Individual ergodic theorem in symmetric ideals of compact operators

Let $H$ be a complex separable infinite-dimensional Hilbert space, let $E$ be a fully symmetric sequence space, and let $\mathcal{C}_E$ be a symmetric ideal of compact operators in $H$ associated with $E$. It is proved that the averages $A_n(T) =\frac1{n + 1}\sum\limits_{k = 0}^n T^k $ for any positive Dunford-Schwartz operator $T: \mathcal{C}_E \to \mathcal{C}_E$ converge in $\mathcal{C}_E$ with respect to the uniform norm. In addition, we show that for every non-compact bounded linear operator $x$ acting in $H$ there exists a positive Dunford-Schwartz operator $T$ such that the averages $A_n(T)$ do not convergence with respect to the uniform norm.