On compact embeddings in $\mathbf{L^p}$ and fractional spaces

Let $X,Y$ be Hilbert spaces and $\mathcal{A}\colon X\to X'$ a continuous and symmetric elliptic operator. We suppose that $X$ is dense in $Y$ and that the embedding $X\subset Y$ is compact. In this paper we show some consequences of this setting on the study of the fractional operator attached to $\mathcal{A}$ in the extension setting $\mathbb{R}^N\times (0, \infty)$. Being more specific, we will give some examples where the embedding $H(\mathbb{R}^{N+1}_+)\subset L^2(\mathbb{R}^N)$ is compact, with the space $H(\mathbb{R}^{N+1}_+)$ depending on the operator $\mathcal{A}$.