Ergodic theorems for flows in ideals of compact operators

Let $\{T_t\}_{t\ge 0}$ a strongly continuous semigroup of Dunford-Schwartz operators acting in fully symmetric ideal $\mathcal C_E$ of compact operators, and let $A_t(x)=\frac1t\int_0^tT_s(x)ds, \ x \in \mathcal C_E, t>0,$ be the corresponding ergodic averages. In this note we establish that the net $A_t(x)$ converge to some $\widehat{x} \in \mathcal C_E $ with respect to the uniform norm $\|\cdot\|_\infty$ as $t\to \infty$ for all $x\in \mathcal C_E$. Besides, we show that if $\mathcal C_E\neq \mathcal C_1$ and $(E, \|\cdot\|_{E})\subset c_0$ is separable space, then $\|A_t(x)-\widehat{x}\|_{\mathcal C_E}\to 0$ as $t\to \infty$.
Also we consider actions of the semigroup $ \mathbb R_+$ in Banach ideals $E$ of compact operators (i.e. in noncommutative atomic symmetric spaces) and show that the corresponding ergodic averages $A_t(x)$ converge to $x$ uniformly as $t\to 0^+$ for all $x\in E$. Besides, we show that if $E$ has order continuous norm $\|\cdot\|_E$, then $\|A_t(x)-x\|_E\to 0$ as $t\to 0^+$.