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**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].

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].

Reviewer: C.D.Aliprantis (MR 87k:47083)

### 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 |