On the iterates of the Laguerre operator

We use the iterates of the Laguerre operator to introduce Pilipovi\'c spaces on positive orthants. It is shown that such spaces coincide with $G-$type spaces $g_\alpha^\alpha(\mathbb{R}^d_+)$ and $G_\alpha^\alpha(\mathbb{R}^d_+)$, when $\alpha > 1$, and $\alpha \geq 1$, respectively. However, in contrast to $G$-type spaces, Pilipovi\'c spaces on positive orthants are nontrivial below the critical index $\alpha = 1$. We also remark that there is a natural isomorphism between subspaces of Pilipovi\'c spaces on $\mathbb{R}^d$ consisting of even functions, and Pilipovi\'c spaces on positive orthants.