Let $(c_{jk})_{j,k \in \mathbb{N}} \subset \mathbb{C}$ be such that $a:=\sup_{k \in \mathbb{N}} \sum_{j \in \mathbb{N}}|c_{jk}|<\infty$ and $b:=\sup_{j \in \mathbb{N}} \sum_{k \in \mathbb{N}}|c_{jk}|<\infty$ Prove that $$T:l^p \to l^p,(Tx)_j:=\sum_{k \in \mathbb{N}}c_{jk}x_k$$ defines a bounded linear map with $\|T\|\leq a^{\frac{1}{p}}b^{\frac{1}{q}}$ where $p \in (1,\infty)$ and $q$ is its Hölder conjugate
I have spent a couple of hours trying to prove this inequality but nothing seems to be working . I can't even prove that T is a bounded operator. Any hints on how could I go about solving this ? I started by writing the definition of the norm as
$\|T\|=\sup_{\|x\|_{p}=1}\big(\|Tx\|\big)=\sup_{\|x\|_{p}=1}\big(\sum_{j \in \mathbb{N}} \big(\sum_{j \in \mathbb{N}}c_{jk}x_k \big)^{p} \big)^{1/p}$ Then I tried to use Holder's inequality inside bracket but nothing seems to be working . I mean I can try to write down what i tried to do, but none of my attempts seem to go anywhere Any hint would be appreciated.
Note that $$|T(x)(j)|^p \leqslant \left(\sum_{l\in\mathbf N}|c_{j,l}| \right)^p \left(\sum_{k\in\mathbf N} \alpha_{j,k}|x_k| \right)^p,$$ where $$\alpha_{j,k}=\frac{|c_{j,k}|}{\sum_{l\in\mathbf N}|c_{j,l}|},$$ hence by Jensen's inequality, $$|T(x)(j)|^p \leqslant \left(\sum_{l\in\mathbf N}|c_{j,l}| \right)^p \sum_{k\in\mathbf N} \alpha_{j,k}|x_k|^p=\left(\sum_{l\in\mathbf N}|c_{j,l}| \right)^{p-1} \sum_{k\in\mathbf N} |c_{j,k} ||x_k|^p \leqslant b^{p-1} \sum_{k\in\mathbf N} |c_{j,k}||x_k|^p.$$ Using the definition of $a$, we derive the bound $$\sum_{j\in\mathbf N} |T(x)(j)|^p \leqslant b^{p-1}a\lVert x\rVert_p^p.$$