Counterexample $||S+T||_{\mathbb{B}(E)}<||S||_{\mathbb{B}(E)}+||T||_{\mathbb{B}(E)}$

Question

Let $S,T\in \mathbb{B}(E),\ \mathbb{B}(E)=\left\{T:E\to E:T\ linear\ bounded\right\}$

Give a countraexamples such that:

(a) $||S+T||_{\mathbb{B}(E)}<||S||_{\mathbb{B}(E)}+||T||_{\mathbb{B}(E)}$

(b)$||ST||_{\mathbb{B}(E)}<||S||_{\mathbb{B}(E)}||T||_{\mathbb{B}(E)}$

Some countraexample? thanks!

Answer 1

Hints:

(a) Any $S\ne 0$, $T=-S$.

(b) Two projections s.t. $ST=0$.