Characterizations of a class of Pilipovi{\'c} spaces by powers of harmonic oscillator

We show that a smooth function $f$ on ${\mathbf{R}}^d$ belongs to the Pilipovi{'c} space ${\mathcal{H}}_{{\mathrm{♭}}_{\sigma }}({\mathbf{R}}^d)$ or the Pilipovi{'c} space ${\mathcal{H}}_{0,{\mathrm{♭}}_{\sigma }}({\mathbf{R}}^d)$, if and only if the $L^p$ norm of $H_d^{N}f$ for $N\ge 0$, satisfy certain types of estimates. Here $H_d=\mathrm{\mid }x{\mathrm{\mid }}^2-{\mathrm{\Delta }}_x$ is the harmonic oscillator.