  Let $\mathcal H$ be an infinite-dimensional Hilbert space, and let $\mathcal
B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^*$-algebra of bounded
(respectively, 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 positive Dunford-Schwartz operator $T$ and $x\in\mathcal
C_E$. Besides, if $x\in\mathcal B(\mathcal H)\setminus\mathcal K(\mathcal H)$,
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.
