Entropy numbers in $γ$-Banach spaces

Abstract: Let $X$ be a quasi-Banach space, $Y$ a $\gamma$-Banach space $(0<\gamma \leq 1)$ and $T$ a bounded linear operator from $X$ into $Y$. In this paper, we prove that the first outer entropy number of $T$ lies between $2^{1-1/\gamma}|T|$ and $|T|$; more precisely, $2^{1-1/\gamma}|T| \leq e_1(T) \leq |T|,$ and the constant $2^{1-1/\gamma}$ is sharp. Moreover, we show that there exist a Banach space $X_0$, a $\gamma$-Banach space $Y_0$ and a bounded linear operator $T_0:X_0 \rightarrow Y_0$ such that $0 \neq e_k(T_0) = 2^{1-1/\gamma}|T_0| $ for all positive integers $k.$ Finally, the paper also provides two-sided estimates for entropy numbers of embeddings between finite dimensional symmetric $\gamma$-Banach spaces.