Averaged operator ($T=(1-\alpha )I+\alpha R$)

An operator is $\alpha$-averaged for $0<\alpha <1$ when $T=(1-\alpha )I+\alpha R$ for a nonexpansive mapping $R$. Averaging controls the update residual and enables fixed-point iteration. A firmly nonexpansive mapping is exactly a $1/2$-averaged operator.

In finite-dimensional Euclidean space, iterating an averaged operator with a nonempty fixed-point set converges in norm to a fixed point from every start. The averaged-operator inequality proves boundedness and a vanishing residual; compactness produces a fixed cluster point and Fejér monotonicity turns subsequential convergence into convergence of the full sequence. A fixed point must exist.

The defining averaging representation is equivalent to $\Vert Tz-Tw{\Vert }^2\le \Vert z-w{\Vert }^2-(1-\alpha )\Vert (I-T)z-(I-T)w{\Vert }^2/\alpha$. Expanding the norm of the convex combination proves both directions. Substituting a fixed point for $w$ gives Fejér monotonicity and summability of squared residuals.