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 $D=\mbox{diag}(\alpha_1,\alpha_2,\alpha_3,\dots)$.

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.

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 $\{\alpha_{n}\}_{n\in\mathbb{N}}$?

Some remarks: 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 $D=\mbox{diag}(\alpha_1,\alpha_2,\alpha_3,\dots)$. Further note that $C$ is positive trace class operator, hence $\|C\|\leq \mbox{Tr}\,C=\sum_{n=1}^{\infty}\alpha_n$.

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 $D=\mbox{diag}(\alpha_1,\alpha_2,\alpha_3,\dots)$.