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When studying $p$-Schatten embeddings I considered spaces of the following kind: $$s_{*,\alpha}=\{a\in\ell^2(\mathbb{N}):\{j^{\alpha}a_j\}_{j=1}^{\infty}\in\ell^2(\mathbb{N})\}$$ for some $\alpha>0$.

In this paper Hanche-Olsen and Holden provide a lemma that can be used to show that $$s_{*,\alpha}\hookrightarrow \ell^2(\mathbb{N})$$ is compact for all $\alpha>0$. By slightly modifying the results of Hanche-Olsen and Holden I could show that a subset $M\subset s_{*,\beta}$ for $0<\beta<\alpha$ is totally bounded if and only if it is pointwise bounded and if for every $\varepsilon>0$ there exists an $n(\varepsilon)\in\mathbb{N}$ so that for all $a\in M$: $$\sum_{j=n(\varepsilon)+1}^{\infty}|j^{\beta}a_j|^2<\varepsilon^2.$$ From this I could conclude that $s_{*,\alpha}\hookrightarrow s_{*,\beta}$ are compact embeddings for $\alpha-\beta>0$. I could show that if $0<p\leq 2$ and $(\alpha-\beta)p>1$ these embeddings are even of $p$-Schatten class. My question is if someone has an idea how to generalize this result to $0<p<\infty$ and if someone encountered these spaces in some other context. Many thanks in advance!

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