] where the last equality holds since \(T_2^*=V_2^*|_{\mathcal{H}}\) and \(V_1^{m+1}T_2^*x \in \mathcal{K}_1\). Furthermore, \[\begin{align} W_1W_2^*V_1^mx=W_1P_{\mathcal{K}_1}V_2^*V_1^{m}x=\overline{q}_{21}^mW_1P_{\mathcal{K}_1}V_1^{m}V_2^*x =\overline{q}_{21}^{m}W_1P_{\mathcal{K}_1}V_1^{m}T_2^*x =\overline{q}_{21}^{m}W_1V_1^{m}T_2^*x =\overline{q}_{21}^{m}V_1^{m+1}T_2^*x, \end{align}\] where the last equality holds as \(V_1^{m}T_2^*x \in \mathcal{K}_1\). Thus, \(W_2^*W_1V_1^mx=\overline{q}_{21}W_1W_2^*V_1^mx\) and by continuity, \(W_2^*W_1=\overline{q}_{21}W_1W_2^*\). Next, we show that \(W_2^*W_1^*=q_{21}W_1^*W_2^*\). Let \(x, h \in \mathcal{H}\) and \(n \geq 1\). Then \[\begin{align} W_2^*W_1^*(h+V_1^nx) &=T_2^*T_1^*h+W_2^*W_1^*W_1^nx \quad [\text{by \eqref{eqn95WT}}]\\ &=q_{21}T_1^*T_2^*h+W_2^*W_1^{n-1}x\\ &=q_{21}T_1^*T_2^*h+\overline{q}_{21}^{n-1}W_1^{n-1}W_2^*x \quad [\text{since W_2^*W_1=\overline{q}_{21}W_1W_2^*}]\\ &=q_{21}W_1^*W_2^*h+\overline{q}_{21}^{n-1}W_1^*(W_1^{n}W_2^*)x \quad [\text{by \eqref{eqn95WT}}]\\ &=q_{21}W_1^*W_2^*h+\overline{q}_{21}^{n-1}q_{21}^nW_1^*W_2^*W_1^{n}x \quad [\text{since W_1W_2^*=q_{21}W_2^*W_1}]\\ &=q_{21}W_1^*W_2^*(h+V_1^nx), \end{align}\] and thus by continuity arguments, it follows that \(W_2^*W_1^*=q_{21}W_1^*W_2^*\). Consequently, \((W_1, W_2)\) is a doubly \(q\)-commuting pair of contractions on \(\mathcal{K}_1\). It follows from Wold type decomposition for doubly \(q\)-commuting contractions (see [11], [12]) that there exist \(4\) joint reducing subspaces \(\{\mathcal{K}_{1\eta}: \eta \subseteq \{1,2 \}\}\) for \(W_1^*, W_2^*\) such that \[\mathcal{K}=\bigoplus_{\eta \subseteq \{1,2 \}} \mathcal{K}_{1\eta} \quad \text{and} \quad \underline{W}^*=\bigoplus_{\eta \subseteq \{1,2 \}} \underline{W}^*_\eta,\] where \(\underline{W}_\eta^*=(W_1^*|_{\mathcal{K}_{1\eta}}, W_2^*|_{\mathcal{K}_{1\eta}})\). Moreover, \(W_i^*|_{\mathcal{K}_{1\eta}}\) is a unitary if \(i \in \eta\), and a c.n.u. contraction if \(i \notin \eta\) for each \(\eta\). Also, \((W_1^*|_{\mathcal{K}_{1\eta}}, W_2^*|_{\mathcal{K}_{1\eta}})\) is doubly \(q\)-commuting for every \(\eta \subseteq \{1,2 \}\).

