## Positive linear operators on $$L^p$$ and the Doeblin condition.(English)Zbl 1167.47303

Aspects of positivity in functional analysis, Proc. Conf. Occas. H. H. Schaefer’s Birthday, Tübingen 1985, North-Holland Math. Stud. 122, 137-156 (1986).
Inspired by a classical result in the theory of Markov processes, due to W. Doeblin [Bull. Math. Soc. Roum. Sci. 39, No. 1, 57–115 et No. 2, 3–61 (1937; Zbl 0019.17503)], the author says that a positive (linear) operator $$T: E\to E$$ on a Banach lattice satisfies the Doeblin condition whenever there exist $$m\in\mathbb{N}, 0\leq\mu\in E'$$ and a real number $$\eta<1$$ such that $$\|T^mx\|\leq\mu(x)+\eta\|x\|$$ holds for all $$x\in E^+$$. This well-written paper studies positive operators with the Doeblin property.
The author considers a Banach lattice $$E$$, a positive operator $$T: E\to E$$ satisfying the Doeblin condition, and defines $$T_n=(1/n)(I+T+\dots+T^{n-1})$$ for each $$n$$. The major results of the paper are the following: (1) If the sequence $$\{T_n\}$$ is uniformly bounded, then $$\lim\|T_n\|/n=0$$. (2) If the spectral radius of $$T$$ satisfies $$r(T)\geq1$$, then $$r(T)$$ is an eigenvalue of $$T^{\prime\prime}$$ with a positive eigenvector. (3) If $$T$$ is also weakly compact with spectral radius $$r(T)\geq1$$, then $$r(T)$$ is an eigenvalue of $$T$$ and $$T^\prime$$ with positive eigenvectors. (4) If $$E^\prime$$ (the norm dual of $$E$$) has order continuous norm and $$\{T_n\}$$ is uniformly bounded, then $$\{T_n\}$$ converges strongly to a positive projection $$P$$ of finite rank. Moreover, $$\{T_n^\prime\}$$ and $$\{T_n^{\prime\prime}\}$$ converge strongly to $$P^\prime$$ and $$P^{\prime\prime}$$, respectively. (5) If $$E$$ satisfies an upper $$p$$-estimate for some $$p>1$$ and $$\{T_n\}$$ is uniformly bounded, then $$\{T_n\}$$ converges uniformly to a positive projection of finite rank. (6) If $$\{T_t: t\geq0\}$$ is a uniformly bounded semigroup of positive operators and $$T_t$$ is quasicompact for some $$t$$, then $$\{T_t\}$$ converges uniformly to a positive projection as $$t$$ tends to infinity.
Various important concrete consequences of the above results are presented. The paper also studies a “dual” type Doeblin condition. For details, we refer the reader to this interesting paper.
For the entire collection see [Zbl 0595.00016].

### MSC:

 47B65 Positive linear operators and order-bounded operators 46B42 Banach lattices 47A35 Ergodic theory of linear operators 47D03 Groups and semigroups of linear operators 60J05 Discrete-time Markov processes on general state spaces

Zbl 0019.17503