IMCS/Publications/BASM/Issues/BASM n.1 (29), 1999/

Interpolation properties of entropy ideals. (English)

Authors: Antonescu C.


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.

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