Conversely, let \(W_1\) acting on a Hilbert space \(\mathcal{K}_1\) be the minimal isometric dilation of \(T_1\), and let \((W_1, W_2)\) be a doubly \(q\)-commuting pair of contractions on \(\mathcal{K}_1\) such that \((T_1^*, T_2^*)=(W_1^*|_{\mathcal{H}}, W_2^*|_{\mathcal{H}})\). By minimality, \(\mathcal{K}_1=\overline{\text{span}}\{W_1^mx: x \in \mathcal{H}, m \in \mathbb{Z}_+\}\). For \(m \in \mathbb{Z}_+\) and \(x \in \mathcal{H}\), it is evident that \[T_1^{*m}T_2^{*}=W_1^{*m}W_2^{*}|_{\mathcal{H}} \quad \text{and so,} \quad P_{\mathcal{H}}W_2(W_1^mx)=T_2T_1^mx=T_2P_{\mathcal{H}}W_1^mx.\] Thus, \(P_{\mathcal{H}}W_2=T_2P_{\mathcal{H}}\) and \(P_{\mathcal{H}}W_2W_1=T_2T_1P_{\mathcal{H}}\). By doubly \(q\)-commutativity of \((W_1, W_2)\), we have \[\begin{align} \label{eqn95W12} P_{\mathcal{H}}W_1W_1^*W_2W_2^*x=P_{\mathcal{H}}W_2W_1W_1^*W_2^*x=T_2T_1P_{\mathcal{H}}W_1^*W_2^*x =T_2T_1T_1^*T_2^*x. \end{align}\tag{4}\] The Brehmer’s positivity condition for the \(q\)-commuting pair \((T_1^*, T_2^*)\) reduces to \[\begin{align} \Delta_{\{1, 2\}}(T_1^*, T_2^*)&=I-T_1T_1^*-T_2T_2^*+T_2T_1T_1^*T_2^* =P_{\mathcal{H}}(I-W_1W_1^*)(I-W_2W_2^*)|_{\mathcal{H}}, \end{align}\] where the last equality follows from 4 and thus, \((T_1^*, T_2^*)\) satisfies the Brehmer’s positivity condition. It follows from Theorem 2 that \((T_1^*, T_2^*)\) admits a minimal regular \(q\)-unitary dilation. We have by Corollary 2.10 in [15] that \((T_1, T_2)\) admits a minimal \(*\)-regular \(q\)-isometric dilation. ◻

Proof. The minimality of \(\underline{U}\) and \(\underline{U}'\) gives that \[\mathcal{K}=\overline{\text{span}}\left\{U_1^mU_2^nx: x \in \mathcal{H}, m, n \in \mathbb{Z}_+ \right\} \quad \text{and} \quad \mathcal{K}'=\overline{\text{span}}\left\{U_1'^mU_2'^nx': x' \in \mathcal{H}', m, n \in \mathbb{Z}_+ \right\}.\] Furthermore, it follows from Theorem 3 that \(\underline{U}\) and \(\underline{U}'\) are doubly \(q\)-commuting. For \(\mathcal{K}_1=\overline{\text{span}}\left\{U_1^mx: x \in \mathcal{H}, m \in \mathbb{Z}_+\right\}\) and \(\mathcal{K}_1'=\overline{\text{span}}\left\{U_1'^m x': x' \in \mathcal{H}', m \in \mathbb{Z}_+\right\}\), we define \[(S_1, S_2)=(U_1|_{\mathcal{K}_1}, P_{\mathcal{K}_1}U_2|_{\mathcal{K}_1}) \quad \text{and} \quad (S_1', S_2')=(U_1'|_{\mathcal{K}'_1}, P_{\mathcal{K}'_1}U_2'|_{\mathcal{K}'_1}).\] Evidently, \(S_1\) and \(S_1'\) are the minimal isometric dilations of \(T_1\) and \(T_1'\) respectively. It follows from the proof of Theorem 4 that \((S_1, S_2)\) and \((S_1', S_2')\) are doubly \(q\)-commuting pair of contractions and the following holds: \[\begin{align} \label{eqn95203} (S_1^*|_{\mathcal{H}}, S_2^*|_{\mathcal{H}})=(T_1^*, T_2^*)=(U_1^*|_{\mathcal{H}}, U_2^*|_{\mathcal{H}}) \;\;\text{and} \;\; (S_1'^*|_{\mathcal{H}'}, S_2'^*|_{\mathcal{H}'})=(T_1'^*, T_2'^*)=(U_1'^*|_{\mathcal{H}'}, U_2'^*|_{\mathcal{H}'}), \end{align}\tag{5}\] where both the last equalities follow from Lemma 2.11 in [15]. From here onwards, the proof is divided into several steps for better understanding.

