The essential norm, block sizes, and some Generalized Hilbert operators on lp

We examine the generalized Hilbert (matrix) operators $$ H_{g,γ} : (a_n) \mapsto \sum_{n=1}^{\infty} \bigg(\frac{k}{n}\bigg)^γ \frac{g_k a_n}{n+k} $$ on the $\ell^p$ spaces, $1<p<\infty$, where $g=(g_n)$ is a sequence and $-1/p<γ<1-1/p$. Given a partition of the natural numbers $\bigcup_j I_j = \mat