On a multi-parametric generalization of the uniform zero-two law in \(L^1\)-spaces. (English) Zbl 1331.47059
Summary: Following an idea of D. Ornstein and L. Sucheston [Ann. Math. Stat. 41, 1631–1639 (1970; Zbl 0284.60068)], S. R. Foguel [Isr. J. Math. 10, 275–280 (1971; Zbl 0229.60056)] proved the so-called uniform “zero-two” law: let \(T:L^1(X,\mathcal{F},\mu) \to L^1(X,\mathcal{F},\mu)\) be a positive contraction. If for some \(m\in\mathbb N\cup\{0\}\) one has \(\| T^{m+1}-T^m\|<2\), then
\[
\lim_{n\to\infty}\| T^{n+1}-T^n\|=0.
\]
There are many papers devoted to generalizations of this law. In the present paper, we provide a multi-parametric generalization of the uniform zero-two law for \(L^1\)-contractions.
MSC:
47B65 | Positive linear operators and order-bounded operators |
47A13 | Several-variable operator theory (spectral, Fredholm, etc.) |
47A35 | Ergodic theory of linear operators |
17C65 | Jordan structures on Banach spaces and algebras |
46L70 | Nonassociative selfadjoint operator algebras |
46L52 | Noncommutative function spaces |
28D05 | Measure-preserving transformations |