Step 1. In this step, we construct a contraction \(A_0\) between the minimal isometric dilation spaces of \(T_1\) and \(T_1'\), and show that \(A_0\) intertwines the corresponding isometric dilations. The minimal isometric dilations spaces can be written (see [17], Chapter II ) as follows: \[\mathcal{K}_1=\mathcal{H}\oplus \bigoplus_{p \in \mathbb{Z}_+}S_1^p \;\overline{(S_1-T_1)\mathcal{H}} \quad \text{and} \quad \mathcal{K}_1'=\mathcal{H}' \oplus \bigoplus_{p \in \mathbb{Z}_+}S_1'^p \;\overline{(S_1'-T_1')\mathcal{H}'}.\] Let \(D_{T_1}=I-T_1^*T_1\) and \(D_{T_1'}=I-T_1'^*T_1'\). By the intertwining property of \(A\), we have that \(D_{T_1'}^2A=AD_{T_1}^2\) and so, \(D_{T_1'}A=AD_{T_1}\). For \(x \in \mathcal{H}\) and \(p \geq 0\), it now follows that \[\begin{align} \|S_1^p(S_1-T_1)x\|^2=\|(S_1-T_1)x\|^2 &=\left\langle(S_1-T_1)x, (S_1-T_1)x \right\rangle\\ &=\|x\|^2-\left\langle x, S_1^*T_1x\right\rangle-\left\langle S_1^*T_1x, x\right\rangle+\|T_1x\|^2\quad [\text{since S_1 is an isometry}]\\ &= \|x\|^2-\left\langle x, T_1^*T_1x\right\rangle-\left\langle T_1^*T_1x, x\right\rangle+\|T_1x\|^2 \quad [\text{as S_1^*=T_1^*|_{\mathcal{H}}}]\\ &=\|x\|^2-\|T_1x\|^2\\ &=\|D_{T_1}x\|^2. \end{align}\] Similarly, one can show that \(\|S_1'^p(S_1'-T_1')Ax\|^2=\|(S_1'-T_1')Ax\|^2=\|D_{T_1'}Ax\|^2=\|AD_{T_1x}\|^2\), where the last equality holds since \(D_{T_1'}A=AD_{T_1}\). Consequently, we have \[\begin{align} \label{eqn95204} \|S_1'^p(S_1'-T_1')Ax\|^2=\|AD_{T_1x}\|^2 \leq \|D_{T_1}x\|^2=\|S_1^p(S_1-T_1)x\|^2. \end{align}\tag{6}\] Consider the operator \(A_0: \mathcal{K}_1 \to \mathcal{K}_1'\) given by \[A_0\left(x+ \sum_{p \in \mathbb{Z}_+}S_1^p(S_1-T_1)x_p\right)=Ax+ \sum_{p \in \mathbb{Z}_+}S_1'^p(S_1'-T_1')Ax_p.\] It follows from 6 that \(A_0\) is a well-defined contraction. Evidently, \(A_0|_{\mathcal{H}}=A\). Moreover, \(\|A_0\|=\|A\|\). To see this, let \(\{x, x_p : p \in \mathbb{Z}_+\} \subset \mathcal{H}\). Then we have by 6 that \[\begin{align} \left\|A_0\left(x+ \sum_{p \in \mathbb{Z}_+}S_1^p(S_1-T_1)x_p\right)\right\|^2 &=\left\|Ax\right\|^2+\sum_{p \in \mathbb{Z}_+}\left\|S_1'^p(S_1'-T_1')Ax_p\right\|^2\\ &=\left\|Ax\right\|^2+\sum_{p \in \mathbb{Z}_+}\left\|AD_{T_1}x_p\right\|^2\\ & \leq \|A\| \left(\|x\|^2+\sum_{p \in \mathbb{Z}_+}\left\|D_{T_1}x_p\right\|^2\right)\\ &=\|A\| \left\|x+ \sum_{p \in \mathbb{Z}_+}S_1^p(S_1-T_1)x_p\right\|^2. \end{align}\] We show that \(A_0\) intertwines \(\underline{S}\) and \(\underline{S}'\), and \(A_0S_1^*=S_1'^*A_0\). For \(p \in \mathbb{Z}_+\) and \(x, x_p \in \mathcal{H}\), note that \[\begin{align} A_0S_1\left(x+ S_1^p(S_1-T_1)x_p\right) &=A_0\left(T_1x+ (S_1-T_1)x+ S_1^{p+1}(S_1-T_1)x_p\right)\\ &=AT_1x+(S_1'-T_1')Ax+S_1'^{p+1}(S_1'-T_1')Ax_p\\ &=T_1'Ax+(S_1'-T_1')Ax+S_1'^{p+1}(S_1'-T_1')Ax_p \quad [\text{as AT_1=T_1'A}]\\ &=S_1'Ax+S_1'^{p+1}(S_1'-T_1')Ax_p\\ &=S_1'A_0\left(x+ S_1^p(S_1-T_1)x_p\right). \end{align}\] Let \(m \in \mathbb{N}\) and \(x, x_0, x_m \in \mathcal{H}\). Since \(A_0|_\mathcal{H}=A, S_1^*|_\mathcal{H}=T_1^*\) and \(S_1\) is an isometry, we have \[\begin{align} A_0S_1^*\left(x+ (S_1-T_1)x_0+S_1^m(S_1-T_1)x_m\right) &=A_0T_1^*x+A_0x_0-A_0T_1^*T_1x_0+A_0S_1^{m-1}(S_1-T_1)x_m\\ &=AT_1^*x+Ax_0-AT_1^*T_1x_0+A_0S_1^{m-1}(S_1-T_1)x_m\\ &=T_1'^*Ax+Ax_0-T_1'^*T_1'Ax_0+S_1'^{(m-1)}(S_1'-T_1')Ax_m\\ & \qquad [\text{sinceAT_1^*=T_1'^*A, \;AT_1=T_1'A, \;A_0S_1=S_1'A_0}]\\ &=S_1'^*Ax+S_1'^*(S_1'-T_1')Ax_0+S_1'^*S_1'^m(S_1'-T_1')Ax_m\\ &=S_1'^*A_0\left(x+ (S_1-T_1)x_0+S_1^m(S_1-T_1)x_m\right). \end{align}\] Next, we show that \(A_0^*|_{\mathcal{H}'}=A^*\). For \(x' \in \mathcal{H}'\) and \(x, x_p \in \mathcal{H}\) with \(p \in \mathbb{Z}_+\), we have \[\begin{align} \left\langle A_0^*x', x+ S_1^p(S_1-T_1)x_p \right\rangle &= \left\langle x', A_0 x\right\rangle+ \left\langle x', A_0S_1^p(S_1-T_1)x_p\right\rangle\\ &=\left\langle x', Ax\right\rangle+ \left\langle x', S_1'^p(S_1'-T_1')Ax_p\right\rangle\quad [\text{as A_0|_{\mathcal{H}}=A}]\\ &=\left\langle x', Ax\right\rangle+ \left\langle A^*(S_1'^*-T_1'^*)S_1'^{*p}x',x_p\right\rangle\\ &=\left\langle x', Ax\right\rangle+ \left\langle(S_1^*-T_1^*)S_1^{*p}A^*x',x_p\right\rangle \quad [\text{since S_1'^*|_{\mathcal{H}'}=T_1'^* and AT_1=T_1'A}]\\ &=\left\langle A^*x', x+ S_1^p(S_1-T_1)x_p \right\rangle. \end{align}\] We prove that \(A_0S_2=S_2'A_0\). Let \(x', x_p' \in \mathcal{H}'\) and \(p \in \mathbb{Z}_+\). Since \((A_0^*|_{\mathcal{H}'}, S_2'^*|_{\mathcal{H}'})=(A^*, T_2'^*)\), it follows that \(A_0^*S_2'^*x'=A^*T_2'^*x'=T_2^*A^*x'=S_2^*A_0^*x'.\) Since \((S_1', S_2')\) is doubly \(q\)-commuting, we have \[\begin{align} A_0^*S_2'^*\left(S_1'^pS_1'x_p'\right) &=\overline{q}_{21}^{p+1}A_0^*S_1'^{(p+1)}S_2'^*x_p'\\ &=\overline{q}_{21}^{p+1}A_0^*S_1'^{(p+1)}T_2'^*x_p'\\ &=\overline{q}_{21}^{p+1}S_1^{(p+1)}S_2^*A^*x_p' \quad [\text{as S_1'^*A_0=A_0S_1^*, A_0^*|_{\mathcal{H}'}=A^* and T_2'A=AT_2}]\\ &=S_2^*S_1^{(p+1)}A_0^*x_p' \quad [\text{since A_0|_{\mathcal{H}'}=A and (S_1', S_2') is a doubly q-commuting pair}]\\ &=S_2^*A_0^*S_1'^{(p+1)}x_p',\quad \text{and} \\ A_0^*S_2'^*\left(S_1'^pT_1'x_p'\right) &=\overline{q}_{21}^{p}A_0^*S_1'^{p}S_2'^*T_1'x_p'\\ &=\overline{q}_{21}^{p}S_1^{p}A_0^*T_2'^*T_1'x_p' \quad [\text{since S_2'^*|_{\mathcal{H}'}=T_2'^* and A_0S_1^*=S_1'^*A_0}]\\ &=\overline{q}_{21}^{p}S_1^{p}T_2^*A^*T_1'x_p' \quad [\text{as A_0^*|_{\mathcal{H}'}=A^* and AT_2=T_2'A}]\\ &=\overline{q}_{21}^{p}S_1^{p}S_2^*A^*T_1'x_p' \\ &=S_2^*S_1^{p}A_0^*T_1'x_p' \quad [\text{as A_0^*|_{\mathcal{H}'}=A^* and (S_1', S_2') is a doubly q-commuting pair}]\\ &=S_2^*A_0^*S_1'^{p}T_1'x_p'. \end{align}\] Consequently, \(S_2^*A_0^*=A_0^*S_2'^*\) and so, \(A_0S_2=S_2'A_0\). Putting everything together, we have that \[\begin{align} \label{eqn95step2} A_0S_1=S_1'A_0, \;\;A_0S_1^*=S_1'^*A_0, \;\; A_0S_2=S_2'A_0, \;\;(A_0|_{\mathcal{H}}, A_0^*|_{\mathcal{H}'})=(A, A^*) \;\;\text{and} \;\;\|A_0\|=\|A\|. \end{align}\tag{7}\]

