I am studying some issues relative to operators, especially bounded operators, but my professor asked me the following and I couldn’t answer him: Let $$B=\left[\begin{array}{cc}a &b \\[4 pt] b & a\end{array}\right],$$ be a matrix in $M_2(\mathbb{C})$, and $x\in \mathbb{C}$. We define $$ C \left[\begin{array}{c}x \\[4 pt] x\end{array}\right] = \left(0,0,0,B \left[\begin{array}{c}x \\[4 pt] x\end{array}\right]^T \right)\in \mathbb{C}\times H^1((a,b)\cup(c,d))\times L^2((a,b)\cup(c,d)) \times \mathbb{C} \times \mathbb{C}. $$
Is $C$ a bounded linear operator? why ?.