Norm of a matrix operator with a special structure Let $\{\alpha_{n}\}_{n\in\mathbb{N}}$ be positive sequence such that
$$\sum_{n=1}^{\infty}\alpha_n<\infty.$$ 
Question: 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 $\{\alpha_{n}\}_{n\in\mathbb{N}}$?
A remark: 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 $D=\mbox{diag}(\alpha_1,\alpha_2,\alpha_3,\dots)$.
An additional comment concerning boundedness of $C$: 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.
 A: I think that this operator might not be bounded.
Let $e_i$ denote the i-th vector in the standard orthonormal basis form $\ell^2(\mathbb{N})$. Then 
$$
C\cdot e_i=\alpha_i\sum_{j=1}^i e_j
$$
thus $||C||\ge\sup_{n}{(\sqrt{n}\alpha_n)}$. However, the rightmost amount might be infinite even if you assume that $\alpha_n$ are positive and that the sum $\sum \alpha_n$ converges. 
Indeed, take the sequence defined by $\alpha_n = \frac{1}{log_2(n)^2}$ if $n$ is a power of 2 and $\alpha = 0$ otherwise. The sum $\sum \alpha_n$ becomes $\sum_{k=1}^\infty \frac{1}{k^2}$, which does converge. However, it's clear that
$$
\sup_{n}{(\sqrt{n}\alpha_n)} = \sup_{k} (2^{k/2}*\frac{1}{k^2})=\infty
$$
if you want $\alpha_n$ to be strictly positive, then replace $\alpha_n=0$ by $\alpha_n=\frac{1}{2^n}$ (or any rapidly decaying sequence).
A: Let $\sum_{p\geq 1}p{\alpha_p}^2=\tau$ and assume that $\tau<\infty$. From $trace(C^TC)=\tau$, we deduce that $||C||=\sqrt{\max(spectrum(C^TC))}\leq\sqrt{\tau}$.
Moreover, if the $(\alpha_i)_i$ are $0$ except $\alpha_k$, then $||C||=\sqrt{\tau}$. 