Step 2. In this step, we prove the existence of an operator \(B : \mathcal{K}\to \mathcal{K}'\) with \(\|B\|=\|A\|\) such that \(B\) intertwines \(U_2\) and \(U_2'\), and \(A^*=B^*|_{\mathcal{H}'}\). To begin with, it follows from 7 that \(AP_{\mathcal{H}}=P_{\mathcal{H}'}A_0\). For \(x \in \mathcal{H}\) and \(m \in \mathbb{Z}_+\), we have by 5 that \[\begin{align} S_2^*U_1^mx=P_{\mathcal{K}_1}U_2^*U_1^mx =\overline{q}_{21}^mP_{\mathcal{K}_1}U_1^mU_2^*x =\overline{q}_{21}^mP_{\mathcal{K}_1}U_1^mT_2^*x =\overline{q}_{21}^mU_1^mT_2^*x =\overline{q}_{21}^mU_1^mU_2^*x =U_2^*U_1^mx \end{align}\] and so, \(S_2^*=U_2^*|_{\mathcal{K}_1}\). It follows from the definitions of \(\mathcal{K}_1\) and \(\mathcal{K}\) that \[\mathcal{K}=\overline{\text{span}}\left\{U_1^mU_2^nx: x \in \mathcal{H}, m, n \in \mathbb{Z}_+ \right\}=\overline{\text{span}}\left\{U_2^nk_1: k_1 \in \mathcal{K}_1, n \in \mathbb{Z}_+\right\}.\] Therefore, \(U_2\) on \(\mathcal{K}\) is the minimal isometric dilation of \(S_2\), and similarly one can show that \(U_2'\) on \(\mathcal{K}'\) is the minimal isometric dilation of \(S_2'\). For \(x, x_m \in \mathcal{H}\) with \(m \in \mathbb{N}\), it follows from 5 that \[\begin{align} S_1^*\left(x+U_1^mx_m\right) =P_{\mathcal{K}_1}U_1^*\left(x+U_1^mx_m\right) =P_{\mathcal{K}_1}T_1^*x+P_{\mathcal{K}_1}U_1^{m-1}x_m =T_1^*x+U_1^{m-1}x_m =U_1^*(x+U_1^mx_m) \end{align}\] and so, \(S_1^*=U_1^*|_{\mathcal{K}_1}\). Similarly, it follows that \(S_1'^*=U_1'^*|_{\mathcal{K}_1'}\). Since \(U_2'\) on \(\mathcal{K}'\) is the minimal isometric dilation of \(S_2'\), we can re-write the structure of \(\mathcal{K}'\) as follows: \[\mathcal{K}'=\mathcal{K}_1' \oplus \bigoplus_{n \in \mathbb{Z}_+}U_2'^n\overline{(U_2'-S_2')\mathcal{K}'_1}.\] Putting everything together, \(S_2\) on \(\mathcal{K}_1\) and \(S_2'\) on \(\mathcal{K}_1'\) are contractions, and \(U_2\) on \(\mathcal{K}\) and \(U_2'\) on \(\mathcal{K}'\) are their minimal isometric dilations respectively. Also, the operator \(A_0: \mathcal{K}_1 \to \mathcal{K}_1'\) intertwines \(S_2\) and \(S_2'\). It follows from Theorem 2.3 of Chapter II in [17] that there exists an operator \(B : \mathcal{K}\to \mathcal{K}'\) such that \(\|B\|=\|A_0\|, BU_2=U_2'B\) and \(A_0^*=B^*|_{\mathcal{K}_1'}\). By 7 , \(\|B\|=\|A\|\) and \(A^*=B^*|_{\mathcal{H}'}\).

Step 3. We now explicitly recall the construction of the operator \(B\) from Section 2 of Chapter II of [17], with further related discussion in Section 3 of [7]. The operator \(B:\mathcal{K}\to\mathcal{K}'\) is defined as \[Bk=A_0P_{\mathcal{K}_1}k+\overset{\infty}{\underset{n=0}{\sum}}U_2'^nB_nk \quad \text{(k \in \mathcal{K})}.\] Here, the contractions \(B_n: \mathcal{K}\to \overline{(U_2'-S_2')\mathcal{K}_1'}\) are inductively defined as \(B_n=D_nC_n\) with \[\begin{align} & C_n: \mathcal{K}\to \mathcal{K}, \quad C_0=(I_\mathcal{K}-A_0^*A_0P_{\mathcal{K}_1})^{1\slash 2} \quad \text{and} \quad C_n=(C_0^2-B_0^*B_0-\dotsc-B_{n-1}^*B_{n-1})^{1\slash 2}, \quad \text{and} \\ & D_n: \mathcal{K}\to \overline{(U_2'-S_2')\mathcal{K}_1'}, \quad D_n(C_nU_2k)=B_{n-1}k \;\;(k \in \mathcal{K}) \quad \text{and} \quad D_n=0 \;\text{on} \;\mathcal{K}\ominus \overline{C_nU_2\mathcal{K}}, \end{align}\] where \(B_{-1}=(U_2'-S_2')A_0P_{\mathcal{K}_1}\). It is not difficult to see that \(B(\mathcal{K}\ominus \mathcal{H}) \subseteq \mathcal{K}' \ominus \mathcal{H}', P_{\mathcal{H}'}B=AP_{\mathcal{H}}\) and \(\|B\|=\|A\|\). Since \(B_nU_2=B_{n-1}\) for \(n \in \mathbb{N}\), it follows that \(BU_2=U_2'B\).

