Ergodic theorems in Banach ideals of compact operators (1902.00759v2)

Abstract: 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.