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The first part of the paper is devoted to the $G$-type spaces i.e. the spaces$G^\alpha_\alpha (\mathbb R^d_+)$, $\alpha\geq 1$ and their duals which can bedescribed as analogous to the Gelfand-Shilov spaces and their duals but withcompletely new justification of obtained results. The Laguerre type expansionsof the elements in $G^\alpha_\alpha(\mathbb R^d_+)$, $\alpha\geq 1$ and theirduals characterise these spaces through the exponential and sub-exponentialgrowth of coefficients. We provide the full topological description and by thenuclearity of $G_\alpha^\alpha(\mathbb{R}^d_+)$, $\alpha\geq 1$ the kerneltheorem is proved. The second part is devoted to the class of the Weyloperators with radial symbols belonging to the $G$-type spaces. The continuityproperties of this class of pseudo-differential operators over theGelfand-Shilov type spaces and their duals are proved. In this way the class ofthe Weyl pseudo-differential operators is extended to the one with the radialsymbols with the exponential and sub-exponential growth rate.

$G$-type Spaces of Ultradistributions over $\mathbb{R}^d_+$ and the Weyl Pseudo-differential Operators with Radial Symbols