Dobrushin coefficients of ergodicity and asymptotically stable \(L^1\)-contractions
Let \((X,{\mathcal A},\mu)\), \((Y,{\mathcal B},\nu)\) be two measure spaces, let \(T:L^1(X,{\mathcal A},\mu)\to L^1(Y,{\mathcal B},\nu)\) be a linear bounded operator, set \[ {\mathcal X}= \Biggl\{f\in L^1(X,{\mathcal A},\mu) \mid \int fd\mu=0\Biggr\}, \] \[ \overline{\alpha}(T)=\sup_{f\in{\mathcal X},f\neq 0}\frac{\|Tf\|}{\|f\|}, \qquad\alpha(T)=\|T\|-\overline{\alpha}(T). \] The number \(\alpha(T)\) is called the Dobrushin coefficient of ergodicity of \(T\). One of the main result in the paper is the following theorem. Let \(T:L^1(X,{\mathcal A},\mu)\to L^1(Y,{\mathcal B},\nu)\) be a linear bounded operator. Then \[ \|Tf\|\leq \overline{\alpha}(T)\|f\|+\alpha(T) \Biggl|\int fd\mu \Biggr|\quad \text{for every } f\in L^1(X,{\mathcal A},\mu). \] Using this theorem the authors give a simple proof of \textit{W. Bartoszek}'s result [Ann. Pol. Math. 52, No. 2, 165-173 (1990; Zbl 0719.47022)] and generalize his result. The ``asymptotic'' approach to the Dobrushin coefficient of ergodicity is also considered. This approach enables them to give necessary and suffcient conditions of strongly asymptotical stability of positive contractions of \(L^1(X,{\mathcal A},\mu)\) in terms of complete mixing and smoothing.
scientific article; zbMATH DE number 1418044