Two-sided estimates of norms of a class of matrix operators

In the article, we establish necessary and sufficient conditions for the validity of a discrete Hardy type inequality
$$ \left(\sum\limits_{n=1}^{\infty}|(Af)_n|^q\right)^{\frac{1}{q}} \le C\left(\sum\limits_{k=1}^{\infty}|f_k|^p\right)^{\frac{1}{p}} $$
for one class of matrix operators
$$(Af)_n=\sum\limits_{k=1}^{n}a_{n,k}f_k, n\ge 1,$$
for $1<p,q<\infty$.