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We consider the counter images $\maclJ (\rr d)$ and $\maclJ _0(\rr d)$ ofentire functions with exponential and almost exponential bounds, respectively,under the Bargmann transform, and we characterize them by estimates of powersof the harmonic oscillator. We also consider the Pilipovi{\'c} spaces $\bsycalS_s(\rr d)$ and $\bsySig _s(\rr d)$ when $0<s<1/2$ and deduce their images underthe Bargmann transform.

The Bargmann transform and powers of harmonic oscillator on Gelfand-Shilov subspaces