Some new restricted maximal operators of Fejér means of Walsh-Fourier series in the space $H_{1\mathrm{/}2}$

In this paper we derive the maximal subspace of natural numbers $\{n_k:k\ge 0\}$, such that the restricted maximal operator, defined by ${\sup }_{k\in \mathbb{N}}\mid {\sigma }_{n_k}F\mid$ on this subspace of Fejér means of Walsh-Fourier series is bounded from the martingale Hardy space $H_{1\mathrm{/}2}$ to the Lebesgue space $L_{1\mathrm{/}2}$. The sharpness of this result is also proved.