On the iterates of the Laguerre operator

We use the iterates of the Laguerre operator to introduce Pilipović 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 $\alpha \ge 1$, respectively. However, in contrast to $G$-type spaces, Pilipović spaces on positive orthants are nontrivial below the critical index $\alpha =1$. We also remark that there is a natural isomorphism between subspaces of Pilipović spaces on ${\mathbb{R}}^d$ consisting of even functions, and Pilipović spaces on positive orthants.