About certain convergence of Cesaro means

Question

If we consider a bounded operator $T: X\to Y$ between Banach spaces, is there any condition on the spaces $X, Y$ or on the operator that can guarantee the following?

$$\lim_{n\to\infty}\sup_{\|x\|=1}\| \sigma_n(T(x))-Tx\|_Y=0$$

Answer 1

This is equivalent to asking whether $\sigma_n(T)\rightarrow T$ in the norm topology. Recall that an operator $P:X\rightarrow X$ is called a projection if $P^2=P$. Clearly for any projection $P$, $\sigma_n(P)\rightarrow P$. The converse is also true and it follows by taking the limit as $n\rightarrow \infty$ in the elementary identity $$\sigma_n(T)T=\frac{n+1}{n}\sigma_{n+1}(T)-\frac{1}{n}I.$$

So an operator satisfies your property if and only if it is a projection.