Let
\(\pi _1\) and
\(\pi _2\) be two projective representation of Möb on the Hilbert spaces
\(H_1\) and
\(H_2,\) respectively, such that one of the following holds:
 (I)

\(\pi _1\) and \(\pi _2\) are from the irreducible Continuous series representation.
 (II)

\(\pi _{1}\) is from the Continuous series representations and \(\pi _{2} = D_{\lambda + 2m}^{+} \oplus D_{2  \lambda + 2k}^{},\) \(1 < \lambda \le 1;\) m, k are integers.
 (III)

\(\pi _{1} = D_{\lambda + 2a}^{+} \oplus D_{2\lambda + 2b}^{}\) and \(\pi _{2} = D_{\lambda + 2m}^{+} \oplus D_{2\lambda + 2p}^{}\) where \(\lambda \in (0, 2]\) and a, b, m, p are any nonnegative integers.
Suppose
\(T_1\) and
\(T_2\) are bounded operators on
\(H_1\) and
\(H_2\), respectively and
\(S_1 : H_2 \rightarrow H_1\) and
\(S_2 : H_1 \rightarrow H_2\) be operators which satisfies the following relations
$$\begin{aligned} S_{1}\pi _{2}(\phi )  e^{i \theta } \pi _{1}(\phi ) S_{1} = {\overline{a}}T_{1}\pi _{1}(\phi )S_{1} + {\overline{a}} S_{1} \pi _{2}(\phi ) T_{2},\,\,\phi \in \hbox {M}\ddot{\mathrm{o}}\hbox {b} \end{aligned}$$
(A.1)
and
$$\begin{aligned} S_{2}\pi _{1}(\phi )  e^{i \theta } \pi _{2}(\phi ) S_{2} = {\overline{a}}S_{2}\pi _{1}(\phi )T_{1} + {\overline{a}} T_{2} \pi _{2}(\phi ) S_{2},\,\,\phi \in \hbox {M}\ddot{\mathrm{o}}\hbox {b}. \end{aligned}$$
(A.2)
(I). We know that
\(\{z^{n} : n \in {\mathbb {Z}}\}\) is an orthogonal basis of
\(H_{i}\). Let
\(e_n^i = \frac{z^n}{\Vert z^n\Vert _i},\,\,i = 1, 2,\) where
\(\Vert \cdot \Vert _i\) denote the inner product of
\(H_i.\) The set of vectors
\(\{e_n^i : n \in {\mathbb {Z}}\}\) is an orthonormal basis of
\(H_i.\) Let
\(\phi _{\theta } \in \) Möb be such that
\(\phi _{\theta }(z) = e^{i \theta }z\). Evaluating Eq. (
A.1) on
\(z^n\) and putting
\(\phi =\phi _{\theta },\) we obtain
$$\begin{aligned} \pi _{1}(\phi _{\theta }) S_{1}z^{n} = e^{i\left( n+1+\frac{\lambda _{2}}{2}\right) \theta } S_{1} z^{n},\,\,n \in {\mathbb {Z}}. \end{aligned}$$
(A.3)
Thus the existence of a sequence
\(\{\alpha _{n} : n \in {\mathbb {Z}} \}\) such
\(S_{1} e_{n}^{2} = \alpha _{n} e_{n+1}^{1}\) follows. Suppose
\(T_i\)’s are weighted shift with respect to the orthonormal basis
\(\{e_n^i : n \in {\mathbb {Z}}\}\). Then evaluating Eq. (
A.1) on the vector
\(e_{m}^{2},\) putting
\(\phi =\phi _{a},\) taking inner product with
\(e_n^1\) and finally using the matrix coefficient of
\(\pi _i(\phi _a)\) (see [
3, p. 316]), we obtain
$$\begin{aligned}&\alpha _{n1} \frac{\Vert z^{n1}\Vert _{2}}{\Vert z^{m}\Vert _{2}} \phi '_{a}(0)^{\mu _{2}} \displaystyle \sum _{k \ge (mn+1)^{+}} C_{k}^{2}(m, n1)r^{k} \nonumber \\&\qquad \, \alpha _{m} \frac{\Vert z^{n}\Vert _{1}}{\Vert z^{m+1}\Vert _{1}} \phi '_{a}(0)^{\mu _{1}} \displaystyle \sum _{k \ge (mn+1)^{+}} C_{k}^{1}(m+1, n)r^{k} \nonumber \\&\quad = v_{m} \alpha _{n1} \frac{\Vert z^{n1}\Vert _{2}}{\Vert z^{m+1}\Vert _{2}} \phi '_{a}(0)^{\mu _{2}} \displaystyle \sum _{k \ge (mn+2)^{+}} C_{k}^{2}(m+1, n1)r^{k} \nonumber \\&\qquad +\, \alpha _{m} u_{n1} \frac{\Vert z^{n1}\Vert _{1}}{\Vert z^{m+1}\Vert _{1}} \phi '_{a}(0)^{\mu _{1}} \displaystyle \sum _{k \ge (mn+2)^{+}} C_{k}^{1}(m+1, n1)r^{k}, \end{aligned}$$
