Ergodic theorems in Banach ideals of compact operators

Let $\mathcal H$ be an infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^\star$–algebra of all bounded (compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a fully symmetric sequence space. If $\{s_n(x)\}_{n=1}^\infty$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{s_n(x)\}\in E\}$ with $\|x\|_{\mathcal C_E}=\|\{s_n(x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. We show that the averages $A_n(T)(x)=\frac1{n+1}\sum\limits_{k = 0}^n T^k(x) $ converge uniformly in $\mathcal C_E$ for any Dunford-Schwartz operator $T$ and $x\in\mathcal C_E$. Besides, if $0\leq x\in\mathcal B(\mathcal H)\setminus\mathcal K(\mathcal H)$, there exists a Dunford-Schwartz operator $T$ such that the sequence $\{A_n(T)(x)\}$ does not converge uniformly. We also show that the averages $A_n(T)$ converge strongly in $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$ if and only if $E$ is separable and $E \neq l^1$ as sets.