Averaged-operator inequality
= Averaged-operator inequality
{title2=$\|Tz-Tw\|^2\leq\|z-w\|^2-\frac{1-\alpha}{\alpha}\|(I-T)z-(I-T)w\|^2$}
The defining averaging representation is equivalent to $\|Tz-Tw\|^2\leq\|z-w\|^2-(1-\alpha)\|(I-T)z-(I-T)w\|^2/\alpha$. Expanding the norm of the convex combination proves both directions. Substituting a fixed point for $w$ gives <Fejér monotonicity> and summability of squared residuals.