(A.4)
where
\(C_{k}^i(m, n) = \left( \begin{array}{cc} \lambda  \mu _i  m\\ k + n m \end{array}\right) \left( \begin{array}{cc}  \mu _i + m\\ k \end{array}\right) ,\) \(i = 1, 2\) and
\(u_n, v_n\) are weights of
\(T_1, T_2,\) respectively. Similar conclusions are true for
\(S_2\) as well. When
\(\pi _1 = \pi _2,\) we denote
\(C_{k}^i\) by
\(C_k.\)(II). Let
\(H^{(\lambda + 2m)}\) be the representation space of
\(D_{\lambda + 2m}^{+}\) and
\(H^{(2\lambda + 2k)}\) be the representation space of
\(D_{2\lambda + 2k}^{}\). Let
\(H_{2} = H^{(\lambda + 2m)} \oplus H^{(2\lambda + 2k)}\). Define
$$\begin{aligned} e_{n}^{2} := \left( \begin{array}{c} \frac{z^{n}}{\Vert z^{n}\Vert _{\lambda + 2m}}\\ 0 \end{array}\right) ,\,\, n \ge 0\,\,\text{ and }\,\,e_{n}^{2} := \left( \begin{array}{c} 0\\ \frac{z^{n1}}{\Vert z^{n1}\Vert _{2\lambda + 2k}} \end{array}\right) ,\,\,n \ge 1. \end{aligned}$$
(A.5)
The set of vectors
\(\{e_{n}^{2} : n \in {\mathbb {Z}} \}\) is an orthonormal basis of
\(H_{2}\). Let
\(\phi _{\theta }\) be a rotation in Möb. Then
$$\begin{aligned} \pi _{2}(\phi _{\theta }) e_{n}^{2} = e^{i \left( n + m + \frac{\lambda }{2} \right) \theta }e_{n}^{2},\,n \ge 0\,\,\text{ and }\,\,\pi _{2}(\phi _{\theta }) e_{n}^{2} = e^{i \left( n + k  \frac{\lambda }{2} \right) \theta }e_{n}^{2},\,n \ge 1. \end{aligned}$$
Substituting
\(\phi =\phi _{\theta }\) in the Eqs. (
A.1) and (
A.2), respectively, we obtain
$$\begin{aligned} \pi _{1}(\phi _{\theta })Se_{n}^{2}\! =\! e^{i \left( n + 1 + m + \frac{\lambda }{2} \right) \theta } Se_{n}^{2},\,\, n\! \ge \! 0;\,\, \pi _{1}(\phi _{\theta })Se_{n}^{2} \!=\! e^{i \left( n  1 + k  \frac{\lambda }{2} \right) \theta } Se_{n}^{2},\,\, n\!\ge 1\nonumber \\ \end{aligned}$$
(A.6)
and
$$\begin{aligned} \pi _{2}(\phi _{\theta }) S_{2} e_{n}^{1} = e^{i \left( n + 1 + \frac{\lambda }{2} \right) \theta } S_{2}e_{n}^{1},\,\,n \in {\mathbb {Z}}. \end{aligned}$$
(A.7)
(III). Substituting
\(\phi =\phi _{\theta }\) in Eqs. (
A.1) and (
A.2), respectively, we obtain
$$\begin{aligned}&\pi _{1}(\phi _{\theta })S_{1}e_{n}^{2} = e^{i \left( n + 1 + m + \frac{\lambda }{2} \right) \theta } S_{1} e_{n}^{2},\quad n \ge 0;\,\,\, \nonumber \\&\quad \pi _{1}(\phi _{\theta })S_{1}e_{n}^{2} = e^{i \left( n  1 + p  \frac{\lambda }{2} \right) \theta } S_{1} e_{n}^{2}, \quad n \ge 1 \end{aligned}$$
(A.8)
and
$$\begin{aligned}&\pi _{2}(\phi _{\theta }) S_{2} e_{n}^{1} = e^{i \left( n + 1 + a + \frac{\lambda }{2} \right) \theta } S_{2} e_{n}^{1}, \quad n \ge 0;\,\,\,\nonumber \\&\quad \pi _{2}(\phi _{\theta }) S_{2} e_{n}^{1} = e^{i \left( n  1 + b  \frac{\lambda }{2} \right) \theta } S_{2} e_{n}^{1}, \quad n \ge 1. \end{aligned}$$
(A.9)
where
\(e_n^1\) and
\(e_n^2\) are defined in a similar way as in (
A.5).
The following two algorithms have been used in Sects. 5 and 6 repeatedly: