Ergodic theorems in Banach ideals of compact operators

Abstract

Let $\mathcal{H}$ be an infinite-dimensional Hilbert space, and let $\mathcal{B}(\mathcal{H})$ ($\mathcal{K}(\mathcal{H})$) be the $C^{\ast }$-algebra of bounded (respectively, compact) linear operators in $\mathcal{H}$. Let $(E,\Vert \cdot {\Vert }_E)$ 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 $\Vert x{\Vert }_{{\mathcal{C}}_E}=\Vert \{s_n(x)\}{\Vert }_E$, $x\in {\mathcal{C}}_E$, be the Banach ideal of compact operators generated by $E$. We show that the averages $A_n(T)(x)=\frac{1}{n+1}\sum _{k=0}^n{T}^k(x)$ 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 $({\mathcal{C}}_E,\Vert \cdot {\Vert }_{{\mathcal{C}}_E})$ if and only if $E$ is separable and $E\ne l^1$, as sets.