Ergodic theorems for flows in the ideals of compact operators

A. N. Azizov ; V. I. Chilin

Taurida Journal of Computer Science Theory and Mathematics, no. 4 (2020), pp. 7-17

Résumé

Let $\mathcal H$ be an infinite-dimensional complex Hilbert space, let $(\mathcal B(\mathcal H), \|\cdot\|_\infty)$ be the $C^\star$-algebra of all bounded linear operators acting in $\mathcal H$, and let $\mathcal C_E $ be the symmetric ideal of compact operators in $\mathcal H$ generated by the fully symmetric sequence space $ E \subset c_0$. If $T_t: \mathcal B(\mathcal H) \to \mathcal B(\mathcal H), \ t \geq 0$, is a semigroup of positive Dunford-Schwartz operators, which is strongly continuous on $C_{1} $, then the following versions of individual and mean ergodic theorems are true: For each $x \in \mathcal C_E$ the net $A_t(x) = \frac1t \int_0^tT_s(x) ds$ converges to some $\widehat{x} \in \mathcal C_E $ with respect to the norm $\|\cdot\|_\infty $, as $t \to \infty $; moreover, if $E$ is separable and $E \neq l_1$ (as a set), then $\lim\limits_{t \to \infty}\|A_t (x)-\widehat{x}\|_{\mathcal C_E} = 0$.

Détail

Keywords: symmetric sequence space, Banach ideal of compact operators, Dunford-Schwartz operator, individual ergodic theorem, mean ergodic theorem.