First, we assume that
\(\{\Lambda_{j}\}_{j\in J}\) and
\(\{ \Gamma_{j}\}_{j\in J}\) are approximately dual g-frames. By Lemmas
3.3 and
3.4 we have
$$ \Gamma_{j}^{\ast}e_{j,k}= \bigl(S_{\Lambda}^{-1}T_{\Lambda}+W\bigl(I-T_{\Lambda}^{\ast}QS_{\Lambda}^{-1}T_{\Lambda}\bigr)\bigr)\tilde {e}_{j,k}, $$
(3.2)
where
I is the identity operator on
\(\bigoplus_{j\in J}V_{j}\), and
\(W: \bigoplus_{j\in J}V_{j}\rightarrow H\) and
\(Q: H\rightarrow H\) are linear bounded operators satisfying
\(\|WT_{\Lambda}^{\ast}(I_{U}-Q)\|<1\). Set
\(z_{j,k}=W\tilde{e}_{j,k}\). We know that
\(\{z_{j,k}\}_{j\in J, k\in K_{j}}\) is a Bessel sequence for
H. Using the notations
\(u_{j,k}:=\Lambda_{j}^{\ast}e_{j,k}\) and
\(v_{j,k}:=\Gamma_{j}^{\ast }e_{j,k}\), we have
$$\bigl\{ \bigl\langle QS_{\Lambda}^{-1}u_{j,k}, u_{j',k'}\bigr\rangle \bigr\} _{j'\in J,k'\in K_{j}}\in l^{2} $$
for any
\(j\in J\) and
\(k\in K_{j}\). So
\(\sum_{j'\in J}\sum_{k'\in K_{j}}\langle QS_{\Lambda}^{-1}u_{j,k}, u_{j',k'}\rangle z_{j',k'}\) converges unconditionally. By (
3.2) we have
$$\begin{aligned} v_{j,k}&=S_{\Lambda}^{-1}T_{\Lambda}\tilde{e}_{j,k}+W\tilde {e}_{j,k}-WT_{\Lambda}^{\ast}QS_{\Lambda}^{-1}T_{\Lambda}\tilde{e}_{j,k} \\ &=S_{\Lambda}^{-1}u_{j,k}+z_{j,k}-WT_{\Lambda}^{\ast}QS_{\Lambda}^{-1}u_{j,k} \\ &=S_{\Lambda}^{-1}u_{j,k}+z_{j,k}-W \biggl(\sum _{j'\in J}\sum_{k'\in K_{j}}\bigl\langle \Lambda_{j'}QS_{\Lambda}^{-1}u_{j,k}, e_{j',k'}\bigr\rangle \tilde{e}_{j',k'} \biggr) \\ &= S_{\Lambda}^{-1}u_{j,k}+z_{j,k}-\sum _{j'\in J}\sum_{k'\in K_{j}}\bigl\langle QS_{\Lambda}^{-1}u_{j,k}, \Lambda_{j'}^{\ast}e_{j',k'} \bigr\rangle W\tilde {e}_{j',k'} \\ &=S_{\Lambda}^{-1}u_{j,k}+z_{j,k}-\sum _{j'\in J}\sum_{k'\in K_{j}}\bigl\langle QS_{\Lambda}^{-1}u_{j,k}, u_{j',k'}\bigr\rangle z_{j',k'}, \end{aligned}$$
that is,
$$\Gamma_{j}^{\ast}e_{j,k}=S_{\Lambda}^{-1} \Lambda_{j}^{\ast }e_{j,k}+W\tilde{e}_{j,k}- \sum_{j'\in J}\sum_{k'\in K_{j}}\bigl\langle QS_{\Lambda}^{-1}\Lambda _{j}^{\ast}e_{j,k}, \Lambda_{j'}^{\ast}e_{j',k'}\bigr\rangle W \tilde{e}_{j',k'} $$
for all
\(j\in J\),
\(k\in K_{j}\).
Now we prove the converse. Assume that (
3.1) holds. For any
\(f\in H\), using the notations
\(u_{j,k}:=\Lambda_{j}^{\ast}e_{j,k}\),
\(v_{j,k}:=\Gamma_{j}^{\ast}e_{j,k}\), and
\(z_{j,k}:=W\tilde{e}_{j,k}\), by a standard argument we get that
\(\sum_{j\in J}\sum_{k\in K_{j}}\langle f, u_{j,k}\rangle S^{-1}_{\Lambda}u_{j,k}\) converges unconditionally to
f. Therefore
$$\begin{aligned} \sum_{j\in\mathcal{J}}\Gamma_{j}^{\ast} \Lambda_{j}f&= \sum_{j\in\mathcal{J}} \Gamma_{j}^{\ast} \sum_{k\in\mathcal{K}_{j}}\langle \Lambda_{j}f, e_{j,k}\rangle e_{j,k} \\ &=\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \bigl\langle f, \Lambda_{j}^{\ast }e_{j,k}\bigr\rangle \Gamma_{j}^{\ast}e_{j,k} \\ &= \sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle v_{j,k} \\ &=\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle \biggl(S^{-1}_{\Lambda}u_{j,k}+z_{j,k}- \sum_{j^{\prime}\in\mathcal{J}} \sum_{k^{\prime}\in\mathcal{K}_{j}}\bigl\langle QS^{-1}_{\Lambda }u_{j,k}, u_{j^{\prime},k^{\prime}}\bigr\rangle z_{j^{\prime},k^{\prime }} \biggr) \\ &= \sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle S^{-1}_{\Lambda }u_{j,k}+ \sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle z_{j,k} \\ &\quad{} - \sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle\sum_{j^{\prime }\in\mathcal{J}} \sum _{k^{\prime}\in\mathcal{K}_{j}}\bigl\langle QS^{-1}_{\Lambda }u_{j,k}, u_{j^{\prime},k^{\prime}}\bigr\rangle z_{j^{\prime},k^{\prime }} \\ &= f+\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle z_{j,k}-\sum _{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \biggl\langle Q \sum_{j^{\prime}\in \mathcal{J}} \sum_{k^{\prime}\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle S^{-1}_{\Lambda}u_{j,k}, u_{j^{\prime},k^{\prime}} \biggr\rangle z_{j^{\prime},k^{\prime}} \\ &=f+\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle z_{j,k}- \sum _{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}}\langle Qf, u_{j^{\prime},k^{\prime }}\rangle z_{j^{\prime},k^{\prime}} \\ &= f+\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f-Qf, u_{j,k}\rangle z_{j,k}\end{aligned} $$
for all
\(f\in H\). Next, we prove that
\(\{\Gamma_{j}\}_{j\in J}\) is a g-Bessel sequence for
H w.r.t.
