**Authors:** Antonescu C.

### Abstract

Generalizations have been obtained for the interpolation properties of operator ideals
$$
L_{\overline{\Phi }_{\left( p\right) }}^{\left (\varepsilon \right) }
\left (E,F\right) =\left\{ T\in L\left( E,F\right) :\Phi \left
(\left\{\frac{\varepsilon _{n}^{p}(T)}{n}\right\} \right)^{\frac{1}{p}}<
\infty \right\},\quad 0<p<\infty
$$
and
$$
L_{\Phi _{\left( p\right) }}^{\left( e\right) }\left( E,F\right) =
\left\{T\in L\left( E,F\right) :\Phi \left( \left\{ e_{n}^{p}(T)
\right\}\right) ^{\frac{1}{p}}<\infty \right\}, \quad 0<p<\infty
$$
established in [1] where $\varepsilon _{n}(T)$ is the $n$-th $\varepsilon$-entropy number of the linear and bounded operator $T$ [2, 3], $e_{n}(T)$ is the $n$-th dyadic entropy number of $T$ [2, 3] and $\Phi $ is a symmetric norming function [3, 4]. Considering the following relations proved in [1]:
$$
L_{\overline{\Phi }_{\left( r\right) }}^{\left( \varepsilon \right) }
\left (E_{0},F\right) \cap L_{\overline{\Phi }_{\left(p\right) }}^
{\left (\varepsilon \right) }\left( E_{1},F\right) \subseteq
L_{\overline{\Phi }_{\left( s\right) }}^{\left (\varepsilon \right) }
\left( E,F\right)
$$
and
$$
L_{\Phi _{\left( p\right) }}^{\left( e\right) }\left( E_{0},F\right)
\cap L_{\Phi _{\left( p\right) }}^{\left( e\right) }\left
( E_{1},F\right) \subseteq L_{\Phi _{\left( s\right) }}^{\left
(\varepsilon \right) }\left (E,F\right)
$$
where $E$ is an interpolation space of $k$-type $\theta$, for the Banach interpolation couple $\left\{ E_{0},E_{1}\right\}$, and $p,r,s$ being such that $(1/p)=(\theta/r)+(1-\theta)/s$, in the present paper we show that, under certain conditions, the space $F$ can also be replaced by an interpolation space relative to an interpolation pair
$\left\{ F_{0},F_{1}\right\}$, the above inclusions remaining true.

Faculty of Mathematics and Informatics "Babes-Bolyai" University

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