Ultradifferentiable functions via the Laguerre operator

We introduce spaces of test functions Gαα( R ᵈ₊) and gαα( R ᵈ₊), α > 0, by using the iterates of the Laguerre operator, and show that these spaces represent a natural counterpart of global Pilipovi\'c spaces on R ᵈ. Moreover, we show that Gαα( Rᵈ₊) and gαα( R ᵈ₊) coincide with $G-$type spaces when α ≥ 1, and α > 1, respectively. We also consider the associated spaces of ultradistributions. The main tool in our study is the use of sequence spaces α( N₀ ᵈ) and 0,α( N₀ ᵈ) arising from the corresponding Laguerre expansions. In addition, by considering sequence spaces _σ( Nᵈ₀) and _0,σ( Nᵈ₀), we introduce and study flat Pilipovi\'c spaces on orthants.