\(\{V_{j}\}_{j\in J}\). Indeed,
$$\begin{aligned} \sum_{j\in J} \Vert \Gamma_{j}f \Vert ^{2}&=\sum_{j\in\mathcal{J}} \sum _{k\in K_{j}} \bigl\vert \langle\Gamma_{j}f, e_{j,k}\rangle \bigr\vert ^{2} \\ &=\sum_{j\in J} \sum_{k\in K_{j}} \bigl\vert \langle f, v_{j,k}\rangle \bigr\vert ^{2} \\ &= \sum_{j\in J} \sum_{k\in K_{j}} \biggl\vert \biggl\langle f, S^{-1}_{\Lambda }u_{j,k}+z_{j,k}- \sum_{j^{\prime}\in J} \sum_{k^{\prime}\in K_{j}}\bigl\langle QS^{-1}_{\Lambda}u_{j,k}, u_{j^{\prime},k^{\prime}}\bigr\rangle z_{j^{\prime},k^{\prime}} \biggr\rangle \biggr\vert ^{2} \\ &\leq C_{1} \biggl(\sum_{j\in\mathcal{J}} \sum _{k\in\mathcal{K}_{j}} \bigl\vert \bigl\langle f, S^{-1}_{\Lambda }u_{j,k}\bigr\rangle \bigr\vert ^{2}+\sum_{j\in J} \sum _{k\in K_{j}} \bigl\vert \langle f, z_{j,k}\rangle \bigr\vert ^{2} \\ &\quad{} +\sum_{j\in J} \sum_{k\in K_{j}} \biggl\vert \biggl\langle Q^{\ast}\sum_{j^{\prime}\in J} \sum_{k^{\prime}\in K_{j}}\langle f, z_{j^{\prime},k^{\prime }}\rangle u_{j^{\prime},k^{\prime}},S^{-1}_{\Lambda}u_{j,k}\biggr\rangle \biggr\vert ^{2} \biggr) \\ &\leq C_{2} \biggl( \Vert f \Vert ^{2}+ \biggl\Vert Q^{\ast}\sum_{j^{\prime}\in J} \sum _{k^{\prime}\in K_{j}}\langle f, z_{j^{\prime},k^{\prime }}\rangle u_{j^{\prime},k^{\prime}} \biggr\Vert ^{2} \biggr) \\ &\leq C_{3} \biggl( \Vert f \Vert ^{2}+\sum _{j^{\prime}\in J} \sum_{k^{\prime}\in K_{j}} \bigl\vert \langle f, z_{j^{\prime}k^{\prime }}\rangle \bigr\vert ^{2} \biggr) \\ &\leq C_{4} \Vert f \Vert ^{2} \end{aligned}$$
for all
\(f\in H\), where
\(C_{1}\),
\(C_{2}\),
\(C_{3}\), and
\(C_{4}\) are different positive constants. Let
\(T_{\Gamma}\) be the synthesis operator of
\(\{\Gamma_{j}\}_{j\in J}\). Then
$$\begin{aligned} \bigl\Vert \bigl(I_{H}-T_{\Gamma}T^{\ast}_{\Lambda} \bigr)f \bigr\Vert &= \biggl\Vert \sum_{j\in J} \sum _{k\in K_{j}}\langle f-Qf, u_{j,k}\rangle z_{j,k} \biggr\Vert \\ &= \biggl\Vert \sum_{j\in J} \sum _{k\in K_{j}}\langle f-Qf, u_{j,k}\rangle W \tilde{e}_{j,k} \biggr\Vert \\ &= \biggl\Vert W\sum_{j\in J} \sum _{k\in K_{j}}\langle f-Qf, u_{j,k}\rangle \tilde{e}_{j,k} \biggr\Vert \\ &= \biggl\Vert W\sum_{j\in J} \sum _{k\in K_{j}}\bigl\langle \Lambda_{j}(f-Qf),e_{j,k} \bigr\rangle \tilde {e}_{j,k} \biggr\Vert \\ &= \bigl\Vert WT^{\ast}_{\Lambda}(f-Qf) \bigr\Vert \\ &\leq \bigl\Vert WT^{\ast}_{\Lambda}(I_{H}-Q) \bigr\Vert \Vert f \Vert \end{aligned}$$
for all
\(f\in H\). Therefore
\(\|I_{H}-T_{\Gamma}T^{\ast}_{\Lambda}\|<1\), and thus
\(\{ \Lambda_{j}\}_{j\in J}\) and
\(\{\Gamma_{j}\}_{j\in J}\) are approximately dual g-frames. □