Step 4. Finally, we establish that \(B\) is the desired operator and it only remains to show that \(BU_1=U_1'B\) and \(BU_1^*=U_1'^*B\). To do so, we consider the intertwining relations of \(B\) with \(U_n\) and \(U_n'\), and with \(U_n^*\) and \(U_n'^*\) for each \(n\). Since \(S_1=U_1|_{\mathcal{K}_1}, S_1^*=U_1^*|_{\mathcal{K}_1}, A_0S_1=S_1'A_0\) and \(A_0S_1^*=S_1'^*A_0\), it follows that \(C_0U_1=U_1C_0\) and \(C_0U_1^*=U_1^*C_0\). Using \(S_1'=U_1'|_{\mathcal{K}_1'}\) and \(S_1'^*=U_1'^*|_{\mathcal{K}_1'}\), we have \[\begin{align} U_1'(U_2'-S_2')k_1' &=U_1'U_2'k_1'-U_1'S_2'k_1'=q_{12}U_2'U_1'k_1'-S_1'S_2'k_1'=q_{12}(U_2'-S_2')S_1'k_1'\quad \text{and} \\ U_1'^*(U_2'-S_2')k_1' &=U_1'^*U_2'k_1'-U_1'^*S_2'k_1'=\overline{q}_{12}U_2'U_1'^*k_1'-S_1'^*S_2'k_1'=\overline{q}_{12}(U_2'-S_2')S_1'^*k_1' \end{align}\] for all \(k_1' \in \mathcal{K}'_1\). Consequently, \(U_1'\overline{(U_2'-S_2')\mathcal{K}'_1} \subseteq \overline{(U_2'-S_2')\mathcal{K}'_1}, \;U_1'^*\overline{(U_2'-S_2')\mathcal{K}'_1} \subseteq \overline{(U_2'-S_2')\mathcal{K}'_1}\) and so, the subspace \(\overline{(U_2'-S_2')\mathcal{K}'_1}\) is a reducing subspace for \(U_1'\). Then \[\begin{align} B_{-1}U_1 =(U_2'-S_2')A_0P_{\mathcal{K}_1}U_1 &=(U_2'-S_2')A_0S_1P_{\mathcal{K}_1} \quad [\text{since S_1^*=U_1^*|_{\mathcal{K}_1}}]\\ &=(U_2'-S_2')S_1'A_0P_{\mathcal{K}_1} \quad [\text{as A_0S_1=S_1'A_0}]\\ &=U_2'U_1'A_0P_{\mathcal{K}_1}-S_2'S_1'A_0P_{\mathcal{K}_1} \quad [\text{because S_1'=U_1'|_{\mathcal{K}_1'}}]\\ &=q_{21}(U_1'U_2'-S_1'S_2')A_0P_{\mathcal{K}_1}\\ &=q_{21}U_1'(U_2'-S_2')A_0P_{\mathcal{K}_1} \quad [\text{as S_1'=U_1'|_{\mathcal{K}_1'}}]\\ &=q_{21}U_1'B_{-1} \end{align}\] and \[\begin{align} B_{-1}U_1^* =(U_2'-S_2')A_0P_{\mathcal{K}_1}U_1^* &=(U_2'-S_2')A_0S_1^*P_{\mathcal{K}_1} \quad [\text{since S_1=U_1|_{\mathcal{K}_1}}]\\ &=(U_2'-S_2')S_1'^*A_0P_{\mathcal{K}_1} \quad [\text{as A_0S_1^*=S_1'^*A_0}]\\ &=U_2'U_1'^*A_0P_{\mathcal{K}_1}-S_2'S_1'^*A_0P_{\mathcal{K}_1} \quad [\text{because S_1'^*=U_1'^*|_{\mathcal{K}_1'}}]\\ &=\overline{q}_{21}(U_1'^*U_2'-S_1'^*S_2')A_0P_{\mathcal{K}_1}\\ &=\overline{q}_{21}U_1'^*(U_2'-S_2')A_0P_{\mathcal{K}_1} \quad [\text{as S_1'^*=U_1'^*|_{\mathcal{K}_1'}}]\\ &=\overline{q}_{21}U_1'^*B_{-1}. \end{align}\]

Next, using \(U_1U_2=q_{12}U_2U_1, U_1U_2^*=\overline{q}_{12}U_2^*U_1, C_0U_1=U_1C_0\) and \(C_0U_1^*=U_1^*C_0\), we have \[\begin{align} U_1\overline{C_0U_2\mathcal{K}} &\subseteq \overline{U_1C_0U_2\mathcal{K}}=\overline{C_0U_1U_2\mathcal{K}} \subseteq \overline{C_0U_2U_1\mathcal{K}} \subseteq \overline{C_0U_2\mathcal{K}} \quad \text{and} \\ U_1^*\overline{C_0U_2\mathcal{K}} &\subseteq \overline{U_1^*C_0U_2\mathcal{K}}=\overline{C_0U_1^*U_2\mathcal{K}} \subseteq \overline{C_0U_2U_1^*\mathcal{K}} \subseteq \overline{C_0U_2\mathcal{K}}. \end{align}\] Therefore, \(\overline{C_0U_2\mathcal{K}}\) is a reducing subspace for \(U_1\) and thus, \(U_1\) doubly commutes with \(P_0\), where \(P_0\) is the orthogonal projection of \(\mathcal{K}\) onto \(\overline{C_0U_2\mathcal{K}}\). Then \[D_0U_1(I-P_0)=D_0(I-P_0)U_1=0=U_1'D_0(I-P_0) \; \;\text{and similarly, } \;\;D_0U_1^*(I-P_0)=0=U_1'^*D_0(I-P_0) .\] So, \(D_0U_1=U_1'D_0\) and \(D_0U_1^*=U_1'^*D_0\) on \(\mathcal{K}\ominus \overline{C_0U_2\mathcal{K}}\). For \(k \in \mathcal{K}\), it follows that \[\begin{align} D_0U_1(C_0U_2k) &=D_0C_0U_1U_2k \quad [\text{as C_0U_1=U_1C_0}]\\ &=q_{12}D_0C_0U_2U_1k\\ &=q_{12}B_{-1}U_1k\\ &=U_1'B_{-1}k \quad [\text{since B_{-1}U_1=q_{21}U_1'B_{-1}}]\\ &=U_1'D_0C_0U_2k \end{align}\] and \[\begin{align} D_0U_1^*(C_0U_2k) &=D_0C_0U_1^*U_2k \quad [\text{as C_0U_1^*=U_1^*C_0}]\\ &=\overline{q}_{12}D_0C_0U_2U_1^*k\\ &=\overline{q}_{12}B_{-1}U_1^*k\\ &=U_1'^*B_{-1}k \quad [\text{since B_{-1}U_1^*=\overline{q}_{21}U_1'^*B_{-1}}]\\ &=U_1'^*D_0C_0U_2k. \end{align}\] Thus, \(D_0U_1=U_1'D_0\) and \(D_0U_1^*=U_1'^*D_0\) on \(\mathcal{K}\). It now follows that \[B_0U_1=D_0C_0U_1=D_0U_1C_0=U_1'D_0C_0=U_1'B_0 \quad \text{and} \quad B_0U_1^*=D_0C_0U_1^*=D_0U_1^*C_0=U_1'^*B_0.\] Note that \(B_{-1}U_1=q_{12}^{-1}U_1'B_{-1}, B_{-1}U_1^*=\overline{q}_{12}^{-1}U_1'^*B_{-1}, B_0U_1=U_1'B_0\) and \(B_0U_1^*=U_1'^*B_0\). Following the similar techniques as above, one can easily show that \[\begin{align} \label{eqn95205} B_nU_1=(q_{12})^nU_1'B_n \quad \text{and} \quad B_nU_1^*=(\overline{q}_{12})^nU_1'^*B_n \end{align}\tag{8}\] for \(n \geq -1\). Since \(P_{\mathcal{K}_1}U_1=S_1P_{\mathcal{K}_1}\) and \(P_{\mathcal{K}_1}U_1^*=S_1^*P_{\mathcal{K}_1}\), we have that \[\begin{align} BU_1 &=A_0P_{\mathcal{K}_1}U_1+\overset{\infty}{\underset{n=0}{\sum}}U_2'^nB_nU_1\\ &=A_0S_1P_{\mathcal{K}_1}+\overset{\infty}{\underset{n=0}{\sum}}(q_{12})^nU_2'^nU_1'B_n \quad [\text{by \eqref{eqn95205}}]\\ &=S_1'A_0P_{\mathcal{K}_1}+\overset{\infty}{\underset{n=0}{\sum}}(q_{12})^n (q_{21})^nU_1'U_2'^nB_n \quad [\text{since (U_1', U_2') is doubly q-commuting}]\\ &=U_1'A_0P_{\mathcal{K}_1}+\overset{\infty}{\underset{n=0}{\sum}}U_1'U_2'^nB_n \quad [\text{as S_1'=U_1'|_{\mathcal{K}_1'}}]\\ &=U_1'B \end{align}\] and similarly, it follows that \(BU_1^*=U_1'^*B\). The proof is now complete. ◻

Proof. It is not difficult to see that \((V_1, V_2, V_3)\) satisfies the Brehmer’s positivity condition. The desired conclusion now follows directly from Theorem 2. ◻

Proof. Let \(S_1\) on a Hilbert space \(\mathcal{K}_1\) be the minimal isometric dilation of \(T_1\). By minimality, \(\mathcal{K}_1=\overline{\text{span}}\left\{S_1^mx: x \in \mathcal{H}, m \in \mathbb{Z}_+\right\}\). By Theorem 4, there exist contractions \(S_2, S_3\) on \(\mathcal{K}_1\) such that \((S_1, S_2)\) is doubly \(q_{12}\)-commuting and \((S_1, S_3)\) is doubly \(q_{13}\)-commuting, and that \((T_1^*, T_2^*, T_3^*)=(S_1^*|_{\mathcal{H}}, S_2^*|_{\mathcal{H}}, S_3^*|_{\mathcal{H}})\). Then \(S_2^*S_3^*S_1^mx=\overline{q}_{21}^m\overline{q}_{31}^mS_1^mS_2^*S_3^*x=\overline{q}_{21}^m\overline{q}_{31}^mS_1^mT_2^*T_3^*x=q_{23}\overline{q}_{21}^m\overline{q}_{31}^mS_1^mT_3^*T_2^*x=q_{23}S_3^*S_2^*S_1^mx\) for all \(m \in \mathbb{Z}_+\) and \(x \in \mathcal{H}\). By continuity arguments, \(S_2S_3=q_{23}S_3S_2\) on \(\mathcal{K}_1\). Since \((S_1, S_2)\) is a doubly \(q_{12}\)-commuting pair of contractions on \(\mathcal{K}_1\), we have by Theorem 3 that it admits a minimal \(*\)-regular \(q_{12}\)-isometric dilation \((U_1, U_2)\) on a Hilbert space \(\mathcal{K}_2\), which is doubly \(q_{12}\)-commuting. By minimality, \((U_1^*|_{\mathcal{H}}, U_2^*|_{\mathcal{H}})=(T_1^*, T_2^*)\). Let us define \[A:=S_3, \quad (T_1', T_2'):=(q_{31}S_1, q_{32}S_2) \;\text{on} \;\mathcal{K}_1 \quad \text{and} \quad (U_1', U_2'):=(q_{31}U_1, q_{32}U_2) \;\text{on} \;\mathcal{K}_2.\] Evidently, \((T_1', T_2')\) is a \(q_{12}\)-commuting pair of contractions and \((U_1', U_2')\) is a doubly \(q_{12}\)-commuting pair of isometries, which is a minimal \(*\)-regular \(q_{12}\)-isometric dilation of \((T_1', T_2')\). Note that \(A\) is a contraction that satisfies \(AS_1=T_1'A, AS_2=T_2'A\) and \(AS_1^*=T_1'^*A\). By Theorem 6, there exists a contraction \(U_3: \mathcal{K}_2 \to \mathcal{K}_2\) such that \(U_3U_1=U_1'U_3, U_3U_2=U_2'U_3, U_3U_1^*=U_1'^*U_3\) and \(U_3^*|_{\mathcal{K}_1}=S_3^*\). Consequently, \(U_3U_1=q_{31}U_1U_3, \;U_3U_2=q_{32}U_2U_3, U_3U_1^*=\overline{q}_{31}U_1^*U_3\) and \(U_3^*|_{\mathcal{H}}=S_3^*|_{\mathcal{H}}=T_3^*\). Hence, the \(q\)-commuting triple \(\underline{U}\) satisfies the hypothesis of Lemma 7 and so, \((U_1, U_2, U_3)\) admits a minimal \(q\)-isometric dilation. Then \((U_1^*, U_2^*, U_3^*)\) has a \(q\)-unitary dilation and so, \((T_1^*, T_2^*, T_3^*)=(U_1^*|_{\mathcal{H}}, U_2^*|_{\mathcal{H}}, U_3^*|_{\mathcal{H}})\) admits a \(q\)-unitary dilation. Thus, \((T_1, T_2, T_3)\) has a \(q\)-isometric dilation. ◻

Proof. It is easy to see that the pairs \((T_1^*, T_2^*)\) and \((T_1^*, T_3^*)\) admit minimal \(*\)-regular \(q_{12}\)-isometric and \(q_{13}\)-isometric dilations respectively (see Lemma 2.9 and Corollary 2.10 in [15]). By Theorem 8, \((T_1^*, T_2^*, T_3^*)\) has a \(q\)-isometric dilation. Since a \(q\)-commuting family of isometries admits a dilation to a \(q\)-commuting family of unitaries (see [13]), the desired conclusion follows. ◻

Proof. By Theorem 3, the pairs \((T_1, T_2)\) and \((T_1, T_3)\) have minimal \(*\)-regular \(q_{12}\)-isometric and \(q_{13}\)-isometric dilations respectively. The desired conclusion now follows from Theorem 8. ◻