Why are these two operators similar? Let $X$ be a Hilbert space with ON-basis $\lbrace e_n : ~ n \in \mathbb{N} \rbrace$. Furthermore let $A, ~ \Gamma : X \to X$ be  linear operators with $A e_n = \alpha_n e_n$ and $\Gamma e_n = \gamma_n e_n$ respectively. Let $c,\lambda \in \mathbb{C}$.
My professor stated, that

$\left[ \begin{array}{ll} 0 & \Gamma \\ \lambda-A & c \Gamma \end{array} \right]$
is similar to the operator of multiplication by the sequence of matrices
$\left[ \begin{array}{ll} 0 & \gamma_n \\ \lambda-\alpha_n & c \gamma_n \end{array} \right] \in \mathbb{C}^{2 \times 2}$

Why is that correct? And what is even ment by "the operator of multiplication by the sequence of matrices"?

If you start calculating with some $v =(v_1, v_2) \in X \times X$ you get
$\left[ \begin{array}{ll} 0 & \Gamma \\ \lambda-A & c \Gamma \end{array} \right]v = \left[ \begin{array}{l} \Gamma v_2 \\ (\lambda -A) v_1 - c \Gamma v_1 \end{array} \right]$
and if you let $v_1 = \sum_{n \in \mathbb{N}} r_n e_n$, $v_2 = \sum_{n \in \mathbb{N}} s_n e_n$ this yields
$\left[ \begin{array}{ll} 0 & \Gamma \\ \lambda-A & c \Gamma \end{array} \right]v = \left[ \begin{array}{l} \sum_{n \in \mathbb{N}} s_n \gamma_n e_n \\ \lambda \sum_{n \in \mathbb{N}} r_n  e_n - \sum_{n \in \mathbb{N}} r_n \alpha_n e_n - c \sum_{n \in \mathbb{N}} s_n \gamma_n e_n \end{array} \right]$
and this yields
$\left[ \begin{array}{ll} 0 & \Gamma \\ \lambda-A & c \Gamma \end{array} \right]v = \sum_{ n \in \mathbb{N}} \left[ \begin{array}{l} s_n \gamma_n e_n \\ \lambda  r_n  e_n -  r_n \alpha_n e_n - c  s_n \gamma_n e_n \end{array} \right]$.
But this doesn't help me at all. Any hint?
 A: My guess is that $A$ and $\Gamma$ share eigenvectors, $e_n$. The "big" matrix is written according to the product $X \times X$. (Bases of the two copies of $X$ need not be chosen here.) The small matrices (with $\alpha_n$ and $\gamma_n$) are only $2$-by-$2$ the way they are written, and they correspond to subspaces of $X \times X$ generated by $(e_n, 0)$ and $(0, e_n)$. They have to be extended to cover $X \times X$ by adding identity to properly act on $X \times X$. A formal definition may look like this:
Let $P_n:X\times X \to X\times X$ be defined as follows. Suppose $\left(u, v\right) = \left(\sum_{i=1}^\infty u_i e_i, \sum_{i=1}^\infty v_i e_i\right)$ and $(x, y) = \left(\sum_{i=1}^\infty x_i e_i, \sum_{i=1}^\infty y_i e_i\right) = P_n(u, v)$.
Then
\begin{align*}
x_i & =
\begin{cases}
\gamma_i v_i & ; i = n\\
u_i & ; i \ne n
\end{cases}\\
y_i & =
\begin{cases}
(\lambda - \alpha_i)u_i + c \gamma_i v_i & ; i = n \\
v_i & ; i \ne n
\end{cases}
\end{align*}
These $P_n$ are operators defined by the small $2$-by-$2$ matrix you wrote above.
The big operator should be the same as $\prod_{i=1}^\infty P_i$, which is at first glance informal but can be made formal by considering how it operates. Its action is exactly what you wrote.
