Sur la convergence forte des moyennes des opérateurs affines positifs. (Strong convergence of means of positive affine operators.). (French) Zbl 0729.47039
The main result of this paper is the following:
If \(\{T_ i\}^ k_{i=1}\) are k positive affine operators on a reflexive Banach lattice B such that \[ \sup_{n}\| A(n,T_ i)\| <\infty,\quad \sup_{n}\| A(n,T_ i)(0)\| <\infty \] for all \(1\leq i\leq k\) \((A(n,T)=\frac{1}{n}\sum^{n-1}_{m=0}T^ m)\), then, for all \(f\in B,\)
\[ \frac{1}{n_ 1...n_ k}\sum^{n_ 1-1}_{m_ 1=0}...\sum^{n_ k-1}_{m_ k=0}T_ 1^{m_ 1}\circ...\circ T_ k^{m_ k}f \] converges to a fixed point in B as \(n_ 1,...,n_ k\to \infty\).
If \(\{T_ i\}^ k_{i=1}\) are k positive affine operators on a reflexive Banach lattice B such that \[ \sup_{n}\| A(n,T_ i)\| <\infty,\quad \sup_{n}\| A(n,T_ i)(0)\| <\infty \] for all \(1\leq i\leq k\) \((A(n,T)=\frac{1}{n}\sum^{n-1}_{m=0}T^ m)\), then, for all \(f\in B,\)
\[ \frac{1}{n_ 1...n_ k}\sum^{n_ 1-1}_{m_ 1=0}...\sum^{n_ k-1}_{m_ k=0}T_ 1^{m_ 1}\circ...\circ T_ k^{m_ k}f \] converges to a fixed point in B as \(n_ 1,...,n_ k\to \infty\).
Reviewer: C.B.Huijsmans (Leiden)
Keywords:
mean bounded; convergence to a fixed point; positive affine operators on a reflexive Banach latticeReferences:
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