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

Abstract

We prove the inequalities:$$\sum _{n=1}^k{a}_n\left(\sum _{i=1}^r{S}_i\right)\le r\sum _{n=1}^k\sum _{i=1}^r{a}_n(S_i),$$$$\sum _{n=1}^k{a}_n\left(\prod _{i=1}^r{S}_i\right)\le r\sum _{n=1}^k\prod _{i=1}^r{a}_n(S_i),k=1,2,...,r\ge 2,$$and$$\prod _{n=1}^k{a}_n\left(\prod _{i=1}^r{S}_i\right)\le \prod _{n=1}^k\prod _{i=1}^r{a}_n^r(S_i),k=1,2,...,r\ge 2,$$where $\{a_n(S)\}$ is the sequence of the approximation numbers of the linear and bounded operators $S:X\to X$ $(S\in L(X))$. $X$ is a Banach space.