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The aim of the paper is two-fold. First, we investigate the ψ-Bessel potential spaces on ${ℝ}_{0+}^{n+1}$ and study some of their properties. Secondly, we consider the fractional powers of an operator of the form $-A_{\pm }=-\psi \left(D_{x^{\text{'}}}\right)\pm \partial /\left(\partial x_{n+1}\right)$, $(x^{\text{'}},x_{n+1})\in {ℝ}_{0+}^{n+1}$, where $\psi \left(D_{x^{\text{'}}}\right)$ is an operator with real continuous negative definite symbol ψ: ℝⁿ → ℝ. We define the domain of the operator $-{(-A_{\pm })}^{\alpha }$ and prove that with this domain it generates an $L_p$-sub-Markovian semigroup.