On some inequalities for the approximation numbers of the sum and product of operators

Abstract

We prove the inequalities:\begin{equation}\textstyle\sum\limits_{n=1}^{k} a_{n} \left(\textstyle\sum\limits_{i=1}^{r} S_{i}\right) \le r\textstyle\sum\limits_{n=1} ^{k}\, \textstyle\sum\limits_{i=1}^{r}a_{n}(S_{i}),\end{equation}\begin{equation}\textstyle\sum\limits_{n=1}^{k} a_{n}\left(\textstyle\prod\limits_{i=1}^{r} S_{i}\right) \leq r\textstyle\sum\limits_{n=1}^{k} \textstyle\prod\limits_{i=1}^{r}a_{n}(S_{i})\;,\;k=1,2,...,\;\;r\geq2,\end{equation}and\begin{equation}\textstyle\prod\limits_{n=1}^{k} a_{n}\left(\textstyle\prod\limits_{i=1}^{r} S_{i}\right)\leq\textstyle\prod\limits_{n=1}^{k}\textstyle\prod\limits_{i=1}^{r}a_{n}^{r}(S_{i})\;,\;k=1,2,...,\;\;\;r \geq2,\end{equation}where \(\left\{ a_{n}(S)\right\} \;\) is the sequence of the approximation numbers of the linear and bounded operators \(S: X\rightarrow X\) \((S\in L(X))\). \(X\) is a Banach space.