Let $\{\alpha_{n}\}_{n\in\mathbb{N}}$ be positive sequence such that
$$\sum_{n=1}^{\infty}\alpha_n<\infty.$$ 

<b>Question:</b> Is there any chance to evaluate the operator norm of the matrix operator
$$C=\begin{pmatrix}
\alpha_1 & \alpha_2 & \alpha_3 & \dots\\
0 & \alpha_2 & \alpha_3 & \dots\\
0 & 0 & \alpha_3 & \dots\\
\vdots & \vdots & \vdots & \ddots\\
\end{pmatrix},$$
acting on $\ell^{2}(\mathbb{N})$, in terms of the sequence $\{\alpha_{n}\}_{n\in\mathbb{N}}$?

<b>A remark</b>: Note $C$ can be decomposed as $C=VD$ where $V_{i,j}=1$, if $i\leq j$, $V_{i,j}=0$, if $i>j$, and $D=\mbox{diag}(\alpha_1,\alpha_2,\alpha_3,\dots)$.

<b>An additional comment concerning boundedness of $C$</b>: As noticed by Miel Sharf below, the assumptions do not guarantee the boundedness of $C$. Operator $C$ is bounded, for instance, under the following additional assumption:
$$\sup_{n}\frac{1}{\alpha_{n}}\sum_{k\geq n}\alpha_{k}^{2}<\infty.$$
This is, in fact, the case I am interested in.